Polygons: Formula and Examples

Exterior Angles and Interior Angles

Interior Angle Sum Theorem

What is true about the sum of angles inside a polygon (ie interior angles)?

The sum of the measures of the interior angles of a convex polygon with n sides is (n-2)180

Examples:
  • Triangle or ( '3-gon')
  • Quadrilateral which has four sides ( ' 4-gon')
    • sum of interior angles: (4-2)180 = 360°
  • Hexagon which has six sides ( '6-gon')
    • sum of interior angles: (6-2)180 = 720°

Video Tutorial

on Interior Angles of a Polygon

Definition of a Regular Polygon: A regular polygon is simply a polygon whose sides all have the same length and whose angles all have the same measure. The most well known example of a regular polygon is the equilateral triangle

Measure of a Single Interior Angle

What about when you just want 1 interior angle?

In order to find the measure of a single interior angle of a regular polygon  (a polygon with sides of equal length and angles of equal measure) with n sides, we just divide the sum of the interior angles or (n-2) × 180 by the number of sides or n

The Formula

An interior angle of a regular polygon with n sides is $ \frac{ (n -2) \cdot 180^{\circ} }{n} $

Example:

To find the measure of an interior angle of a regular octagon, which has 8 sides, apply the formula above as follows: ((8-2) × 180) / 8 = 135°

Problem 1

What is the total number degrees of all interior angles of a triangle?

180°

Problem 2

What is the total number of degrees of all interior angles of the polygon on the left?

360° since the polygon on the left is really just two triangles and each triangle has 180°

Problem 3

What is the sum measure of the interior angles of the polygon (a pentagon) on the left?

Use Interior Angle Theorem: (5 - 2) × 180 = 3 × 180 = 540°

Problem 4

What is sum of the measures of the interior angles of the polygon (a hexagon) on the left?

Use Interior Angle Theorem: (6 - 2) × 180 = 4 × 180 = 720°

Finding 1 interior angle of a regular Polygon

Problem 5

What is the measure of 1 interior angle of a regular octagon?

Substitute 8 (an octagon has 8 sides) into the formula to find a single interior angle

poly1
Problem 6

Calculate the measure of 1 interior angle of a regular dodecagon (12 sided polygon)?

Substitute 12 (a dodecagon has 12 sides) into the formula to find a single interior angle

poly2
Problem 7

Calculate the measure of 1 interior angle of a regular hexadecagon (16 sided polygon)?

Substitute 16 (a hexadecagon has 16 sides) into the formula to find a single interior angle

poly2
Challenge Problem challenge problem

What is the measure of 1 interior angle of a pentagon?

This question cannot be answered because the shape is not a regular polygon. You can only use the formula to find a single interior angle if the polygon is regular!

Consider, for instance, the irregular pentagon drawn below.

You can tell, just by looking at the picture, that $$ \angle A    and    \angle B $$ are not congruent.

pentagon irregular

How about the measure of an exterior angle?

Exterior Angles of a Polygon

Formula for sum of exterior angles:
The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360°.

Measure of a Single Exterior Angle

Formula to find 1 angle of a regular convex polygon of n sides = a

Exterior Angles of Triangle

$$ \angle1 + \angle2 + \angle3 = 360° $$


Exterior Angles of Polygon

$$ \angle1 + \angle2 + \angle3 + \angle4 = 360° $$


Exterior Angles of Pentagon

$$ \angle1 + \angle2 + \angle3 + \angle4 + \angle5 = 360° $$

Practice Problems

Problem 8

Calculate the measure of 1 exterior angle of a regular pentagon?

Substitute 5 (a pentagon has 5sides) into the formula to find a single exterior angle

poly2
Problem 9

What is the measure of 1 exterior angle of a regular decagon (10 sided polygon)?

Substitute 10 (a decagon has 10 sides) into the formula to find a single exterior angle

poly2
Problem 10

What is the measure of 1 exterior angle of a regular dodecagon (12 sided polygon)?

Substitute 12 (a dodecagon has 12 sides) into the formula to find a single exterior angle

poly2
Challenge Problem challenge problem

What is the measure of 1 exterior angle of a pentagon?

This question cannot be answered because the shape is not a regular polygon. Although you know that sum of the exterior angles is 360, you can only use formula to find a single exterior angle if the polygon is regular!

Consider, for instance, the pentagon pictured below. Even though we know that all the exterior angles add up to 360 °, we can see, by just looking, that each $$ \angle A    and    \angle B $$ are not congruent..

Determine Number of Sides from Angles

It's possible to figure out how many sides a polygon has based on how many degrees are in its exterior or interior angles.

Problem 11

If each exterior angle measures 10°, how many sides does this polygon have?

Use formula to find a single exterior angle in reverse and solve for 'n'.

equation
Problem 12

If each exterior angle measures 20°, how many sides does this polygon have?

Use formula to find a single exterior angle in reverse and solve for 'n'.

equation
Problem 13

If each exterior angle measures 15°, how many sides does this polygon have?

Use formula to find a single exterior angle in reverse and solve for 'n'.

equation
Challenge Problem challenge problem

If each exterior angle measures 80°, how many sides does this polygon have?

There is no solution to this question.

When you use formula to find a single exterior angle to solve for the number of sides , you get a decimal (4.5), which is impossible. Think about it: How could a polygon have 4.5 sides? A quadrilateral has 4 sides. A pentagon has 5 S is nothing in between.

challenge