# data[].attributes.attachment_link.readable_text
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

# data[].attributes.attachment_link.description
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.

# data[].attributes.attachment_link.readable_text
What is a Hexagon?

In geometry, a hexagon can be defined as a closed two-dimensional polygon with six sides.

Hexagon has 6 vertices and 6 angles also.

Hexa means six and gonia means angles.
Hexagon in Real Life
We can find the shape of a hexagon in a honeycomb, a football, face of pencil, and floor tiles.

Types of Hexagons

Hexagonal shape is classified into several types based on the measure of sides and angles.
Classification of hexagons based on their sides 1. Regular Hexagon
When the length of all the sides and measure of all the angles are equal, it is a regular hexagon. All interior angles of a regular hexagon are 120 degrees each.

Properties of a Regular Hexagon

All the sides are equal in length.

All the interior angles measure 120°.

All the exterior angles measure 60°.

Since all angles are equal in a regular hexagon, each angle is 120o and the sum of all the interior angles is 720o.

A regular hexagon can be divided into six equilateral triangles.

The opposite sides of a regular hexagon are parallel to each other.

A regular hexagon is also a convex hexagon.
Symmetry in regular hexagon:
A regular hexagon has 6 lines of symmetry and a rotational symmetry of order 6
2. Irregular Hexagon
In an irregular hexagon, the length of sides and measure of angles do not have the same measure.

Classification of hexagons based on their angles 1. Convex Hexagon
For convex hexagons, all of its interior angles must be less than 180 degrees, and all the vertices are pointed outwards. Convex hexagons can be regular or irregular.

2. Concave Hexagon
For concave hexagons, at least one of its interior angles must be greater than 180° and at least one of its vertex points inwards.

Properties of Hexagon

It is two-dimensional.

It has six sides, six edges and six vertices.

The sum of the interior angles is 720 degrees.

It has nine diagonals.

Perimeter of a Hexagon:

The perimeter of a hexagon is the sum of the length of all 6 sides.

Perimeter = AB + BC + CD + DE +EF + FA

In regular hexagons, all sides are equal in length. So, the perimeter of a regular hexagon is six times the length of one side.

Solved Examples on Hexagon:
Example 1: Find the perimeter of a regular hexagon having each side measure 20 cm. Solution:
Perimeter of regular hexagon = 6 × length of side

= 6 × 20 cm

= 120 cm

So, the perimeter of the hexagon is 120 cm.
Example 2: The perimeter of a regular hexagon is 36 cm. What is the length of its sides? Solution:
Perimeter of regular hexagon = 6 × length of side

36 = 6 × length of side

Length of side = 366 cm = 6 cm

So, the length of its sides is 6 cm.
Example 3: Five angles of a hexagon measure 110° each. What is the measure of the sixth angle? Solution:
Sum of the interior angles in hexagon = 720°

Sum of five angles = 5 × 110° = 550°

Sixth angle = 720° - 550° = 170°

So, the measure of the sixth angle is 170°.

Practice Problems

Frequently Asked Questions

# data[].attributes.attachment_link.title
What is a Hexagon? Definition, Properties, Area, Perimeter, Facts

# data[].attributes.attachment_link.description
In geometry, a hexagon can be defined as a closed two-dimensional polygon with six sides. Hexagon has 6 vertices and 6 angles also. Hexa means six and gonia means angles. Hexagon in Real Life We can find the shape of a hexagon in a honeycomb, a football, face of pencil, and floor tiles.

# data[].attributes.attachment_link.readable_text
Did you know that there are different kinds of triangles? Knowing how to identify these triangles is an important part of solving many problems involving these triangles. Check out this tutorial and learn about some of the different kinds of triangles!

# data[].attributes.attachment_link.title
What are Acute, Obtuse, and Right Triangles?

# data[].attributes.attachment_link.description
Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. In this non-linear system, users are free to take whatever path through the material best serves their needs. These unique features make Virtual Nerd a viable alternative to private tutoring.

# data[].attributes.attachment_link.readable_text
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.

9121, 9122, 9123, 9124, 9125, 9126, 9127, 9128, 9129, 9130

# data[].attributes.attachment_link.description
Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.

# data[].attributes.attachment_link.title
What is Trapezoid? [Definition, Facts & Example]

# data[].attributes.attachment_link.description
Definition of Trapezoid explained with real life illustrated examples. Also learn the facts to easily understand math glossary with fun math worksheet online at SplashLearn.

# data[].attributes.attachment_link.title
Scalene Triangle -- from Wolfram MathWorld

# data[].attributes.attachment_link.description
A scalene triangle is a triangle that has three unequal sides, such as those illustrated above.

# data[].attributes.attachment_link.readable_text
What Is an Isosceles Triangle?

A triangle with two sides of equal length is an isosceles triangle.

Examples of Isosceles Triangle:

Not an Isosceles Triangle:

Examples of Isosceles Triangles in Real Life:

Many things in the world have the shape of an isosceles triangle. Some popular examples of these triangles in real life are:

Parts of an Isosceles Triangle

Parts of an isosceles triangle

1. Legs: The two equal sides of an isosceles triangle are known as 'legs'. In the triangle ABC (given above), AB and AC are the two legs of the isosceles triangle.

2. Base: The 'base' of an isosceles triangle is the third and unequal side. In the triangle ABC, BC is the base of the isosceles triangle.

3. Vertex angle: The 'vertex angle' is the angle formed by two equal sides of an isosceles triangle. ∠BAC is a vertex angle of the isosceles triangle.

4. Base angles: The 'base angles' are the angles that involve the base of an isosceles triangle. ∠ABC and ∠ACB are the two base angles of the isosceles triangle.

Properties of an Isosceles Triangle

Here is a list of some properties of isosceles triangles:

In an isosceles triangle, if two sides are equal, then the angles opposite to the two sides correspond to each other and are also always equal.

In the isosceles triangle given above, the two angles ∠B and ∠C, opposite to the equal sides AB and AC are equal to each other.

The isosceles triangle has three acute angles, meaning that the angles are less than 90°.

The sum of three angles of an isosceles triangle is always 180°.

Types of Isosceles Triangles

Generally, isosceles triangles are classified into three different types:

Isosceles acute triangle: An isosceles acute triangle is a triangle in which all three angles are less than 90°, and at least two of its angles are equal in measurement. One example of the angles of an isosceles acute triangle is 50°, 50°, and 80°.

Isosceles right triangle: The following is an example of a right triangle with two legs (and their corresponding angles) of equal measure.

Isosceles obtuse triangle: An isosceles obtuse triangle is a triangle in which one of the three angles is obtuse (lies between 90° and 180°), and the other two acute angles are equal in measurement. One example of isosceles obtuse triangle angles is 30°, 30°, and 120°.

Area and Perimeter of Isosceles Triangle

The area of an isosceles triangle is given by the following formula:

Area (A) = ½ × base (b) × height (h)

The perimeter of the isosceles triangle is given by the formula:

Perimeter (P) = 2a + base (b)

Here, 'a' refers to the length of the equal sides of the isosceles triangle and 'b' refers to the length of the third unequal side.

Solved Examples

Example 1

What is the height of an isosceles triangle with an area of 12 sq. cm and a base of 6 cm?
Solution:
Area of isosceles triangle = ½ x base x height

i.e. 12 = ½ x 6 x height

i.e. 12 = 3 x height

i.e. height = 4 cm

Example 2

What is the perimeter of an isosceles triangle, if equal sides are 'a' cm each and the unequal side is 'b' cm?
Solution:
Perimeter of an isosceles triangle = sum of its sides

Perimeter of an isosceles triangle = (a + a + b) cm, i.e., (2a + b) cm

Example 3

Find the perimeter of an isosceles triangle if the base is 16 cm and the equal sides are 24 cm each.
Solution:
Formula of the perimeter of an isosceles triangle, P = 2a + b

Here, a (sides) = 24 cm and b (base) = 16 cm

Therefore, perimeter of an isosceles triangle, P = 2(24) + 16 = 64 cm.

Hence, the perimeter is 64 cm.

Triangle Games

With SplashLearn, there are several games about triangles for children to try. Let us look at a few of them:

Identify Types of Triangles: In this game, your child will identify various types of triangles. They will use the attributes given to determine the correct triangle and learn the characteristics of a triangle. Students will choose the correct answer from the options.

Classify Triangles: The game challenges your child to sharpen their skills by solving a series of problems involving two-dimensional shapes to identify different types of triangles. Students will analyze a triangle's sides and angle measurements and classify each into the appropriate category.

Other Games

Classify Triangles and Rectangles as Closed Shapes: This game will help children classify different types of shapes and help memorize them. It will help your child recognize various shapes quicker and easier.

Sort the Shapes by Name: This will turn out to be a fun game, with your child learning all about different shapes at the end! The game involves sorting shapes based on their names, and by doing so, your young mathematician will gain more practice with 2D shape concepts. This game will push your child towards mastery while developing overall mathematical capabilities.

Students may also find it a bit overwhelming to remember the properties of isosceles triangles. But that's precisely where you will require a great deal of patience while teaching your kid. Allow your child to shine bright with SplashLearn.

Practice Problems on Isosceles Triangles

Frequently Asked Questions

# data[].attributes.attachment_link.title
Isosceles Triangle - Definition, Angles, Properties, Examples

# data[].attributes.attachment_link.description
A triangle with two sides of equal length is an isosceles triangle. Many things in the world have the shape of an isosceles triangle. Some popular examples of these triangles in real life are: Parts of an isosceles triangle 1. Legs: The two equal sides of an isosceles triangle are known as 'legs'.

# data[].attributes.attachment_link.readable_text
An equilateral triangle is a triangle with all three sides of equal length , corresponding to what could also be known as a "regular" triangle. An equilateral triangle is therefore a special case of an isosceles triangle having not just two, but all three sides equal. An equilateral triangle also has three equal angles.

The altitude of an equilateral triangle is

where is the side length, so the area is

The inradius , circumradius , and area can be computed directly from the formulas for a general regular polygon with side length and sides,

The areas of the incircle and circumcircle are

Central triangles that are equilateral include the circumnormal triangle, circumtangential triangle, first Morley triangle, inner Napoleon triangle, outer Napoleon triangle, second Morley triangle, Stammler triangle, and third Morley triangle.

An equation giving an equilateral triangle with is given by

Geometric construction of an equilateral consists of drawing a diameter of a circle and then constructing its perpendicular bisector . Bisect in point , and extend the line through . The resulting figure is then an equilateral triangle. An equilateral triangle may also be constructed from the intersections of the angle trisectors of the three interior angles of any triangles ( Morley's theorem).

Napoleon's theorem states that if three equilateral triangles are drawn on the legs of any triangle (either all drawn inwards or outwards) and the centers of these triangles are connected, the result is another equilateral triangle.

Given the distances of a point from the three corners of an equilateral triangle, , , and , the length of a side is given by

(Gardner 1977, pp. 56-57 and 63). There are infinitely many solutions for which , , and are integers. In these cases, one of , , , and is divisible by 3, one by 5, one by 7, and one by 8 (Guy 1994, p. 183).

Begin with an arbitrary triangle and find the excentral triangle. Then find the excentral triangle of that triangle, and so on. Then the resulting triangle approaches an equilateral triangle. The only rational triangle is the equilateral triangle (Conway and Guy 1996). A polyhedron composed of only equilateral triangles is known as a deltahedron.

Let any rectangle be circumscribed about an equilateral triangle. Then

where , , and are the areas of the triangles in the figure (Honsberger 1985).

The smallest equilateral triangle which can be inscribed in a unit square (left figure) has side length and area

The largest equilateral triangle which can be inscribed (right figure) is oriented at an angle of and has side length and area

(Madachy 1979).

Triangle line picking for points in an equilateral triangle with side lengths gives a mean line segment length of

See also

30-60-90 Triangle, Acute Triangle, Deltahedron, Equilic Quadrilateral, Fermat Points, Gyroelongated Square Dipyramid, Icosahedron, Isosceles Right Triangle, Isosceles Triangle, Morley's Theorem, Octahedron, Pentagonal Dipyramid, Polygon, Regular Polygon, Reuleaux Triangle, Right Triangle, Scalene Triangle, Snub Disphenoid, Tetrahedron, Triangle, Triangle Packing, Triangular Dipyramid, Triaugmented Triangular Prism, Viviani's Theorem Explore this topic in the MathWorld classroom

Explore with Wolfram|Alpha

References

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 121, 1987. Conway, J. H. and Guy, R. K. "The Only Rational Triangle." In The Book of Numbers. New York: Springer-Verlag, pp. 201 and 228-239, 1996. Dixon, R. Mathographics. New York: Dover, p. 33, 1991. Fukagawa, H. and Pedoe, D. "Circles and Equilateral Triangles." §2.1 in Japanese Temple Geometry Problems. Winnipeg, Manitoba, Canada: Charles Babbage Research Foundation, pp. 23-25 and 100-102, 1989. Gardner, M. Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. New York: Vintage Books, 1977. Guy, R. K. "Rational Distances from the Corners of a Square." §D19 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 181-185, 1994. Honsberger, R. "Equilateral Triangles." Ch. 3 in Mathematical Gems I. Washington, DC: Math. Assoc. Amer., 1973. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 19-21, 1985. Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 115 and 129-131, 1979.

Cite this as:

Weisstein, Eric W. "Equilateral Triangle." FromMathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EquilateralTriangle.html

Subject classifications

# data[].attributes.attachment_link.title
Equilateral Triangle -- from Wolfram MathWorld

# data[].attributes.attachment_link.description
An equilateral triangle is a triangle with all three sides of equal length a, corresponding to what could also be known as a "regular" triangle. An equilateral triangle is therefore a special case of an isosceles triangle having not just two, but all three sides equal.

# data[].attributes.attachment_link.title
What is Congruent? Definition, Examples, Facts

# data[].attributes.attachment_link.description
Definition of Congruent explained with real life illustrated examples. Also learn the facts to easily understand math glossary with fun math worksheet online at SplashLearn.

# data[].attributes.attachment_link.readable_text
If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# data[].attributes.attachment_link.title
Types of triangles review (article) | Khan Academy

# data[].attributes.attachment_link.description
If you're seeing this message, it means we're having trouble loading external resources on our website. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# data[].attributes.attachment_link.title
Triangles: categorization by angle or equal sides | 4th grade | Khan Academy

# data[].attributes.attachment_link.readable_text
A polygon is aplane shape with straight sides.

Is it a Polygon?

Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up).

Polygon
(straight sides)

Not a Polygon
(has a curve)

Not a Polygon
(open, not closed)

Polygon comes from Greek. Poly- means "many" and -gon means "angle".

Types of Polygons

Regular or Irregular

A regular polygon has all angles equal and all sides equal, otherwise it is irregular

Regular

Irregular

Concave or Convex

A convex polygon has no angles pointing inwards. More precisely, no internal angle can be more than 180°.

If any internal angle is greater than 180° then the polygon is concave. ( Think: concave has a "cave" in it)

Convex

Concave

Simple or Complex

A simple polygon has only one boundary, and it doesn't cross over itself. A complex polygon intersects itself! Many rules about polygons don't work when it is complex.

Simple Polygon
(this one's a Pentagon)

Complex Polygon
(also a Pentagon)

More Examples

Irregular Hexagon

Concave Octagon

Complex Polygon
(a "star polygon",
in this case a pentagram)

Play With Them!

Try Interactive Polygons ... make them regular, concave or complex.

Names of Polygons

If it is a Regular Polygon...

Name

Sides

Shape

Interior Angle

Triangle (or Trigon)
3
60°

Quadrilateral (or Tetragon)
4
90°

Pentagon 5
108°

Hexagon 6
120°

Heptagon (or Septagon)
7
128.571°

Octagon 8
135°

Nonagon (or Enneagon)
9
140°

Decagon
10
144°

Hendecagon (or Undecagon)
11
147.273°

Dodecagon
12
150°

Megagon
1,000,000
~180°

Googolgon
10¹⁰⁰
~180°

n-gon
n
( n−2) × 180 ° / n

You can make names using this method:

BUT, for polygons with 13 or more sides, it is OK (and easier) to write " 13-gon", " 14-gon" ... " 100-gon", etc.

Remembering

Quadrilateral (4 Sides)

A Quad Bike has 4 wheels

Pentagon (5 Sides)

The " Pentagon" in Washington DC has 5 sides

Hexagon (6 Sides)

Honeycomb has H exagons

Septagon (7 Sides)

Think Septagon is a "Seven- agon"

Octagon (8 Sides)

An Octo pus has 8 tentacles

Nonagon (9 Sides)

Think Nonagon is a "Nine- agon"

Decagon (10 Sides)

Think Decagon has 10 sides,
just like our Dec imal system has 10 digits

# data[].attributes.attachment_link.description
Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up). Polygon (straight sides) Not a Polygon (has a curve) Not a Polygon (open, not closed) Polygon comes from Greek. Poly- means "many" and -gon means "angle".