Chapter 3.5, Problem 26WE

a.

To determine

Expression of sum of interior angle of polygon without using n.

Answer to Problem 26WE

Expression of sum of interior angle will be, $S+{180}^0$

Explanation of Solution

Given:

Number of sides is equal to n+1

Sum of interior angle of polygon is given by, $\left(n-2\right)\times {180}^0$

Therefore,

  $\begin{array}{l}S=\left(n-2\right)\times {180}^0\\ S=180n-{360}^0\\ n=\frac{S+{360}^0}{{180}^0}\end{array}$

Now, number of sides is n+1 sum of interior angle is given by,

  $\left((n+1)-2\right)\times {180}^0$

Now, substituting the value of n .

  $\begin{array}{l}\left(\frac{S+{360}^0}{{180}^0}+1-2\right)\times {180}^0\\ =\frac{S+{360}^0-{180}^0}{{180}^0}\times {180}^0\\ =S+{180}^0\end{array}$

b.

To determine

Expression of sum of interior angle of polygon without using n.

Answer to Problem 26WE

Expression of sum of interior angle will be,

Explanation of Solution

Given:

Number of sides is equal to n+1

Sum of interior angle of polygon is given by, $\left(n-2\right)\times {180}^0$

Therefore,

  $\begin{array}{l}S=\left(n-2\right)\times {180}^0\\ S=180n-{360}^0\\ n=\frac{S+{360}^0}{{180}^0}\end{array}$

Now, number of sides is 2n sum of interior angle is given by,

  $\left(2n-2\right)\times {180}^0$

Now, substituting the value of n .

  $\begin{array}{l}\left(2\times \frac{S+{360}^0}{{180}^0}-2\right)\times {180}^0\\ =\frac{2S+{720}^0-{360}^0}{{180}^0}\times {180}^0\\ =2S+{360}^0\end{array}$