Topics
6. Quadrilaterals & Other Polygons
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United States of AmericaFL
Grade 912

6.01 The polygon angle sum theorems

Lesson
Practice
Lesson

Concept summary

A polygon is a closed plane figure formed by at least three straight sides.

A regular polygon is a polygon that is both equilateral (all sides congruent) and equiangular (all angles congruent).

Convex polygon

A polygon with each interior angle measuring less than 180 \degree. All diagonals of a convex polygon lie inside the polygon.

A pentagon with all diagonals interior to the figure

The sum of interior angle measures of a polygon depends on the number of sides of the polygon.

Polygon angle sum theorem

The sum of the interior angles of a convex n-gon is equal to \left(n-2\right)180 \degree

Corollary to the polygon angle sum theorem

The measure of each interior angle of a regular n-gon is \dfrac{\left(n-2\right)180 \degree}{n}

Polygon exterior angle sum theorem

The sum of the exterior angles of any polygon is 360 \degree

Worked examples

Example 1

For a regular 24-gon:

A regular 24 gon.
a

Find the sum of the interior angles.

Approach

We use the polygon angle sum theorem to find the sum of the interior angles. The polygon angle sum theorem states that the sum of the interior angles of a convex polygon is equal to \left (n-2\right)180 \degree for n sides.

A 24-gon is a convex polygon and has 24 sides. So we want to substitute 24 for n in the expression and evaluate.

Solution

\left(24 - 2 \right)180 \degree= 3960 \degree

The sum of interior angles is 3960 \degree

b

Find the measure of a single interior angle.

Approach

Since we want to find the measure of a specific interior angle of a regular polygon, we want to use the corollary to the polygon angle sum theorem. This theorem states that the interior angle is equal to the sum of the interior angles divided by the number of sides, \dfrac{\left(n-2\right)180 \degree}{n}

Now we need to substitute 24 for n and evaluate.

Solution

\dfrac{\left(24-2\right)180 \degree}{24}=165 \degree

The measure of an interior angle is 165 \degree

Example 2

Determine the value of y:

A pentagon with its exterior angles shown. Moving clockwise, the exterior angles are: 68 degrees, y degrees, 42.5 degrees, 65 degrees, and 60 degrees.

Approach

We want to solve for y which is an exterior angle. To do so, we can use the polygon exterior angle theorem to write an equation relating the exterior angles of the polygon.

Solution

\displaystyle 60 + 65 + 42.5 + y + 68\displaystyle =\displaystyle 360Exterior angle formula
\displaystyle y\displaystyle =\displaystyle 124.5Simplify

The measure of angle y is 124.5 \degree.

Outcomes

MA.912.GR.1.4

Prove relationships and theorems about parallelograms. Solve mathematical and real-world problems involving postulates, relationships and theorems of parallelograms.

MA.912.LT.4.8

Construct proofs, including proofs by contradiction.

MA.912.LT.4.10

Judge the validity of arguments and give counterexamples to disprove statements.

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