The base of a jewelry box is shaped like a regular 15-gon. What is the measure of each interior angle of the…

Question

The base of a jewelry box is shaped like a regular 15-gon. What is the measure of each interior angle of the base of the jewelry box.

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the problems.

1. Measure of each interior angle of a regular 15-gon:

Identify the number of sides, $n$ .
For a 15-gon, $n=15$ .
Use the formula for the measure of each interior angle of a regular $n$ -gon
$\text{Each interior angle} = \frac{(n-2) \times 180^\circ}{n}$
Substitute $n=15$ into the formula and calculate
$\text{Each interior angle} = \frac{(15-2) \times 180^\circ}{15} = \frac{13 \times 180^\circ}{15} = \frac{2340^\circ}{15} = 156^\circ$
The measure of each interior angle is $\boxed{\text{156}^\circ}$ .

2. Sum of the measures of the interior angles of a regular undecagon:
Identify the number of sides, $n$ .
For an undecagon, $n=11$ .
Use the formula for the sum of the measures of the interior angles of an $n$ -gon
$S = (n-2) \times 180^\circ$
Substitute $n=11$ into the formula and calculate
$S = (11-2) \times 180^\circ = 9 \times 180^\circ = 1620^\circ$
The sum of the measures of its interior angles is $\boxed{\text{1620}^\circ}$ .

3. Measure of each interior angle of a regular dodecagon:
Identify the number of sides, $n$ .
For a dodecagon, $n=12$ .
Use the formula for the measure of each interior angle of a regular $n$ -gon
$\text{Each interior angle} = \frac{(n-2) \times 180^\circ}{n}$
Substitute $n=12$ into the formula and calculate
$\text{Each interior angle} = \frac{(12-2) \times 180^\circ}{12} = \frac{10 \times 180^\circ}{12} = \frac{1800^\circ}{12} = 150^\circ$
The measure of each interior angle is $\boxed{\text{150}^\circ}$ .

The base of a jewelry box is shaped like a regular 15-gon. What is the measure of each interior angle of the base of the jewelry box.