# Shapes

## What Is Geometry?

Geometry is a branch mathematics that studies shapes and their properties.

## Dictionary Definition

The Oxford English Dictionary defines geometry as "the branch of mathematics concerned with the properties and relations of points, lines, surfaces, and solids."
Here are some examples of common shapes. We might be interested in finding the angle in a triangle, the area of a circle or the volume of a cone. Geometry allows us to do this. ## Where Does the Word Geometry Come From?

Geometry comes from combining the Greek words 'ge' ("earth") and 'metria' ("measurement"). Geometry means measuring the earth or land. This is because initially geometry would have been used practically to measure areas of fields and the lengths of roads.

## Euclid and Geometry Euclid was an ancient Greek mathematician, famous for his work in geometry. His book of geometry, Elements, is one of the most widely read books of all time and has earned Euclid the nickname the Father of Geometry. Euclid has had many great fans. Abraham Lincoln kept a copy of Euclid in his saddlebag and would read it late at night by lamplight to aid his career as a lawyer: "You never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father's house and stayed there till I could give any proposition in the six books of Euclid at sight". Albert Einstein received a copy when he was a boy and claimed it had a great influence on him. He called Elements the "holy little geometry book".

## Plane Shapes Plane shapes are two-dimensional, flat shapes: such as circles, squares and triangles. In this mini-curriculum, you will learn about plane shapes.

Circles
A circle is a shape containing a set of points that are all the same distance from a given point, its center. Ellipses
An ellipse is a flattened circle. Parallelograms
A parallelogram is a four sided shape with opposite sides parallel. Rectangles
A rectangle is a four sided shape with four right angles. Squares
A square is a four sided shape with four equal sides and four right angles. Trapezoids
A trapezoid is a four sided shape with one pair of opposite parallel sides. Triangles
A triangle is a shape with three sides and three angles. ## Areas Area is a measure of the size of a plane shape. Larger shapes have more area. In this mini-curriculum, you will learn the areas of plane shapes.

Area of a Circle
The area of a circle where r is the radius is: Area of an Ellipse
The area of an ellipse where a is the semi-major axis and b is the semi-minor axis is: Area of a Parallelogram
The area of a parallelogram where b is the length of the base and h is the height is: Area of a Rectangle
The area of a rectangle where b is the length of the base and h is the height is: Area of a Square
The area of a square where a is the length of the side is: Area of a Trapezoid
The area of a trapezoid where b1 and b2 are the lengths of the bases (parallel sides) and h is the height of the trapezoid is: Area of a Triangle
The area of a triangle where b is the length of the base and h is the height is: Here is an interactive widget to find the areas of plane shapes.

## Volumes Volume is a measure of a space that a three-dimensional shape contains. In this mini-curriculum, you will learn the volumes of shapes.

Volume of a Cone
The volume of a cone where r is the radius of the circular base and h is the height is: Volume of a Cube
The volume of a cube where a is the length of the side is: Volume of a Cylinder
The volume of a cylinder where r is the radius of the circular base and h is the height is: Volume of a Sphere
The volume of a sphere where r is the radius is: Here is an interactive widget to find the volumes of 3D shapes.

## Circles Circles have an interesting geometry; there are many terms to understand: radius, diameter, circumference, sector and segment. In this mini-curriculum, you will learn about circle geometry.

Center
The center is the point that is the same distance from all the points on the circle. The radius is the line segment from the center of the circle to any point on the circle. Diameter
The diameter is the line segment that contains the centre of the circle and has its endpoints on the circle. Circumference
The circumference is the distance around the circle. Pi
Pi is the ratio of a circle's circumference to its diameter. Arc
An arc is a portion of the circumference.
The length of an arc of a circle where θ is the angle subtended by the arc and r is the radius is: Sector
A sector is a region bounded by two radii and the arc lying between the radii.
The area of a sector of a circle is: Read more about how to find the length of an arc (degrees)
Read more about how to find the area of a sector of a circle (degrees)

Here is an interactive widget to learn about circle geometry. The radius is the line segment from the center of the circle to any point on the circle. The radius also refers to the length of this line segment. In this mini-curriculum, you will learn about how to find the length of the radius from the other dimensions of a circle.

The radius of a circle is half of the diameter. The radius of a circle can be found from the circumference using the formula below: The radius of a circle can be found from the area using the formula below: ## Diameter of a Circle The diameter is the line segment that contains the centre of the circle and has its endpoints on the circle. The diameter also refers to the length of this line segment. In this mini-curriculum, you will learn about how to find the length of the diameter from the other dimensions of a circle.

The diameter of a circle is twice the diameter. Diameter from Circumference
The diameter of a circle can be found from the circumference using the formula below: Diameter from Area
The diameter of a circle can be found from the area using the formula below: ## Area of a Circle The area of a circle is can be found from the diameter. In this mini-curriculum, you will learn about how to find the area of a circle from its diameter.

Area from Diameter
The area of a circle can be found from the diameter using the formula below: ## Circle Theorems Circles have properties related to angles and lines. In this mini-curriculum, you will learn the circle theorems that relate to all circles, such as that tangents meet the radius at 90° and the alternate segment theorem.

Circle Theorems
There are many circle theorems which relate to lines and angles in a circle. The tangent makes 90° with the radius which it meets at the point at which it touches. Two Radii Make an Isosceles Triangle
Two radii form the two equal sides of an isosceles triangle. Perpendicular Bisector of a Chord
If a line cuts through a chord of the circle, such that it crosses it at 90° and cuts it in half, that line passes through the centre of the circle. Angle at Center Twice Angle at Circumference
The angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at the circumference. Angle in Semicircle Is 90°
A triangle drawn from two ends of a diameter makes 90° at the circumference. Angles in Same Segment Are Equal
All triangles drawn from a chord make the same angle at the circumference. A cyclic quadrilateral is a 4 sided shape where each corner touches the circle. Both pairs of opposite angles add up to 180°. Tangents from Point Same Length
Two tangents drawn from the same point outside of the circle are the same length and form two congruent right triangles. Alternate Segment Theorem
The angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment. Read more about the perpendicular bisector of a chord passes through the center
Read more about the angle at the center is twice the angle at the circumference
Read more about angles in the same segment make the same angle at the circumference
Read more about tangents from the same point are the same length

## Triangles A triangle is a shape with three sides and three angles. In this mini-curriculum, you will learn about the different types of triangles and their properties.

Types of Triangles
There are different types of triangles. Equilateral Triangles
An equilateral triangle has three equal angles and three sides the same length. Isosceles Triangles
An isosceles triangle has two equal sides and two equal angles across from them. Scalene Triangles
A scalene triangle has no sides or angles that are equal. Interior Angles of a Triangle
A triangle has three interior angles which determine the type of triangle and add up to 180°. Find Missing Angle in a Triangle
To find the missing angle of a triangle, use the fact that the interior angles of a triangle add up to 180°. Find Missing Angle in an Isosceles Triangle
To find the missing angle of an isosceles triangle, use the facts that the interior angles of a triangle add up to 180° and that two angles are the equal to each other. Read more about how to find a missing angle in a triangle
Read more about how to find a missing angle in an isosceles triangle

Here is an interactive widget to learn about triangle geometry. Here is a game to learn about the different types of triangle.

## Angles An angle is a created when two lines meet. It is also a measure of the rotation between the two lines. In this mini-curriculum, you will learn about the different types of angles and how they are measured.

Angles
An angle is created by two rays that have a common end point, called the vertex. It is also a measure of rotation between the two rays. Degrees
Degrees are a unit of measurement of an angle. There are 360° in a full rotation. Types of Angles
There are different types of angles. Acute Angles
An acute angle is less than 90°. Right Angles
An right angle is 90°. It is a quarter of a revolution. Obtuse Angles
An obtuse angle is greater than 90° and less than 180°. Straight Angles
A straight angle is 180°. It is a half of a revolution. Reflex Angles
A reflex angle is greater than 180° and less than 360°. Full Angles
A full angle is 360°. It is a complete revolution. Radians are a unit of measurement of an angle. There are 2π in a full rotation.
1 radian is the angle found when the radius is wrapped around the circle. ## Angle Properties Two more angles can add up to a right angle, a straight angle or a full angle. In this mini-curriculum, you will learn about complementary angles, supplementary angles and explementary angles.

Complementary Angles
Complementary angles are two angles which add up to a right angle (90°). Supplementary Angles
Supplementary angles are two angles which add up to a straight angle (180°). Explementary Angles
Explementary angles are two angles which add up to a full angle (360°). Here is a game to learn about the 3 main types of angle.

## Polygons A polygon is a 2-dimensional shape with straight sides. In this mini-curriculum, you will learn about different types of polygons and their properties.

Polygons
Polygons are 2-dimensional shapes with straight sides. Interior Angles of a Polygon
The interior angles of a polygon are the angles between two sides, inside the shape. Find the Interior Angle of a Regular Polygon
Regular polygons have sides of equal length and equal angles. The interior angle of a regular polygon where n is the number of sides of the polygon is: Sum of Interior Angles of a Polygon
The sum of the interior angles of a polygon where n is the number of sides of the polygon is: Exterior Angles of a Polygon
The exterior angles of a polygon are the angles outside of the polygon, between a side and the extension of the side next to it. Find the Exterior Angle of a Regular Polygon
Regular polygons have sides of equal length and equal angles. The exterior angle of a regular polygon where n is the number of sides of the polygon is: Sum of Interior and Exterior Angle of a Polygon is 180°
The sum of an interior and exterior angle of a polygon is 180°. 