Polygons - Part #2Regular Polygons

Regular Polygons

Recall:

Regular polygons are polygons that have congruent sides and congruent angles

Irregular polygons are polygons that have either: different side lengths, different angle measurements, or both

Common PolygonsName Number of Sides Diagram

Triangle 33

Quadrilateral 4

Pentagon 5

Hexagon 6

Heptagon 7

Octagon 8

Nonagon 9

Decagon 10

Properties

Central Angle

Interior Angle

Exterior Angle

Measure of Interior AnglesWe can use a formula to find the measure of an interior angle of a regular polygon.

Formula: Measure of an Interior Angle

M = 180 (n-2)

———————n

Where n = the number of sides

ExampleFind the measure of an interior angle in a hexagon

ExampleFind the measure of an interior angle in a nonagon

ExampleFind the measure of an interior angle in a 12 sided regular polygon

Sum of Interior AnglesWe can use a formula to find the sum of the interior angles of a regular polygon.

Formula: Sum of the Interior Angles

S = 180 (n-2)

Where n = the number of sides

ExampleFind the sum of the interior angles of a hexagon

ExampleFind the sum of the interior angles in an octagon

ExampleThe sum of the interior angles of a polygon is 900°. Determine how many sides this polygon has.

Measure of the Central AngleWe can also determine the measure of the central angles in a regular polygon

The central angle is the angle made at the centre of a polygon by any two adjacent vertices of the polygon

Central Angle

Radius

Note: All central angles would add up to 360° (a full circle) so the measure of the central angle is 360 divided by the number of sides.

Formula: Measure of the Central Angle

Measure of the Central Angle

C = 360

———————n

Where n = the number of sides

ExampleWhat is the measure of the central angle in a hexagon?

ExampleWhat is the measure of the central angle in a heptagon?

Determining the DiagonalsDiagonals are straight lines that extend from one vertex to another

You can have multiple diagonals per vertex

Example:

We can also determine the number of diagonals in a regular polygon by using a formula

Determining the Diagonals

Formula: Determine the Number of Diagonals

D = n (n - 3)

———————2

Where n = the number of sides

ExampleDetermine the number of diagonals in a regular pentagon

ExampleDetermine the number of diagonals in a regular octagon