4.1: Polygon Angle Sums | Geometry Honors

How it works

A diagonal fan divides a convex polygon with n sides into n minus two triangles. The exterior angles, taking one consistent turn at each vertex, total a full revolution.

S=(n-2)180^\circ,\quad E_{\text{regular}}=\frac{360^\circ}{n}

Worked examples

Example 1

Find the sum of the interior angles of a hexagon.

$\displaystyle S=(6-2)180^\circ=720^\circ$ .

Example 2

Find each interior angle of a regular octagon.

$\displaystyle S=1080^\circ$ ; divide by $\displaystyle 8$ to get $\displaystyle 135^\circ$ .

Problems

Problem 4.1.1

A regular hexagon is divided from one vertex into triangles. How many triangles appear, and what is each interior angle?

A regular hexagon split from one vertex into four triangles by three diagonals. — A diagonal fan uses n minus 2 triangles.

Show worked solution

There are $\displaystyle 6-2=4$ triangles. The sum is $\displaystyle 720^\circ$ , giving $\displaystyle 120^\circ$ at each vertex.

A regular hexagon split from one vertex into four triangles by three diagonals. The solution adds labeled relationships. — Four triangles give 720 degrees; each regular-hexagon interior angle is 120 degrees.

Problem 4.1.2

A regular polygon has exterior angle $\displaystyle 24^\circ$ . How many sides does it have?

Show worked solution

$\displaystyle n=360/24=15$ .

Problem 4.1.3

Five interior angles of a hexagon are $\displaystyle 100^\circ,110^\circ,120^\circ,130^\circ,140^\circ$ . Find the sixth.

Show worked solution

Their sum is $\displaystyle 600^\circ$ . Subtract from $\displaystyle 720^\circ$ to obtain $\displaystyle 120^\circ$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 4.1.1

Problem 4.1.2

Problem 4.1.3

Need help with this topic?