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Measure Angles at Math Playground! Practice measuring angles with a protractor.

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The complex geometry of Islamic design - Eric Broug

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In Islamic culture, geometric design is everywhere: you can find it in mosques, madrasas, palaces, and private homes. And despite the remarkable complexity of these designs, they can be created with just a compass to draw circles and a ruler to make lines within them. Eric Broug covers the basics of geometric Islamic design.

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- [Voiceover] Which shapes are symmetrical? To answer this, we need to know what it means for a shape to be symmetrical. A shape is symmetrical if it has at least one line of symmetry, a line of symmetry. And now that answer is only helpful if we know what a line of symmetry is. So let's talk about it. A line of symmetry is a line where we can fold the image and have both halves match exactly. Let's look at an example. Let's maybe draw a circle and then we could put a line on that circle. Let's draw a line maybe somewhere like this. This line is a line of symmetry if we can take one side of the line and fold it onto the other and have them match exactly. So let's take one side, doesn't matter which one, let's say the top side, and if we were gonna fold this top side down onto the bottom, would it match exactly what is shown under here? Let's see, it would probably look something like this. And does that match exactly? No, definitely not. So this is not a line of symmetry. Let's try another line. Maybe if we drew a line and we'll try to get as close down the center as we can here like this. Try to be as close to the center as possible and here if we took one side, again it doesn't matter which side, let's say over here, let's say the left side, and we folded this left side onto the right side, would it match exactly and if our line truly was in the center of the circle, then yes it would, which means that this line is a line of symmetry and because we can draw this line of symmetry on our circle, it means that our circle is symmetrical. Shapes are symmetrical if they have at least one line of symmetry and circles have many, many, many lines of symmetry. There was many places we could have drawn a line and folded it so that it worked so the two halves matched exactly. But here's one and as soon as we find one, we know we have a symmetrical shape. So let's go back to the shapes we were given. We can start with the triangle. If we draw a line, maybe a vertical line, let's try to draw it as close to the middle as possible, something like this, and we fold, let's take one side, if we fold this side over, these two lines might match up nicely, but this line here is gonna create something more like this, which does not match what's shown over here, so that's not a line of symmetry, and anywhere else vertical, same thing. We're not gonna have it lined up. So let's try maybe a horizontal line. Is there anywhere horizontally we could draw a line? And again, I think we're gonna see the same thing that the top and the bottom of the line are not gonna match up exactly. So maybe one last thing we could try is a diagonal line, something like this. Maybe this could be our line of symmetry. If we fold this bottom side, this might line up pretty nice here and then this side is gonna do something like this. So it's close, it's the closest we've gotten, but still does not match exactly. For it to be a line of symmetry, it needs to match exactly. So we weren't able to find a way to draw a vertical line or a horizontal line or even the diagonal line. So this shape has no lines of symmetry. So we can say it is not symmetrical. Moving on to the rectangle. Let's try here. Again, this time maybe we'll try a horizontal line. We can draw one right here and if that line truly is in the middle, which is what I've tried for, then this side should match up nicely to this one, across the top should match across the bottom and these sides, if I was right at the halfway point, should fold over each other also. So it has a line of symmetry so it is symmetrical. It has more than one line of symmetry. It has another one in the middle right here. But once we've found one, we know that it's symmetrical. And finally, let's look, we have a pentagon. Here again, trying a line in the middle in some way is usually a good place to start. We can try to draw a line right if this is right down the center here. Then if we folded this side, should line up nicely to this side, this side and this side would overlap and these two would match exactly. So again, has one line of symmetry so it is symmetrical and just like the rectangle, this one had quite a few lines of symmetry. Here's another line of symmetry, here's another line of symmetry, here's one more line of symmetry and so it has quite a few. It has it looks like one, two, three, four lines of symmetry, but as long as it has one, it is symmetrical. So of the shapes we were given, the rectangle and the pentagon were symmetrical.

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Identifying symmetrical figures | Line of symmetry | Math (video) | Khan Academy

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Voiceover] Which shapes are symmetrical? To answer this, we need to know what it means for a shape to be symmetrical. A shape is symmetrical if it has at least one line of symmetry, a line of symmetry. And now that answer is only helpful if we know what a line of symmetry is.

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Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

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After any of those transformations (turn, flip or slide), the shape still has the same size, area, angles and line lengths.

When one shape can become another using only Turns, Flips and/or Slides, then the two shapes are Congruent

When a Resize is neeeded for one shape to become another (we may also Turn, Flip and/or Slide) the two shapes are

In other words:

When one shape becomes another ...

The shapes are ...

... using only Rotate, Reflect and/or Translate

... using a Resize (may also Rotate, Reflect
and/or Translate)

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Learn about the Four Transformations: Rotation, Reflection, Translation and Resizing

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Grade 6 Math #11.8, Transformations - translation, rotation, reflection

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Share to Google Classroom
Or copy and share the URL

https://www.hoodamath.com/games/transformationgolf.html

Transformation Golf Instructions
Get the White Ball into the Black Hole. Pick a Transformation and then a factor choice of that transformation. Translation gives the option to move up,down,left,or right one unit. Rotation gives the option to rotate clockwise or counterclockwise, 90 or 180 degrees. Reflection gives the option to reflect over x-axis, y-axis, y=x, or y=-x. Dilation gives the scale factor option of multiplying the radius by 2 or 3, or by 1/2 or 1/3. There is 9 holes try to get par on each hole, if you are in trouble press restart to redo a hole before you complete it.

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Transformation Golf 🕹 Play Transformation Golf at HoodaMath

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Play Transformation Golf for Free Now. No Pre-Roll Ads. Unblocked Transformation Golf - Safe for School. No Downloads Required. No Pop-Ups. Nothing to Install.

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Moby gives Tim a geometry lesson in symmetry and transformation by translating, rotating, and reflecting him all over the kitchen!

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Can you draw your polygon? Maybe you know the coordinates, or lengths and angles,
either way this can give you a good estimate of the Area.
Note: due to computer rounding errors the last digit is not always correct.

Edit

Use the "Edit" button to manually edit the coordinates, or to enter new coordinates of your own.

What is Regular, Concave, Complex?

When all angles are equal and all sides are equal it is regular, otherwise it is irregular

When any internal angle is greater than 180° (it points inwards) it is concave. ( Think: concave has a "cave" in it), otherwise it is convex.

When the boundary crosses over (so the polygon intersects itself) it is complex, otherwise it is simple like most polygons we deal with. Many rules about polygons don't work when they are complex.

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Can you draw your polygon? Maybe you know the coordinates, or lengths and angles, either way this can give you a good estimate of the Area. Note: due to computer rounding errors the last digit is not always correct. Use the "Edit" button to manually edit the coordinates, or to enter new coordinates of your own.

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Descartes and Cartesian Coordinate System | Genius by Stephen Hawking | PBS LearningMedia

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Descartes discovered a lot about the dimensions of length, width, and height without leaving his room. Learn more about these three dimensions and how we use them in this excerpted clip from Genius by Stephen Hawking.

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Definition: A number of coplanar line segments, each connected end to end to form a closed shape.

Try this Drag the orange dots on each vertex to reshape the polygon. It is initially an irregular polygon; change it by clicking on "make regular", and change the number of sides.

Polygons are one of the most all-encompassing shapes in geometry. From the simple triangle up through squares, rectangles, trapezoids, to dodecagons and beyond.

In the figure above click on "make regular" and then gradually change the number of sides to see familiar shapes.

Types of Polygon

Properties of all Polygons (regular and irregular)

Properties of Regular Polygons

Named Polygons

Many polygons have names based on the number of sides. A 5-sided polygon is called a pentagon for example. There are some that wish to name every possible polygon, but there seems little point in doing so. For example a 42-sided polygon is called a "tetracontakaidigon".

Beyond about 10 sides, most people call them an "n-gon". For example a 15-gon has 15 sides. This seems easier to remember and understand. However, there are some names that do occur in everyday experience:
But if you would prefer to call a heptagon a 7-gon for example, that's fine. Everyone will know what you mean.
Other polygon topics

General

Types of polygon

Area of various polygon types

Perimeter of various polygon types

Angles associated with polygons

Named polygons

(C) 2011 Copyright Math Open Reference.
All rights reserved

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Definition: A number of coplanar line segments, each connected end to end to form a closed shape. Try this Drag the orange dots on each vertex to reshape the polygon. It is initially an irregular polygon; change it by clicking on "make regular", and change the number of sides.

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A polygon is a flat surfaced shape. A quadrilateral is a four-sided polygon.

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- [Voiceover] Which shapes are symmetrical? To answer this, we need to know what it means for a shape to be symmetrical. A shape is symmetrical if it has at least one line of symmetry, a line of symmetry. And now that answer is only helpful if we know what a line of symmetry is. So let's talk about it. A line of symmetry is a line where we can fold the image and have both halves match exactly. Let's look at an example. Let's maybe draw a circle and then we could put a line on that circle. Let's draw a line maybe somewhere like this. This line is a line of symmetry if we can take one side of the line and fold it onto the other and have them match exactly. So let's take one side, doesn't matter which one, let's say the top side, and if we were gonna fold this top side down onto the bottom, would it match exactly what is shown under here? Let's see, it would probably look something like this. And does that match exactly? No, definitely not. So this is not a line of symmetry. Let's try another line. Maybe if we drew a line and we'll try to get as close down the center as we can here like this. Try to be as close to the center as possible and here if we took one side, again it doesn't matter which side, let's say over here, let's say the left side, and we folded this left side onto the right side, would it match exactly and if our line truly was in the center of the circle, then yes it would, which means that this line is a line of symmetry and because we can draw this line of symmetry on our circle, it means that our circle is symmetrical. Shapes are symmetrical if they have at least one line of symmetry and circles have many, many, many lines of symmetry. There was many places we could have drawn a line and folded it so that it worked so the two halves matched exactly. But here's one and as soon as we find one, we know we have a symmetrical shape. So let's go back to the shapes we were given. We can start with the triangle. If we draw a line, maybe a vertical line, let's try to draw it as close to the middle as possible, something like this, and we fold, let's take one side, if we fold this side over, these two lines might match up nicely, but this line here is gonna create something more like this, which does not match what's shown over here, so that's not a line of symmetry, and anywhere else vertical, same thing. We're not gonna have it lined up. So let's try maybe a horizontal line. Is there anywhere horizontally we could draw a line? And again, I think we're gonna see the same thing that the top and the bottom of the line are not gonna match up exactly. So maybe one last thing we could try is a diagonal line, something like this. Maybe this could be our line of symmetry. If we fold this bottom side, this might line up pretty nice here and then this side is gonna do something like this. So it's close, it's the closest we've gotten, but still does not match exactly. For it to be a line of symmetry, it needs to match exactly. So we weren't able to find a way to draw a vertical line or a horizontal line or even the diagonal line. So this shape has no lines of symmetry. So we can say it is not symmetrical. Moving on to the rectangle. Let's try here. Again, this time maybe we'll try a horizontal line. We can draw one right here and if that line truly is in the middle, which is what I've tried for, then this side should match up nicely to this one, across the top should match across the bottom and these sides, if I was right at the halfway point, should fold over each other also. So it has a line of symmetry so it is symmetrical. It has more than one line of symmetry. It has another one in the middle right here. But once we've found one, we know that it's symmetrical. And finally, let's look, we have a pentagon. Here again, trying a line in the middle in some way is usually a good place to start. We can try to draw a line right if this is right down the center here. Then if we folded this side, should line up nicely to this side, this side and this side would overlap and these two would match exactly. So again, has one line of symmetry so it is symmetrical and just like the rectangle, this one had quite a few lines of symmetry. Here's another line of symmetry, here's another line of symmetry, here's one more line of symmetry and so it has quite a few. It has it looks like one, two, three, four lines of symmetry, but as long as it has one, it is symmetrical. So of the shapes we were given, the rectangle and the pentagon were symmetrical.

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Identifying symmetrical figures | Line of symmetry | Math (video) | Khan Academy

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Voiceover] Which shapes are symmetrical? To answer this, we need to know what it means for a shape to be symmetrical. A shape is symmetrical if it has at least one line of symmetry, a line of symmetry. And now that answer is only helpful if we know what a line of symmetry is.

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Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

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3) Line Symmetry and Rotational Symmetry

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3) Line Symmetry and Rotational Symmetry

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Nets Of A Solid - Animations

Demonstrations showing how some shapes are made up from their nets.
Show how the net of a cube folds to form the cube. (all the 11 possible nets of a cube are shown)

Show how the net of a cylinder folds to form the cylinder.

Show how the net of a cone folds to form the cone.

Show how the net of a tetrahedron folds to form a tetrahedron.

For video lessons on how 3D shapes unfold into nets and how nets fold to form geometric solids, see: Nets of Solids - Video Lessons.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

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Nets Of 3d Shapes - Animations (video lessons, diagrams, examples, step-by-step solutions)

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Geometry: Animations of Nets for 3D shapes - cubes, cuboids, rectangular solids, prisms, cylinders, spheres, cones, pyramids, net of solids, What is meant by the net of a solid, net of cylinder, activities and demonstrations, How to use nets to find surface area and volumes, Interactive animations for nets of solids, in video lessons with examples and step-by-step solutions.

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Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

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This tool allows you to manipulate various geometric solids and investigate their properties.

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Select the cube, face, or segment along the left navigation.

Then, place the object on the grid where you want it. If your selection is red, on the grid, then it is a location where you can not place the object.

HINT

Draw your shape from back to front and from bottom to top, to assure proper alignment of cubes.

When adding adjacent cubes, be sure to click on the face of the cube you want to be touching.

Create Mode

There are two ways to move objects:

1. Using the Arrow. Simply select and drag the object(s) to a new location.

2. Unit Movement Buttons

The other buttons along the top navigation serve various functions:

Rotate the entire figure by dragging the image or by using the sliders.

Use the Eraser to erase individual objects. You can also select multiple objects with the Pointer and then use the eraser to erase those objects.

Explode- Change all cubes into faces. If two cubes share a face, the face will not be shown. Note that you can not undo this action.

Use the Paint Brush to select a color before you create an object. You can also change the color once an object is created by selecting the object with the color you wish it to be.

Inspect Mode

In the Inspect Mode, you can:

See a 3D or 2D version of your isometric drawing.

Print your isometric drawing, as it is shown in the workspace.

Rotate your isometric drawing.

Choose to display your isometric drawing with or without axis.

Choose to display a solid or transparent isometric drawing.

You can reset all the view at any point by choosing the Reset View button.

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Use this interactive tool to create dynamic drawings on isometric dot paper.

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Algebra Basics: Graphing On The Coordinate Plane - Math Antics

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Interactivate: Maze Game

Maze Game: Investigate the Cartesian coordinate system by directing a robot through a mine field laid out on the plane. Maze Game is one of the Interactivate assessment explorers.

On a mission to transform learning through computational thinking, Shodor is dedicated to the reform and improvement of mathematics and science education through student enrichment, faculty enhancement, and interactive curriculum development at all levels. Student development of numerical models and simulations integrated with core curriculum provides an opportunity to gain practical experience in computational science. Additionally, the National Computational Science Institute (NCSI) provides nation-wide workshops portraying resources and instructional ideas to middle school, high school, and undergraduate instructors for use in the classroom. Resources and materials offered to these instructors are available free of charge from Shodor's website and are largely developed by Shodor student interns. Shodor's academic program efficiently guides participants from excitement to experience to expertise through computational explorations, research opportunities, and service.

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Interactivate: Maze Game On a mission to transform learning through computational thinking, Shodor is dedicated to the reform and improvement of mathematics and science education through student enrichment, faculty enhancement, and interactive curriculum development at all levels. Student development of numerical models and simulations integrated with core curriculum provides an opportunity to gain practical experience in computational science.

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- A game for developing your angle intuition

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Interior Angles of Polygons Exterior Angles Supplementary Angles Complementary Angles Angles On a Straight Line Angles Around a Point Degrees (Angle) Geometry Index

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Exterior Angles of Polygons Interior Angles Interior Angles of Polygons Supplementary Angles Angles On a Straight Line Straight Angle Degrees (Angle) Geometry Index

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"Vertical" in this case means they share the same Vertex (corner point), not the usual meaning of up-down.

Example: a° and b° are vertically opposite angles.

The interesting thing here is that vertically opposite angles are equal:

a° = b°

(in fact they are congruent angles)

Have a play with them yourself. Notice how the 4 angles are actually two pairs of "vertically opposite angles":

images/geom-vert-angles.js

Example: Find angles a°, b° and c° below:

Because b° is vertically opposite 40°, it must also be 40°

A full circle is 360°, so that leaves 360° − 2×40° = 280°

Angles a° and c° are also vertically opposite angles, so must be equal, which means they are 140° each.

Answer: a = 140°, b = 40° and c = 140°.

Note: They are also called Vertical Angles, which is just another way of saying the same thing.

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"Vertical" in this case means they share the same Vertex (corner point), not the usual meaning of up-down. The interesting thing here is that vertically opposite angles are equal: a° = b° (in fact they are congruent angles) Have a play with them yourself.

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Two angles are Complementary when they
add up to 90 degrees (a Right Angle )

These two angles (40° and 50°) are Complementary Angles, because they add up to 90°:

Notice that together they make a right angle

But the angles don't have to be together.

These two are complementary because 27° + 63° = 90°

Play With It ...

(Drag the points)

images/geom-comp-supp.js?mode=comp

Right Angled Triangle

In a right angled triangle, the two non-right angles are complementary, because in a triangle the three angles add to 180°, and 90° has already been taken by the right angle.

When two angles add to 90°, we say they "Complement" each other.

Complementary comes from Latin completum meaning "completed" ...
because the right angle is thought of as being a complete angle.

Spelling: be careful, it is not "Complimentary Angle" (with an "i") ...
that would be an angle you get for free!

Complementary vs Supplementary

A related idea is Supplementary Angles - those add up to 180°

How to remember which is which? Well, alphabetically they are:

Complementary add to 90°

Supplementary add to 180°

You can also think:

"C" of Complementary is for "Corner" (a Right Angle), and

"S" of Supplementary is for "Straight" (180° is a straight line)

Or you can think:

when you are right you get a compliment (sounds like complement)

"supplement" (like a vitamin supplement) is something extra, so is bigger

5231, 5233, 819, 2109, 820, 2110, 5240, 9063, 3305, 3306

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Two angles are Complementary when they add up to 90 degrees (a Right Angle )

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These two angles (140° and 40°) are Supplementary Angles, because they add up to 180°:

Notice that together they make a straight angle.

But the angles don't have to be together.

These two are supplementary because
60° + 120° = 180°

Play With It ...

(Drag the points)

When the two angles add to 180°, we say they "Supplement" each other.
Supplement comes from Latin supplere, to complete or "supply" what is needed.

Spelling: be careful, it is not "Supplimentary Angle"

Complementary vs Supplementary

A related idea is Complementary Angles, they add up to 90°

How to remember which is which? Well, alphabetically they are:

Complementary add to 90°

Supplementary add to 180°

You can also think:

"C" of Complementary is for "Corner" (a Right Angle), and

"S" of Supplementary is for "Straight" (180° is a straight line)

Or you can think:

when you are right you get a compliment (sounds like complement)

"supplement" (like a vitamin supplement) is something extra, so is bigger

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These two angles (140° and 40°) are Supplementary Angles, because they add up to 180°: Notice that together they make a straight angle. But the angles don't have to be together. These two are supplementary because 60° + 120° = 180° (Drag the points) When the two angles add to 180°, we say they "Supplement" each other.

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Supplementary angles are two angles whose sum is 180 degrees while complementary angles are two angles whose sum is 90 degrees. Supplementary and complementary angles do not have to be adjacent (sharing a vertex and side, or next to), but they can be.

Two concepts that are related but not the same are supplementary angles and complementary angles. The difference is their sum.
Supplementary angles are two angles whose measures sum to a 180 degrees and complementary are the sum have to add up to 90 degrees. And I noted here that these do not have to be adjacent. So supplementary angles could be adjacent so if I had angles one and two those two would be supplementary. But I could also say if we had some angle here that we said three and let's say 3 was equal to 60 degrees and I had some other angle over here, let's say angle four was equal to 120 degrees, I could say that these two angles three and four are supplementary because they sum to 180 degrees.
The same is true for complementary angles. Let's look at a specific example where you might be asked to identify supplementary angles and complementary angles. Here we have five angles; 1, 2, 3, 4 and 5 and we're told that this angle 3 is 90 degrees, now one thing that you can assume is that 1, 2 and 3 are all linear, so if you add up 1, 2 and 3 it would be 180 degrees, which means that 1 and 2 must also sum to 90 degrees so I could label this as a right-angle. So complementary angles could be angles 1 and 2. So I could say angle 1 and angle 2.
Now, a supplementary pair could be angle 4 and angle 5 which are adjacent and they are linear. So notice that for a supplementary and for complementary you can't say that five angles are complementary but we're always talking about pairs or two's. So remember that when you're trying to evaluate your problems that supplementary sum to 180 degrees or they're linear and complementary angles sum to 90 degrees.

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Supplementary and Complementary Angles - Concept - Geometry Video by Brightstorm

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Supplementary angles are two angles whose sum is 180 degrees while complementary angles are two angles whose sum is 90 degrees. Supplementary and complementary angles do not have to be adjacent (sharing a vertex and side, or next to), but they can be.

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Complementary & supplementary angles (video) | Khan Academy

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Learn about complementary and supplementary angles, as well as the definitions of adjacent and straight angles.

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2) Angle Properties (complementary, supplementary, opposite angles, interior, and exterior angles)

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2) Angle Properties (complementary, supplementary, opposite angles, interior, and exterior angles)

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Quadrilateral just means "four sides"
( quad means four, lateral means side).

A Quadrilateral has four-sides, it is 2-dimensional (a flat shape), closed (the lines join up), and has straight sides.

Try it Yourself

geometry/images/geom-quad.js?mode=choose
(Also see this on Interactive Quadrilaterals)
Properties

A quadrilateral has:

four sides (edges)

four vertices (corners)

interior angles that add to 360 degrees:

Try drawing a quadrilateral, and measure the angles. They should add to 360°

Types of Quadrilaterals

There are special types of quadrilateral:

Some types are also included in the definition of other types! For example a square, rhombus and rectangle are also parallelograms. See below for more details.

Let us look at each type in turn:

The Rectangle

the little squares in each corner mean "right angle"
A rectangle is a four-sided shape where every angle is a right angle (90°).

Also opposite sides are parallel and of equal length.

the little squares in each corner mean "right angle"
The Square

A square has equal sides (marked "s") and every angle is a right angle (90°)

Also opposite sides are parallel.

A square also fits the definition of a rectangle (all angles are 90°), and a rhombus (all sides are equal length).

The Rhombus

A rhombus is a four-sided shape where all sides have equal length (marked "s").

Also opposite sides are parallel and opposite angles are equal.

Another interesting thing is that the diagonals (dashed lines) meet in the middle at a right angle. In other words they "bisect" (cut in half) each other at right angles.

A rhombus is sometimes called a rhomb or a diamond.

The Parallelogram

A parallelogram has opposite sides parallel and equal in length. Also opposite angles are equal (angles "A" are the same, and angles "B" are the same).

NOTE: Squares, Rectangles and Rhombuses are all Parallelograms!

Example:

A parallelogram with:

all sides equal and

angles "A" and "B" as right angles

is a square!

The Trapezoid (UK: Trapezium)

A trapezoid (called a trapezium in the UK) has a pair of opposite sides parallel.

And a trapezium (called a trapezoid in the UK) is a quadrilateral with NO parallel sides:

An Isosceles trapezoid, as shown above, has left and right sides of equal length that join to the base at equal angles.

The Kite

Hey, it looks like a kite (usually).

It has two pairs of sides:

Each pair is made of two equal-length sides that join up.

Also:

the angles where the two pairs meet are equal.

the diagonals, shown as dashed lines above, meet at a right angle.

one of the diagonals bisects (cuts equally in half) the other.

... and that's it for the special quadrilaterals.

Irregular Quadrilaterals

The only regular (all sides equal and all angles equal) quadrilateral is a square. So all other quadrilaterals are irregular.

The "Family Tree" Chart

Quadrilateral definitions are inclusive.
(We don't say "Having all 90° angles makes it a rectangle except when all sides are equal then it is a square.")
Example: a square is also a rectangle.

So we include a square in the definition of a rectangle.

This may seem odd, as in daily life we think of a square as not being a rectangle ... but in mathematics it is.

Using the chart below we can answer such questions as:

Is a Square a type of Rectangle? (Yes)

Is a Rectangle a type of Kite? (No)

Complex Quadrilaterals

Oh Yes! when two sides cross over, we call it a "Complex" or "Self-Intersecting" quadrilateral, like these:

They still have 4 sides, but two sides cross over.

Polygon

A quadrilateral is a polygon. In fact it is a 4-sided polygon, just like a triangle is a 3-sided polygon, a pentagon is a 5-sided polygon, and so on.

Play with Them

Now that you know the different types, you can play with the Interactive Quadrilaterals.

Other Names

A quadrilateral can sometimes be called:

a Quadrangle ("four angles"), so it sounds like "triangle"

a Tetragon ("four polygon"), so it sounds like "pentagon", "hexagon", etc.

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Quadrilateral just means four sides(quad means four, lateral means side). ... A Quadrilateral has four-sides, it is 2-dimensional (a flat shape), closed (the lines join up), and

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Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

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A Parallelogram is a flat shape with opposite sides parallel and equal in length.

Play with a Parallelogram:

images/geom-quad.js?mode=parallelogram

NOTE: Squares, Rectangles and Rhombuses are all Parallelograms!

Example:

A parallelogram where all angles are right angles is a rectangle!

Area of a Parallelogram

The Area is the base times the height:

Area = b × h

( h is at right angles to b)

Example: A parallelogram has a base of 6 m and is 3 m high, what is its Area?

Area = 6 m × 3 m = 18 m²

Perimeter of a Parallelogram

The Perimeter is the distance around the edges.

The Perimeter is 2 times the (base + side length):

Perimeter = 2(b+s)

Example: A parallelogram has a base of 12 cm and a side length of 6 cm, what is its Perimeter?

Perimeter = 2 × (12 cm + 6 cm) = 2 × 18 cm = 36 cm

Diagonals of a Parallelogram

The diagonals of a parallelogram bisect each other.

In other words the diagonals intersect each other at the half-way point.

Inside Any Quadrilateral

And there is a parallelogram in any quadrilateral.

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Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

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A Square is a flat shape with 4 equal sides and every angle is a right angle (90°)

the little squares in each corner mean "right angle"

All sides are equal in length

Each internal angle is 90°

Opposite sides are parallel (so it is a Parallelogram).

Play with a square:

images/geom-quad.js?mode=square

A square also fits the definition of a rectangle (all angles are 90°), a rhombus (all sides are equal length), a parallelogram (opposite sides parallel and equal in length) and a regular polygon(all angles equal and all sides equal). What a hero!

Area of a Square

The Area is the side length squared:

Area = a ² = a × a

Example: A square has a side length of 6 m, what is its Area?

Area = 6 m × 6 m = 36 m²

The Area is also half of the diagonal squared:

Area = d² 2

Perimeter of a Square

The Perimeter is the distance around the edge.

The Perimeter is 4 times the side length:

Perimeter = 4a

Example: A square has a side length of 12 cm, what is its Perimeter?

Perimeter = 4 × 12 cm = 48 cm

Diagonals of a Square

A square has two diagonals, they are equal in length and intersect in the middle.

The Diagonal is the side length times the square root of 2:

Diagonal "d" = a × √2

Example: A square has a side length of 5 m, what is the length of a diagonal?

Calculator

Enter the side length, area, diagonal or perimeter and the other values are calculated live.

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Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

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A Rhombus is a flat shape with 4 equal straight sides.

A rhombus looks like a diamond

All sides have equal length

Opposite sides are parallel, and opposite angles are equal (it is a Parallelogram).

The altitude is the right angle distance between opposite sides

And the diagonals "p" and "q" of a rhombus bisect each other at right angles.

Play with a rhombus:

images/geom-quad.js?mode=rhombus

Area of a Rhombus

The Area can be calculated by:

the altitude times the side length:

the side length squared (s²) times the sine of angle A (or angle B):

by multiplying the lengths of the diagonals and then dividing by 2:

Example: A rhombus has diagonals of 6 m and 8 m, what is its Area?

Area = (6 m × 8 m)/2 = 24 m²

If you can draw your Rhombus, try the Area of Polygon by Drawing tool.

Perimeter of a Rhombus

The Perimeter is the distance around the edges.

The Perimeter is 4 times "s" (the side length)
because all sides are equal in length:

Perimeter = 4s

Example: A rhombus has a side length of 12 cm, what is its Perimeter?

Perimeter = 4 × 12 cm = 48 cm

Is a Square a Rhombus?

Yes, because a square is just a rhombus where the angles are all right angles.

Other Names

It is more common to call this shape a rhombus, but some people call it a rhomb or even a diamond.

The plural is rhombi or rhombuses, and, rarely, rhombbi or rhombbuses (with a double b).

The name "rhombus" comes from the Greek word rhombos: a piece of wood whirled on a string to make a roaring noise!

9101, 9102, 9103, 9104, 9105, 9106, 9107, 9108, 9109, 9110

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A Rhombus is a flat shape with 4 equal straight sides. A rhombus looks like a diamond All sides have equal length Opposite sides are parallel, and opposite angles are equal (it is a Parallelogram). The altitude is the right angle distance between opposite sides And the diagonals "p" and "q" of a rhombus bisect each other at right angles.

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A trapezoid is a 4-sided flat shape with straight sides that has a pair of opposite sides parallel (marked with arrows below):

Trapezoid

Isosceles Trapezoid

A trapezoid:

Play with a trapezoid:

images/geom-quad.js?mode=trapezoid

The parallel sides are the "bases"

The other two sides are the "legs"

The distance (at right angles) from one base to the other is called the "altitude"

Area of a Trapezoid

The Area is the average of the two base lengths times the altitude:

Area = a+b 2 × h

Example: A trapezoid's two bases are 6 m and 4m, and it is 3m high. What is its Area?

The Area of Polygon by Drawing tool is helpful when you can draw your Trapezoid.

Perimeter of a Trapezoid

The Perimeter is the distance around the edges.

The Perimeter is the sum of all side lengths:

Perimeter = a+b+c+d

Example: A trapezoid has side lengths of 5 cm, 12 cm, 4 cm and 15 cm, what is its Perimeter?

Perimeter = 5 cm + 12 cm + 4 cm + 15 cm = 36 cm

Median of a Trapezoid

The median (also called a midline or midsegment) is a line segment half-way between the two bases.

The median's length is the average of the two base lengths:

m = a+b 2

You can calculate the area when you know the median, it is just the median times the height:

Area = mh

Trapezium

A trapezium (UK: trapezoid) is a quadrilateral with NO parallel sides.

The US and UK have their definitions swapped over, like this:

9131, 9132, 9133, 9134, 9135, 9136, 9137, 9138, 9139, 9140

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A trapezoid is a 4-sided flat shape with straight sides that has a pair of opposite sides parallel (marked with arrows below): Trapezoid Isosceles Trapezoid A trapezoid: Play with a trapezoid: images/geom-quad.js?mode=trapezoid The parallel sides are the "bases" The other two sides are the "legs" The distance (at right angles) from one base to the other is called the "altitude" The Area is the average of the two base lengths times the altitude: Area = a+b 2 × h The Area of Polygon by Drawing tool is helpful when you can draw your Trapezoid.

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Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

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Estimation - visualize angle sizes

Common Core Connection for 4th Grade

Recognize angles as geometric shapes formed wherever 2 rays share a common endpoint.
Understand concepts of angle measurement.
Measure angles in whole-number degrees using a protractor.

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Play Alien Angles at MathPlayground.com! Estimate each angle. Compare your estimate with a protractor.

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Tim and Moby go to any lengths to get you in line with your compass and protractor!

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What I want to do in this video is give an overview of quadrilaterals. And you can imagine, from this prefix, or, I guess you could say, the beginning of this word, quad-- this involves four of something. And quadrilaterals, as you can imagine, are shapes. And we're going to be talking about two-dimensional shapes that have four sides and four vertices and four angles. So, for example-- one, two, three, four. That is a quadrilateral, although that last side didn't look too straight. One, two, three, four. That is a quadrilateral. One, two, three, four. These are all quadrilaterals. They all have four sides, four vertices, and, clearly, four angles. One angle, two angles, three angles, and four angles. Actually, let me draw this one a little bit bigger, because it's interesting. So in this one right over here, you have one angle, two angles, three angles, and then you have this really big angle right over there. If you look at the interior angles of this quadrilateral. Now, quadrilaterals, as you can imagine, can be subdivided into other groups based on the properties of the quadrilaterals. And the main subdivision of quadrilaterals is between concave and convex quadrilaterals. So you have concave, and you have convex. And the way I remember concave quadrilaterals, or really concave polygons of any number of shapes, is that it looks like something has caved in. So, for example, this is a concave quadrilateral. It looks like this side has been caved in. And one way to define concave quadrilaterals-- so let me draw it a little bit bigger, so this right over here is a concave quadrilateral-- is that it has an interior angle that is larger than 180 degrees. So for example, this interior angle right over here is larger than 180 degrees. And it's an interesting proof. Maybe I'll do a video. It's actually a pretty simple proof to show that, if you have a concave quadrilateral, if at least one of the interior angles has a measure larger than 180 degrees, that none of the sides can be parallel to each other. The other type of quadrilateral, you can imagine, is when all of the interior angles are less than 180 degrees. And you might say, wait-- what happens at 180 degrees? Well, if this angle was 180 degrees, then these wouldn't be two different sides, it would just be one side. And that would look like a triangle. But if all of the interior angles are less than 180 degrees, then you're dealing with a convex quadrilateral. So this convex quadrilateral would involve that one and that one over there. So this right over here is what a convex quadrilateral could look like-- four points, four sides, four angles. Now, within convex quadrilaterals, there are some other interesting categorizations. So now we're just going to focus on convex quadrilaterals, so that's going to be all of this space over here. So one type of convex quadrilateral is a trapezoid. And a trapezoid is a convex quadrilateral, and sometimes the definition here is a little bit-- different people will use different definitions. So some people will say a trapezoid is a quadrilateral that has exactly two sides that are parallel to each other. So, for example, they would say that this right over here is a trapezoid, where this side is parallel to that side. If I give it some letters here, if I call this trapezoid ABCD, we could say that segment AB is parallel to segment DC, and because of that we know that this is a trapezoid. Now I said that the definition is a little fuzzy, because some people say you can have exactly one pair of parallel sides, but some people say at least one pair of parallel sides. So if you use the original definition-- and that's the kind of thing that most people are referring to when they say a trapezoid, exactly one pair of parallel sides-- It might be something like this. But if you use the broader definition of at least one pair of parallel sides, then maybe this could also be considered a trapezoid so you have one pair of parallel sides like that and then you have another pair of parallel sides like that. So this is a question mark where it comes to a trapezoid. A trapezoid is definitely this thing here, where you have exactly one pair of parallel sides. Depending on people's definition, this may or may not be a trapezoid. If you say it's exactly one pair of parallel sides, this is not a trapezoid, because it has two pairs. If you say at least one pair of parallel sides, then this is a trapezoid. So I'll put that in a little question mark there. But there is a name for this, regardless of your definition of what a trapezoid is. If you have a quadrilateral with two pairs of parallel sides, you are then dealing with a parallelogram. So the one thing that you definitely can call this is a parallelogram. And I'll just draw it a little bit bigger. So it's a quadrilateral, and if I have a quadrilateral, and if I have two pairs of parallel sides. So the opposite sides are parallel. So that side is parallel to that side, and then this side is parallel to that side there-- you're dealing with a parallelogram. And then parallelograms can be subdivided even further. If the four angles in a parallelogram are all right angles, you're dealing with a rectangle. So let me draw one like that. This is all in the parallelogram universe, what I'm drawing right over here. This is all the parallelogram universe. So it's a parallelogram, which tells me that opposite sides are parallel. And then if we know that all four angles are 90 degrees. And we've proven in previous videos how to figure out the sum of the interior angles of any polygon. And using that same method you could say that the sum of the interior angles of any quadrilateral is actually 360 degrees. And you see that in this special case as well. But maybe we'll prove it in a separate video. But this right over here we would call a rectangle. Parallelogram-- opposite sides parallel and we have four right angles. Now, if we have a parallelogram where we don't necessarily have four right angles, but where we do have the length of the sides being equal, then we're dealing with a rhombus. So let me draw it like that. So it's a parallelogram. This is a parallelogram, so that side is parallel to that side, this side is parallel to that side. And we also know that all four sides have equal length. So this side's length is equal to that side's length, which is equal to that side's length, which is equal to that side's length. Then we are dealing with a rhombus. So one way to view it-- all rhombi are parallelograms. All rectangles are parallelograms. All parallelograms you cannot assume to be rectangles. All parallelograms you cannot assume to be rhombi. Now, something can be both a rectangle and a rhombus. So let's say that this is the universe of rectangles. So the universe of rectangles-- I'll draw a little bit of a Venn diagram here-- is that set of shapes and the universe of rhombi is this set of shapes right over here. So what would it look like? Well, you would have four right angles and they would all have the same length. So it would look like this. So it'd definitely be a parallelogram. Four right angles and all the sides would have the same length. And this is probably the first of the shapes that you learned, or one of the first shapes. This is clearly a square. So all squares could also be considered a rhombus, and they could also be considered a rectangle, and they could also be considered a parallelogram. But clearly, not all rectangles are squares, and not all rhombi are squares. And definitely not all parallelograms are squares. This one, clearly, right over here, is neither a rectangle nor a rhombi, nor a square. So that's an overview. Just gives you a little bit of taxonomy of quadrilaterals. And then in the next few videos, we can start to explore them and find their interesting properties or just do interesting problems involving them.

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What I want to do in this video is give an overview of quadrilaterals. And you can imagine, from this prefix, or, I guess you could say, the beginning of this word, quad-- this involves four of something. And quadrilaterals, as you can imagine, are shapes.

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Measuring angles in degrees (video) | Khan Academy

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Learn how to measure an angle with a protractor.

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Use this applet to create patterns to cover the screen using regular polygons.

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This activity allows manipulation and investigation of various types of angles. The online protractor tool can be used to practice measuring angles. It can be used at a variety of different grade levels. At its most basic for teaching about types of angles, acute, obtuse or reflex. For more advanced use to create angle problems in which the missing letter angle values have to be found. You can click and drag the handles to change the angles and the values will updated.

The controls

The mode control is used to it selects the type of angle problem.

Next to this is number of lines. But please note this control is disabled for both 'one' and 'intersect' modes.

the random button sets each angle to a random value

the equals button make all the angles equal.

the Protractor button when clicked will show or hide a protractor

Paper can select spots, squares or none for the background.

Use the angles display to choose whether to show or hide angles

In all the modes individual angles can be shown or hidden for example click the angle (a). Notice when the value is hidden the angle value next to the angle arc is replaced by the corresponding letter.

The handles at the end of each line can be clicked and dragged to change the angle.

Mode 'one' Identifying angle types.

The initial mode is unique in that there is only one angle. This mode is great for identifying various types of angles. Click the angle type ctrl to either show or hide the type. So now you can change the angle by dragging a handle or clicking dice for a random value. The angle types definitions are listed in the glossary below.

Complementary Angles mode

In this mode a complementary angle is shown, complementary angles add up to 90° Initially the 90° is split by just one line but this can be changed by selecting a different number of lines. A standard school problem is to have one missing angle that needs to be calculated. You can produce infinite such problems by selecting calculate from angles display, now press dice for a new problem.

Supplementary Angles mode

Supplementary angles are angles that add up to 180°. This mode behaves exactly the same as the previous.

Angles at a point mode

The angles at a point add up to 360°. In this mode press = button to make all the angles equal. This can be good way to start a problem followed by small adjustments to the lines by dragging.

Angles at intersecting lines mode

The final mode deals with angles produced by an intersection of two lines. If you have selected calculate you will notice that only one angle is given and all the other 3 can be worked out using the vertically opposite rule, or spotting the supplementary angles.

Angles Glossary

Acute Angle

Any angle that is less than 90°

Right Angle

An angle that is 90°

Obtuse Angle

An angle greater than 90° but less than 180°

Straight Angle

An angle of 180° which makes a straight line

Reflex Angle

An angle greater than 180°

Related activities

The interactive clock angles activity can also be used in explaining angles.

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Online protractor & angle maker | interactive angle visualiser | visnos

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Easy measure angles, using interactive whiteboard angle simulator. Online protractor or angle problems with acute, obtuse, reflex angles. Further complementary, supplementary and angles at a point. Use on interactive whiteboards, angles can be automatically shown or measured with a protractor.

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This investigation explores using different shapes as the hands of the clock. What things occur as the the hands move.

On a clock the three hands - the second, minute and hour hands - are on the same axis. How often in a 24 hour day will the second hand be parallel to either of the two other hands?

During the third hour after midnight the hands on a clock point in the same direction (so one hand is over the top of the other). At what time, to the nearest second, does this happen?

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md angles polygons and geom proof - stage 3

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(zoom out so all nets fit)

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Image Source: http://benswift.com

The above picture is a brilliant piece of computer art by Adobe Flash Programmer Ben Swift.

Go to the following link to see this artwork running, and changing colour in Flash on your computer. (Does not work on iPad).
http://benswift.com/2011/11/pretty-colors/
Isometric Cubes can be used to create a variety of 3D shapes, and are a great tool for learning how to visulaise in 3D.

Isometric drawing with cubes is often covered as part of Geometry topics in Mathematics.

In this lesson we take a quick look at drawing Isometric Cubes, as well as looking at some great online tools for working with 3D using cubes.

Isometric Drawing with Cubes

Image Copyright 2012 by Passy's World

Making these 3D drawings involves using Isometric Dot Grid Paper, which can easily be obtained from Google Images.

If you want some lined grids then click the following link:
http://mathforum.org/workshops/sum98/participants/sanders/IsomGrid.html
Once we have the Grid Paper we can draw and colour in cubes like those shown above.

This type of drawing is also called "Orthographic Projection".

Isometric Drawing Videos

Here is a nice short video which introduces the concept of Isometric Drawing.

This next short video shows how to make Isometric Alphabet Letters.

The following video shows how to make an Isometric Cubes Drawing when provided with a Front View, a Side View, and a Top View.

Isometric Drawing Worksheets

The links below are to worksheets we found on the Internet for Isometric Drawing.
Isometric Worksheet 1 Isometric Worksheet 2 Isometric Worksheet 3
Isometric Cubes Drawing Tool

The above online tool is great for drawing Isometric cubes.

it even has a 3D Rotatable Viewer which can be opened in a new window once you have placed a few blocks onto the grid.

It can be accessed at the following link:
http://illuminations.nctm.org/ActivityDetail.aspx?ID=125
Included are full instructions and a lesson on using this Isometric Cubes Drawing Tool at the following link:
http://illuminations.nctm.org/ActivityDetail.aspx?ID=125
There is also a ready made lesson about drawing the top view mat for Isometric shapes at the following link:
http://illuminations.nctm.org/LessonDetail.aspx?ID=L611
Isometric Cubes Counting Game

This is a fun game with a timer to count how many cubes are in each random isometric shape.

Play it at the following link:
http://www.mathplayground.com/cube_perspective.html
Building with Cubes Puzzle

There are ten puzzles to be solved.

On the main entry page, go to "Building with 3 Sides" and then choose the figures to do (there are 10 puzzles). You can move the grey grid square around to help you do views.

Once you play around and get used to how build mode works, it is a lot of fun.

Important tip for beginners - Build the top view onto the base mat first, and then work upwards from there.

Click the link below to go to the web page.
http://www.fisme.science.uu.nl/toepassingen/00724/
Classifying Triangles
Angle Sum in a Triangle
Exterior Angle of a Triangle
Angles in Parallel Lines
GeoGebra
Interactives at Mathwarehouse
Jobs that use Geometry

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Image Source: http://benswift.com The above picture is a brilliant piece of computer art by Adobe Flash Programmer Ben Swift. Go to the following link to see this artwork running, and changing colour in Flash on your computer. (Does not work on iPad).

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Maths - What is the  net of a solid - English

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Hello, BodhaGuru Learning proudly presents an animated video in English which explains what net of 3D shapes is. It also explains how to find nets of solid shape. It shows how to find different nets of cube unfolding it in different ways.

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Angles - Acute, Obtuse, Straight and Right

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6) Identify, sketch, and draw nets of solids

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6) Identify, sketch, and draw nets of solids