A **polygon** is a closed plane figure formed by at least three straight sides. Polygons can be classified in many ways.

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

Consider the quadrilateral shown:

- What ways could you break the quadrilateral into the least number of triangles? How many triangles does this create?
- Determine the sum of the interior angles of the quadrilateral.
- Draw a hexagon and determine the least number of triangles you could break the polygon into, and determine the sum of its interior angles.
- What can you say about the relationship between the number of sides on any polygon, the number of triangles it can be divided into, and the sum of its interior angles?

The sum of the interior angle measures of a polygon depends on the number of sides of the polygon. A polygon with n sides (or an n-gon) can always be divided into (n-2) non-overlapping triangles. This fact and the triangle angle sum theorem helps us calculate interior angle sums and individual angle measures of regular polygons.

Consider the polygon angle sum theorem.

a

Prove the interior polygon angle sum theorem works for a pentagon.

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b

Explain why the polygon angle sum theorem will work for any convex polygon.

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For a regular 24-gon:

a

Find the sum of the interior angles.

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b

Find the measure of a single interior angle.

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Find the value of y.

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Determine the number of sides of a regular polygon when each interior angle has a measure of 150\degree.

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Idea summary

We can use the **polygon angle sum theorem** and its corollary to find unknown angles of convex and regular n-gons:

- The sum of the interior angles of a convex n-gon is equal to \left(n-2\right)180 \degree
- The measure of each interior angle of a regular n-gon is \dfrac{\left(n-2\right)180 \degree}{n}

Drag the points to shape the convex polygon. Then, explore the applet.

- How would you describe the angles measured in the applet?
- What do you notice about the angle measurements?

Prove the polygon exterior angle sum theorem.

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Determine the value of y:

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Determine the number of sides of a regular polygon when each exterior angle has a measure of 20\degree.

Worked Solution

Idea summary

We can use the **polygon exterior angle sum theorem** and its corollary to find unknown angles of convex and regular n-gons:

- The sum of the exterior angles of any polygon is 360 \degree
- The measure of each exterior angle of a regular n-gon is \dfrac{1}{n} \cdot 360 \degree