Question

for exercises 1–4, find the sum of the interior angles and the measure of each interior angle for the given regular polygons. round to the nearest tenth as needed.

1. 12 - gon
2. 102 - gon
3. 90 - gon
4. 36 - gon

Explanation:

Problem 1: 12 - gon

Step 1: Recall the formula for the sum of interior angles of a polygon

The formula for the sum of the interior angles of a polygon with \(n\) sides is \((n - 2)\times180^{\circ}\). For a 12 - gon, \(n = 12\).
So, the sum of interior angles \(=(12 - 2)\times180^{\circ}=10\times180^{\circ} = 1800^{\circ}\)

Step 2: Find the measure of each interior angle of a regular polygon

In a regular polygon, all interior angles are equal. The formula for the measure of each interior angle of a regular \(n\) - gon is \(\frac{(n - 2)\times180^{\circ}}{n}\)
For \(n = 12\), each interior angle \(=\frac{(12 - 2)\times180^{\circ}}{12}=\frac{1800^{\circ}}{12}=150^{\circ}\)

Problem 2: 102 - gon

Step 1: Sum of interior angles

Using the formula \((n - 2)\times180^{\circ}\) with \(n = 102\)
Sum \(=(102 - 2)\times180^{\circ}=100\times180^{\circ}=18000^{\circ}\)

Step 2: Measure of each interior angle

Using the formula \(\frac{(n - 2)\times180^{\circ}}{n}\) with \(n = 102\)
Each interior angle \(=\frac{(102 - 2)\times180^{\circ}}{102}=\frac{18000^{\circ}}{102}\approx176.5^{\circ}\) (rounded to the nearest tenth)

Problem 3: 90 - gon

Step 1: Sum of interior angles

Using \((n - 2)\times180^{\circ}\) with \(n = 90\)
Sum \(=(90 - 2)\times180^{\circ}=88\times180^{\circ}=15840^{\circ}\)

Step 2: Measure of each interior angle

Using \(\frac{(n - 2)\times180^{\circ}}{n}\) with \(n = 90\)
Each interior angle \(=\frac{(90 - 2)\times180^{\circ}}{90}=\frac{15840^{\circ}}{90}=176^{\circ}\)

Problem 4: 36 - gon

Answer:

Step 1: Sum of interior angles

Using \((n - 2)\times180^{\circ}\) with \(n = 36\)
Sum \(=(36 - 2)\times180^{\circ}=34\times180^{\circ}=6120^{\circ}\)

Step 2: Measure of each interior angle

Using \(\frac{(n - 2)\times180^{\circ}}{n}\) with \(n = 36\)
Each interior angle \(=\frac{(36 - 2)\times180^{\circ}}{36}=\frac{6120^{\circ}}{36}=170^{\circ}\)

Final Answers:

1. 12 - gon: Sum of interior angles \(=\boldsymbol{1800^{\circ}}\), Each interior angle \(=\boldsymbol{150^{\circ}}\)
2. 102 - gon: Sum of interior angles \(=\boldsymbol{18000^{\circ}}\), Each interior angle \(\approx\boldsymbol{176.5^{\circ}}\)
3. 90 - gon: Sum of interior angles \(=\boldsymbol{15840^{\circ}}\), Each interior angle \(=\boldsymbol{176^{\circ}}\)
4. 36 - gon: Sum of interior angles \(=\boldsymbol{6120^{\circ}}\), Each interior angle \(=\boldsymbol{170^{\circ}}\)