# What Is Geometry?

# 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".

## 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 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.

Angles can be measured in degrees (°) or in radians.

There are different types of angles:

- 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°.

Two angles can be classified as complementary, supplementary or explementary depending on whether they add to 90°, 180° or 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 b
_{1}and b_{2}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 ( interactive widget)

Read more about the area of a circle using the diameter ( interactive widget)

Read more about the area of an ellipse ( interactive widget)

Read more about the area of a parallelogram ( interactive widget)

Read more about the area of a rectangle ( interactive widget)

Read more about the area of a square ( interactive widget)

Read more about the area of a trapezoid ( interactive widget)

Read more about the area of a triangle ( interactive widget)

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 ( interactive widget)

Read more about the volume of a cube ( interactive widget)

Read more about the volume of a cylinder

Read more about the volume of a sphere ( interactive widget)

## 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.

The radius can be found in terms of the diameter, circumference and area.

The diameter can be found in terms of the diameter, circumference and area.

- 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.

Triangles can be classified by their 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°).

The interior angles of a triangle add up to 180°. This means a missing angle can be found if two are known.

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.

Polygons have interior angles. The sum of the interior angles of a polygon with n sides is:

A regular polygon has equal side lengths and angles. Each interior angle in a regular polygon is:

Polygons have exterior angles which add up to 180°. Each exterior angle of a regular polygon is:

An interior and exterior angle of a polygon will add to 180°.

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:

A circle of radius r centered at (a, b) has an equation:

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 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.

The area of a sector of a circle is:

The length of an arc is:

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)