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."

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.
"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".
Shapes
- A circle is a shape containing a set of points that are all the same distance from a given point, its center.
- An ellipse is a shape containing a set of points whose distance from a two fixed points, the foci, add up to a constant.
An ellipse looks like a flattened circle.
- A parallelogram is a four sided shape with opposite sides parallel.
- A rectangle is a four sided shape with four right angles.
- A square is a four sided shape with four equal sides and four right angles.
- A trapezoid is a four sided shape with one pair of opposite parallel sides.
- A triangle is a shape with three sides and three angles.
Read more about circles
Read more about ellipses
Read more about parallelograms
Read more about rectangles
Read more about squares
Read more about trapezoids
Read more about triangles
Angles
An angle is created by two rays that have a common end point, called the vertex. The angle is also a measure of the rotation between the two rays.
- An acute angle is an angle of less than 90°.
- A right angle is an angle of 90°.
- An obtuse angle is an angle greater than 90° and less than 180°.
- A straight angle
is an angle of 180°.
- A reflex angle is an angle greater then 180° and less than 360°.
- A full angle is an angle of 360°.
- Complementary angles are two angles which add up to 90°.
- Supplementary angles are two angles which add up to 180°.
- Explementary angles are two angles which add up to 360°.
Read more about angles
Read more about degrees
Read more about radians
Read more about the types of angles
Read more about acute angles
Read more about right angles
Read more about obtuse angles
Read more about straight angles
Read more about reflex angles
Read more about full angles
Read more about complementary angles
Read about supplementary angles
Read more about explementary angles
Areas
Area is the space contained within a 2-dimensional shape.- The area of a circle of radius r is:
- The area of a circle of diameter d is:
- The area of an ellipse with semi-major axis a and semi-minor axis b is:
- The area of a parallelogram where b is the length of the base and h is the height is:
- The area of a rectangle where b is the length of the base and h is the height is:
- The area of a square where a is the length of the side is:
- 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:
- The area of a triangle where b is the length of the base and h is the height is:
- The area of a triangle, using trigonometry, where a and b are lengths of two sides of the triangle and C is the angle between them is:
Read more about the area of a circle (

Read more about the area of a circle using the diameter (

Read more about the area of an ellipse (

Read more about the area of a parallelogram (

Read more about the area of a rectangle (

Read more about the area of a square (

Read more about the area of a trapezoid (

Read more about the area of a triangle (

Read more about the area of a triangle using trigonometry
Volumes
Volume is the space contained within a 3-dimensional shape.- The volume of a cone of height h and a circular base of radius r is:
- The volume of a cube where a is the length of the side is:
- The volume of a cylinder with radius r and height h is:
- The volume of a sphere of radius r is:
Read more about the volume of a cone (

Read more about the volume of a cube (

Read more about the volume of a cylinder
Read more about the volume of a sphere (

Circles (Basics)
- The center is the point the same distance from the points on the circle.
- The radius is the line segment from the center of the circle to any point on the circle.
- The diameter is the line segment that contains the centre of the circle and has its endpoints on the circle.
- A chord is a line whose endpoints lie on the circle.
- The circumference is the distance around the circle.
The circumference of a circle with radius r and diameter d is:
- An arc is a portion of the circumference.
- A sector is a region bounded by two radii and the arc lying between the radii.
- A segment is a region, not containing the center, bounded by a chord and an arc lying between the chord's endpoints.
- A tangent is a line that touches the circle at one point.
Read more about the radius
Read more about the diameter
Read more about finding the radius from the diameter
Read more about finding the radius from the circumference
Read more about finding the radius from the area
Read more about finding the diameter from the radius
Read more about finding the diameter from the circumference
Read more about finding the diameter from the area
Read more about the circumference of a circle
Triangles
There are different types of triangles. Triangles can be classified by how many sides and angles are equal:- Equilateral triangles have 3 equal side lengths and angles.
- Isosceles triangles have 2 equal side lengths and angles.
- Scalene triangles have 0 equal side lengths and angles.
- Acute triangles have all acute angles (less than 90°).
- Obtuse triangles have one angle that is obtuse (more than 90°, less than 180°).
- Right triangles have one right angle (90°).

Read more about the types of triangle
Read more about equilateral triangles
Read more about isosceles triangles
Read more about scalene triangles
Read more about right triangles
Read more about the interior angles of a triangle
Read more about finding the missing angle in a triangle
Polygons
A polygon is a 2-dimensional shape with straight sides.- A 3 sided polygon is a triangle.
- A 4 sided polygon is a quadrilateral.
- A 5 sided polygon is a pentagon.
- A 6 sided polygon is a hexagon.
- A 7 sided polygon is a hexagon.
- An 8 sided polygon is an octagon.
- A 9 sided polygon is a nonagon.
- A 10 sided polygon is a decagon.
- A 12 sided polygon is a dodecagon.



Read more about polygons
Read more about the interior angles of a polygon
Read more about finding the sum of the interior angles of a polygon
Read more about finding the interior angle of a regular polygon
Read more about the exterior angles of a polygon
Read more about finding the exterior angle of a regular polygon
Read more about the sum of the interior and exterior angle of a polygon
Circles (Advanced)
A circle of radius r centered at the origin has an equation:

- The tangent makes 90° with the radius which it meets at the point at which it touches.
- Two radii form the two equal sides of an isosceles triangle.
- 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.
- The angle subtended by an arc at the center of a circle is twice the angle subtended by the same arc at the circumference.
- A triangle drawn from two ends of a diameter makes 90° at the circumference.
- 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°.
- Two tangents drawn from the same point outside of the circle are the same length and form two congruent right triangles.
- 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 basic equation of a circle
Read more about the equation of a circle
Read more about the circle theorems:
- The tangent meets the radius at 90 degrees
- Two radii make an isosceles triangle
- The perpendicular bisector of a chord passes through center
- The angle at the center is twice that at the circumference
- The angle in a semicircle is 90 degrees
- Angles in the same segment are equal
- Opposite angles in a cyclic quadrilateral add up to 180 degrees
- Tangents from the same point are equal
- The angle in the alternate segment is equal
Read more about the area of a sector of a circle
Read more about the area of a sector of a circle (radians)
Read more about the length of an arc
Read more about the length of an arc (radians)