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Triangles are 3 sided shapes that are polygons. Triangles are equal to 180 degrees and can have multiple different sides. Triangles can have a variety of different lengths of sides. They also have all different types of angles (acute, obtuse, right).

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A square is also a quadrilateral with all four sides being equal and 90 degrees as well. Squares are similar to rectangles but they have all parallel sides. All length in the square have the same length and sides.

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A star is a shape that has angles less than 180 degrees. Stars have points on each end and are known for their equal sides. They typically have 5 points knows as the tips.

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A rectangle is a four sided shape that has 4 right angles. Opposite sides are the same length known as parallel. This is known as a quadrilateral and has corners that are all 90 degrees.

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Circles are 360 degrees. Circles are even from the middle to the edge known as the radius. Also, circles have no points and look like the letter "O".

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Triangle Facts for Kids

Enjoy a range of interesting triangle facts for kids and have fun learning about the 3-sided polygon.

Find information related to equilateral triangles, isosceles triangles, scalene triangles, obtuse triangles, acute triangles, right angle triangles, the hypotenuse, angles of a triangle and more. Improve your geometry knowledge with our interesting triangle facts and trivia.

Triangles are polygons with the least possible number of sides (three).

The three internal angles of a triangle always add to 180 degrees.

An equilateral triangle has three sides of equal length and three equal angles.

An isosceles triangle has two sides of equal length and two equal angles.

A scalene triangle has no sides of equal length and no equal angles.

A right angle triangle has one angle that is 90 degrees.

An obtuse triangle has one angle larger than 90 degrees.

An acute triangle has angles that are all less than 90 degrees.

The longest side of a right angle triangle is called the hypotenuse, it is always found opposite the right angle.

Trigonometry is the study of the relationship between the angles of triangles and their sides.

Triangle shapes are often used in construction because of their great strength.

Now that you're an expert on triangles, try learning about squares, quadrilaterals and other 2D polygon shapes.

Equilateral triangle

Isosceles triangle

Scalene triangle

Right angle triangle

Obtuse triangle

Acute triangle

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Triangle Facts for Kids - Equilateral, Isosceles, Scalene, Obtuse, Acute, Right Angle

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Triangle Facts for Kids Enjoy a range of interesting triangle facts for kids and have fun learning about the 3-sided polygon. Find information related to equilateral triangles, isosceles triangles, scalene triangles, obtuse triangles, acute triangles, right angle triangles, the hypotenuse, angles of a triangle and more.

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The Singing Walrus - English Songs For Kids

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Shapes song for kids | The Singing Walrus

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In Euclidean plane geometry, a rectangle is a shape with for sides and four right angles. A rectangle with four sides of equal length is a square. The term is occasionally used to refer to a non- square rectangle.

The word rectangle comes from the Latin rectangulus, which is a combination of rectus (as an adjective, right, proper) and angulus ( angle).

Other geometries, such as spherical, elliptic, and hyperbolic, have so-called rectangles with opposite sides equal in length and equal angles that are not right angles.

Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons.

Classification

Traditional hierarchy

A rectangle is a special case of a parallelogram in which each pair of adjacent sides is perpendicular.

A parallelogram is a special case of a trapezium (known as a trapezoid in North America) in which both pairs of opposite sides are parallel and equal in length.

A trapezium is a convex quadrilateral which has at least one pair of parallel opposite sides.

A convex quadrilateral is

Simple: The boundary does not cross itself.

Star-shaped: The whole interior is visible from a single point, without crossing any edge.

Properties

A rectangle is cyclic: all corners lie on a single circle.

It is equiangular: all its corner angles are equal (each of 90 degrees).

It is isogonal or vertex-transitive: all corners lie within the same symmetry orbit.

It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).

Rectangle-rhombus duality

The dual polygon of a rectangle is a rhombus, as shown in the table below.

Alternate sides are equal.

Alternate angles are equal.

Its centre is equidistant from its , hence it has a circumcircle.

Its centre is equidistant from its sides, hence it has an incircle.

Two axes of symmetry bisect opposite sides.

Two axes of symmetry bisect opposite angles.

Diagonals are equal in length.

Diagonals intersect at equal angles.

The figure formed by joining, in order, the midpoints of the sides of a rectangle is a rhombus and vice versa.

If a rectangle has length and width

it has area ,

it has perimeter ,

each diagonal has length ,

and when , the rectangle is a square.

The isoperimetric theorem for rectangles states that among all rectangles of a given perimeter, the square has the largest area.

The midpoints of the sides of any quadrilateral with perpendicular diagonals form a rectangle.

A parallelogram with equal diagonals is a rectangle.

The Japanese theorem for cyclic quadrilaterals states that the incentres of the four triangles determined by the vertices of a cyclic quadrilateral taken three at a time form a rectangle.

The British flag theorem states that with vertices denoted A, B, C, and D, for any point P on the same plane of a rectangle:

Crossed rectangles

A crossed quadrilateral (self-intersecting) consists of two opposite sides of a non-self-intersecting quadrilateral along with the two diagonals. Similarly, a crossed rectangle is a crossed quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. It has the same vertex arrangement as the rectangle. It appears as two identical triangles with a common vertex, but the geometric intersection is not considered a vertex.

A crossed quadrilateral is sometimes likened to a bow tie or butterfly, sometimes called an "angular eight". A three-dimensional rectangular wire frame that is twisted can take the shape of a bow tie.

The interior of a crossed rectangle can have a polygon density of ±1 in each triangle, dependent upon the winding orientation as clockwise or counterclockwise.

A crossed rectangle may be considered equiangular if right and left turns are allowed. As with any crossed quadrilateral, the sum of its interior angles is 720°, allowing for internal angles to appear on the outside and exceed 180°.

A rectangle and a crossed rectangle are quadrilaterals with the following properties in common:

Opposite sides are equal in length.

The two diagonals are equal in length.

It has two lines of reflectional symmetry and rotational symmetry of order 2 (through 180°).

Other rectangles

In spherical geometry, a spherical rectangle is a figure whose four edges are great circle arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length. The surface of a sphere in Euclidean solid geometry is a non-Euclidean surface in the sense of elliptic geometry. Spherical geometry is the simplest form of elliptic geometry.

In elliptic geometry, an elliptic rectangle is a figure in the elliptic plane whose four edges are elliptic arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length.

In hyperbolic geometry, a hyperbolic rectangle is a figure in the hyperbolic plane whose four edges are hyperbolic arcs which meet at equal angles less than 90°. Opposite arcs are equal in length.

Squared, perfect, and other tiled rectangles

A rectangle tiled by squares, rectangles, or triangles is said to be a "squared", "rectangled", or "triangulated" (or "triangled") rectangle respectively. The tiled rectangle is perfect if the tiles are similar and finite in number and no two tiles are the same size. If two such tiles are the same size, the tiling is imperfect. In a perfect (or imperfect) triangled rectangle the triangles must be right triangles. A database of all known perfect rectangles, perfect squares and related shapes can be found at squaring.net. The lowest number of squares need for a perfect tiling of a rectangle is 9 and the lowest number needed for a perfect tilling a square is 21, found in 1978 by computer search.

A rectangle has commensurable sides if and only if it is tileable by a finite number of unequal squares. The same is true if the tiles are unequal isosceles right triangles.

The tilings of rectangles by other tiles which have attracted the most attention are those by congruent non-rectangular polyominoes, allowing all rotations and reflections. There are also tilings by congruent polyaboloes.

In Spanish: Rectángulo para niños

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A circle is a two-dimensional shape formed by all the points that are the same distance from the center.

Explore this collection of circle facts. Learn how to find the circumference, diameter, radius, and area of a circle and get definitions of circle terms used in geometry.

Circle Facts

A circle is a two-dimensional shape formed by all the points that are the same distance from a center point.

Technically, only the points equidistant from the center form the circle. The area enclosed within a circle is called a disc.

The word circle comes from the Greek word κρίκος (krikos), meaning "hoop" or "ring".

A circle is the only one-sided shape containing an area. A straight line is a circle containing an infinite area.

Humans have recognized circles since ancient times. Natural circles include the shapes of the Sun and Moon, the human eye, tree cross-sections, some flowers, some shells, etc.

The distance around a circle is its circumference.

The distance from the center to the circle is its radius.

The longest distance between two points on a circle is the diameter, which is a line segment running through the center.

A circle is the shape with the shortest perimeter enclosing an area.

The circle is the most symmetric shape because every line through the center is a line of reflection symmetry. It has rotation symmetry for every angle around its center.

Pi (π) is an irrational number that is the ratio of a circle's circumference to its diameter. It is approximately equal to 3.1415259.

Archimedes proved the area enclosed within a circle is the same as the area of a triangle with a base the length of the circle's circumference and height equal to the circle's radius.

The full arc of circle measures 360 degrees.

A circle is a special type of ellipse where the two foci are in the same location and the eccentricity is 0.

Written in 1700 BCE, the Rhind papyrus describes a method of finding the area of a circle. The result comes out as 256/81, which is about 3.16 (close to pi).

You can draw a special circle inside every triangle, called the incircle, where each of the three triangle sides are tangent to the circle.

How to Find the Circumference of a Circle

The circumference (C) is the distance around a circle. There are a few ways to find the circumference. You can calculate it from either the radius (r) or diameter (d) or you can measure it.

C = 2πr

C = πd

It's easiest to measure a circle's circumference using a string. Shape the string around the circle, mark the length, and then use a ruler or meter stick to measure the length of the string.

How to Find the Diameter of a Circle

The diameter (d) is the length of the line segment with end points on the circle that passes through its center. It is the longest distance across a circle. The diameter is twice the length of the radius.

d = 2r

d = C/π

Measure the diameter by finding the longest line segment across a circle.

How to Find the Radius of a Circle

The radius (r) is the distance from the center of a circle to its border. It is half the length of the diameter.

r = d/2

r = C/2π

If you draw a circle using a compass, the radius is the distance between its two points. Measuring the radius of a circle is a bit tricky unless you know its center. Sometimes its easier to measure the circumference or diameter and calculate the radius.

How to Find the Area of a Circle

The area (A) of a circle is the region enclosed by a circle or the area of its disc.

A = πr²

A = π(d/2)²

A = Cr/2 - You can use Archimedes' proof to find the circle area using its circumference and radius. Set the base of the triangle equal to circumference C and height equal to radius r. The triangle area formula 1/2 bh becomes A = Cr/2

Circle Vocabulary Terms

Here are key circle vocabulary terms to know:

Annulus: An annulus is a ring-shape formed between two concentric circles.

Arc: An arc is any segment of a circle formed by connected points.

Center (Centre): The center is the point that is equidistant from all points on a circle. It is also called the origin.

Chord: A chord is a line segment with endpoints on the circle. The diameter is the longest chord.

Circumference: The circumference is the distance around a circle.

Closed: A region that includes its boundaries.

Diameter: The diameter is the line segment with endpoints on the circle and midpoint at its center. It is the largest distance between any two points on a circle.

Disc: A disc is the area inside a circle.

Lens: A lens is a region shared by two overlapping discs.

Open: Any region, excluding its boundaries.

Passant: A passant is a coplanar line that has no points in common with a circle.

Radius: A radius is a line segment running from the center to the circle.

Sector: A sector is an area within a circle bounded by two radii.

Segment: A segment is an area bounded by an arc and a chord.

Secant: A secant is a chord that extends beyond the circle. In other words, it is a coplanar line that intersects a circle at two points.

Semicircle: A semicircle is an arc which has the diameter as endpoints and center as midpoint. The interior of a semicircle is a half-disc.

Tangent: A tangent is a coplanar line sharing one single point in common with a circle.

Circle Worksheets

Practice finding the circumference and area of circles with these math worksheets.

References

Gamelin, Theodore (1999). Introduction to Topology. Mineola, N.Y: Dover Publications. ISBN 0486406806.

Harkness, James (1898). "Introduction to the theory of analytic functions". Nature. 59 (1530): 30. doi:10.1038/059386a0

Katz, Victor J. (1998). A History of Mathematics / An Introduction (2nd ed.). Addison Wesley Longman. ISBN 978-0-321-01618-8.

Ogilvy, C. Stanley (1969). Excursions in Geometry. Dover.

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Circle Facts - Area, Circumference, Diameter, Radius

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Get interesting circle facts. See how to find the area, circumference, diameter, and radius of a circle in geometry.