Chapter 7.1, Problem 63PS

Solution

To state: The first five terms of the sequence. Write a rule for the sequence giving the total measure $T_n$ of the interior angles in each regular n -sided polygon. Use the rule to find the total measure of the angles in the G. Museum skylight, which is a regular dodecagon.

For a regular n -sided polygon $\left(n\ge 3\right)$ , the measure $a_n$ , of an interior angle is given by this formula:

  $a_n=\frac{180\left(n-2\right)}{n}$

The resultant terms are ${60}^{\circ },{90}^{\circ },{108}^{\circ },{120}^{\circ },\text{}\text{}\text{and}\text{about}{128.57}^{\circ }$ .

The rule is $T_n=180\left(n-2\right)$ and the total measure of angles of a decagon is ${1800}^{\circ }$ .

Given information:

For a regular n -sided polygon $\left(n\ge 3\right)$ , the measure $a_n$ , of an interior angle is given by this formula:

  $a_n=\frac{180\left(n-2\right)}{n}$

Explanation:

Consider the given sequence:

  $a_n=\frac{180\left(n-2\right)}{n},n\ge 3$

Now find the first five terms of the sequence by substituting the value of n as $3,4,5,6,7$ ,

The first term is:

  $\begin{array}{c}{a}_3=\frac{180\left(3-2\right)}{3}\\ =\frac{180}{3}\\ =60\end{array}$

The second term is:

  $\begin{array}{c}{a}_4=\frac{180\left(4-2\right)}{4}\\ =\frac{180\cdot 2}{4}\\ =90\end{array}$

The third term is:

  $\begin{array}{c}{a}_5=\frac{180\left(5-2\right)}{5}\\ =\frac{180\cdot 3}{5}\\ =108\end{array}$

The fourth term is:

  $\begin{array}{c}{a}_6=\frac{180\left(6-2\right)}{6}\\ =\frac{180\cdot 4}{6}\\ =120\end{array}$

The fifth term is:

  $\begin{array}{c}{a}_7=\frac{180\left(7-2\right)}{7}\\ =\frac{180\cdot 5}{7}\\ =\frac{900}{7}\end{array}$

Therefore, the first five terms are ${60}^{\circ },{90}^{\circ },{108}^{\circ },{120}^{\circ },\text{}\text{}\text{and}\text{about}{128.57}^{\circ }$ .

The measure of interior angles is given by: $a_n=\frac{180\left(n-2\right)}{n}$

For an n -sided polygon, there will be n internal angles. The total measure $T_n$ of the interior angles is:

  $\begin{array}{c}{T}_n=n\cdot \left(\frac{180\left(n-2\right)}{n}\right)\\ =180\left(n-2\right)\end{array}$

Thus the rule for the sequence giving the total measures of the angles is $T_n=180\left(n-2\right)$ .

A dodecagon has 12 sides, then substitute n as 12 in $T_n=180\left(n-2\right)$ .

  $\begin{array}{c}{T}_n=180\left(12-2\right)\\ =180\cdot 10\\ =1800\end{array}$

Therefore, the measure of the angles in the museum is ${1800}^{\circ }$ .