Chapter 7.1, Problem 32WE

a

To determine

To calculate: The ratio of the measure of an interior angle to the measure of an exterior angle in a regular polygon, a regular decagon and a regular n -gon.

Answer to Problem 32WE

The ratio of the measure of an interior angle to the measure of an exterior angle of a regular hexagon is $1:2$ .

The ratio of the measure of an interior angle to the measure of an exterior angle of a regular decagon is $2:3$ .

The ratio of the measure of an interior angle to the measure of an exterior angle of a regular n -gon is $x:y$ .

Explanation of Solution

Given information:

A regular polygon, a regular decagon and a regular n -gon..

Formula Used:

Property of hexagon:

The sum of all the angles of a hexagon is $720°$ .

Property of decagon:

The sum of all the angles of a decagon is $1440°$ .

Property of n -gon:

The sum of all the angles of a decagon is $X°$ .

Calculation:

Let $A,B,C,D,EandF$ are the angles of a hexagon.

So, $A+B+C+D+E+F=720°$ .

Know that, all the interior angles of a regular hexagon are same and equal to $120°$ .

Now, find the exterior angle.

  $360°-120°=240°$

Find the ratio of the measure of an interior angle to the measure of an exterior angle of a regular hexagon.

  $\begin{array}{c}R=\frac{i}{e}\\ =\frac{120}{240}\\ =\frac{1}{2}\end{array}$

Thus, the ratio of the measure of an interior angle to the measure of an exterior angle of a regular hexagon is $1:2$ .

Let $A,B,C,D,E,F,G,H,IandJ$ are the angles of a hexagon.

So, $A+B+C+D+E+F+G+H+I+J=1440°$ .

Know that, all the interior angles of a regular decagon are same and equal to $144°$ .

Now, find the exterior angle.

  $360°-144°=216°$

Find the ratio of the measure of an interior angle to the measure of an exterior angle of a regular decagon.

  $\begin{array}{c}R=\frac{i}{e}\\ =\frac{144}{216}\\ =\frac{2}{3}\end{array}$

Thus, the ratio of the measure of an interior angle to the measure of an exterior angle of a regular decagon is $2:3$ .

Let $A,B,C,D,E,F,G,H,IandJ$ are the angles of a hexagon.

So, $A+B+C+D+E+F+G+H+I+J=1440°$ .

Know that, all the interior angles of a regular decagon are same and equal to $144°$ .

Now, find the exterior angle.

  $360°-144°=216°$

Find the ratio of the measure of an interior angle to the measure of an exterior angle of a regular decagon.

  $\begin{array}{c}R=\frac{i}{e}\\ =\frac{144}{216}\\ =\frac{2}{3}\end{array}$

Thus, the ratio of the measure of an interior angle to the measure of an exterior angle of a regular decagon is $2:3$ .

Let $A,B,C,D,E,F,G,H,IandJ$ are the angles of a hexagon.

So, $A+B+C+D+E+F+G+H+I+J=1440°$ .

Know that, all the interior angles of a regular decagon are same and equal to $144°$ .

Now, find the exterior angle.

  $360°-144°=216°$

Find the ratio of the measure of an interior angle to the measure of an exterior angle of a regular decagon.

  $\begin{array}{c}R=\frac{i}{e}\\ =\frac{144}{216}\\ =\frac{2}{3}\end{array}$

Thus, the ratio of the measure of an interior angle to the measure of an exterior angle of a regular decagon is $2:3$ .

Let $n$ be the number of interior angles in n -gon.

So, the interior angle is $\frac{X°}{n°}$ .

Let $x°=\frac{X°}{n°}$ .

Now, find the exterior angle.

Let $360°-x°=y°$

Find the ratio of the measure of an interior angle to the measure of an exterior angle of a regular n -gon.

  $\begin{array}{c}R=\frac{i}{e}\\ =\frac{x°}{y°}\end{array}$

Thus, the ratio of the measure of an interior angle to the measure of an exterior angle of a regular n -gon is $x:y$ .