Geometry Reference Guide

Geometry is the branch of mathematics that deals with points, lines, and shapes. Here you will find formulas, definitions, and examples to help you understand geometry. For a student in middle or upper level math, this handy guide will prove useful for math homework or as a study tool.

POINTS, LINES, AND RAYS

A line segment is a line that has two endpoints. A line extends indefinitely in both directions. A ray is a line that extends in one direction forever. A vertical line segment is a line that extends vertically and has two endpoints. A horizontal line segment is a line with two endpoints that extends horizontally. Intersecting lines meet at a point. Parallel lines are in the same plane and never intersect. Perpendicular lines intersect and form a right angle.

ANGLES

An angle is formed when two line segments come together. They are measured in degrees with a protractor. A half-circle has 180 degrees. An acute angle has less than 90 degrees. An obtuse angle has more than 90 degrees. A right angle measures exactly 90 degrees. A straight angle equals exactly 180 degrees. Supplementary angles combine to make a 180 degree angle. Complementary angles combine to make a 90 degree angle. Adjacent angles share a vertex and a ray. Vertical angles are not adjacent; they share a vertex and are congruent.

POLYGONS

A polygon is a shape that means "many angles" and has 3 properties. It starts and stops at the same place. It can be traced without lifting the pencil or crossing/retracing any part. It is made of at least 3 line segments. A regular polygon has sides that are all the same length.

TRIANGLES

Triangles are three-sided polygons. They have three sides and three angles. Scalene triangles have no equal sides. Isosceles triangles have two sides of equal length. Equilateral triangles have three sides of equal length.

QUADRILATERALS

A quadrilateral has four sides and four angles. The sum of all the angles in a quadrilateral is 360 degrees. Quadrilaterals are categorized by their sides and angles. A square has four parallel sides of equal length and four right angles. A rectangle has four parallel sides but only opposite sides are equal in length, and four right angles. A parallelogram has four parallel sides, with opposite sides of equal length, and 4 angles are more than or less than 90 degrees. A trapezoid has two opposite sides that are parallel. Its sides may or may not be equal and it may have 0, 1, or two angles that are 90 degrees.

CIRCLES

Circles are shapes on which all the points on it are the same distance from a given point. The center is the point from which all points on a circle are the same distance. Diameter is a line segment that connects two points on a circle and passes through the center. Radius is a line segment that connects the center with any point on the circle. A chord is a line segment that connects two points on a circle but does not pass through the center point.

SYMMETRY

A shape has symmetry if it can be separated into two identical parts. If the shape is folded along the line of symmetry, the two halves will match. A shape can have any number of lines of symmetry. There are horizontal and vertical lines of symmetry.

CONGRUENT AND SIMILAR

Shapes are congruent if they are exactly the same shape and size. They are similar if they are about the same relative size.

PYTHAGOREAN THEOREM

In right triangles, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the legs (the shorter sides). Use the following formula a² + b² = c² where a and b are the legs and c is the hypotenuse.

PERIMIETER

Perimeter is the distance around a shape. It is found by adding the lengths of each side of a polygon. To find the perimeter of a square you multiply the length of a side times 4. For rectangles, the formula is 2l +2w where l is the length and w is the width.

CIRCUMFERENCE

The distance around a circle is the circumference. The formula for finding the circumference of a circle is pi (3.14) x d (diameter). The diameter equals two times the radius. You can also find the circumference of a circle by multiplying pi (3.14) times 2r (2 times the radius) since the radius is the distance from the center of circle to its outside edge.

C = 3.14 x d or

C= 3.14 x 2r