Question

geometry
name:
worksheet: polygon angle measures
period:
date:
use the given information to complete the table. round to the nearest tenth if necessary.
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)

Explanation:

Step1: Recall formulas

- Interior angle sum formula: \(S=(n - 2)\times180^{\circ}\)
- Measure of one interior angle of a regular polygon: \(I=\frac{(n - 2)\times180^{\circ}}{n}\)
- Exterior angle sum of any polygon: \(360^{\circ}\)
- Measure of one exterior angle of a regular polygon: \(E=\frac{360^{\circ}}{n}\)

Step2: Solve for row 1

- Interior angle sum: \((n - 2)\times180^{\circ}\)
- Measure of one interior angle: \(\frac{(n - 2)\times180^{\circ}}{n}\)
- Exterior angle sum: \(360^{\circ}\)
- Measure of one exterior angle: \(\frac{360^{\circ}}{n}\)

Step3: Solve for row 2 (\(n = 14\))

- Interior angle sum: \((14 - 2)\times180^{\circ}=2160^{\circ}\)
- Measure of one interior angle: \(\frac{(14 - 2)\times180^{\circ}}{14}\approx154.3^{\circ}\)
- Exterior angle sum: \(360^{\circ}\)
- Measure of one exterior angle: \(\frac{360^{\circ}}{14}\approx25.7^{\circ}\)

Step4: Solve for row 3 (\(n = 24\))

- Interior angle sum: \((24 - 2)\times180^{\circ}=3960^{\circ}\)
- Measure of one interior angle: \(\frac{(24 - 2)\times180^{\circ}}{24}=165^{\circ}\)
- Exterior angle sum: \(360^{\circ}\)
- Measure of one exterior angle: \(\frac{360^{\circ}}{24}=15^{\circ}\)

Step5: Solve for row 4 (\(n = 17\))

- Interior angle sum: \((17 - 2)\times180^{\circ}=2700^{\circ}\)
- Measure of one interior angle: \(\frac{(17 - 2)\times180^{\circ}}{17}\approx158.8^{\circ}\)
- Exterior angle sum: \(360^{\circ}\)
- Measure of one exterior angle: \(\frac{360^{\circ}}{17}\approx21.2^{\circ}\)

Step6: Solve for row 5 (\(S=1080^{\circ}\))

- Use \(S=(n - 2)\times180^{\circ}\), so \(n=\frac{1080^{\circ}}{180^{\circ}}+ 2=8\)
- Measure of one interior angle: \(\frac{1080^{\circ}}{8}=135^{\circ}\)
- Exterior angle sum: \(360^{\circ}\)
- Measure of one exterior angle: \(\frac{360^{\circ}}{8}=45^{\circ}\)

Step7: Solve for row 6 (\(S = 900^{\circ}\))

- Use \(S=(n - 2)\times180^{\circ}\), so \(n=\frac{900^{\circ}}{180^{\circ}}+2 = 7\)
- Measure of one interior angle: \(\frac{900^{\circ}}{7}\approx128.6^{\circ}\)
- Exterior angle sum: \(360^{\circ}\)
- Measure of one exterior angle: \(\frac{360^{\circ}}{7}\approx51.4^{\circ}\)

Step8: Solve for row 7 (\(S = 5040^{\circ}\))

- Use \(S=(n - 2)\times180^{\circ}\), so \(n=\frac{5040^{\circ}}{180^{\circ}}+2=30\)
- Measure of one interior angle: \(\frac{5040^{\circ}}{30}=168^{\circ}\)
- Exterior angle sum: \(360^{\circ}\)
- Measure of one exterior angle: \(\frac{360^{\circ}}{30}=12^{\circ}\)

Step9: Solve for row 8 (\(S = 1620^{\circ}\))

- Use \(S=(n - 2)\times180^{\circ}\), so \(n=\frac{1620^{\circ}}{180^{\circ}}+2 = 11\)
- Measure of one interior angle: \(\frac{1620^{\circ}}{11}\approx147.3^{\circ}\)
- Exterior angle sum: \(360^{\circ}\)
- Measure of one exterior angle: \(\frac{360^{\circ}}{11}\approx32.7^{\circ}\)

Step10: Solve for row 9 (\(I = 150^{\circ}\))

- Use \(I=\frac{(n - 2)\times180^{\circ}}{n}\), so \(150n=(n - 2)\times180\), \(150n=180n-360\), \(30n = 360\), \(n = 12\)
- Interior angle sum: \((12 - 2)\times180^{\circ}=1800^{\circ}\)
- Exterior angle sum: \(360^{\circ}\)
- Measure of one exterior angle: \(\frac{360^{\circ}}{12}=30^{\circ}\)

Step11: Solve for row 10 (\(I = 120^{\circ}\))

- Use \(I=\frac{(n - 2)\times180^{\circ}}{n}\), so \(120n=(n - 2)\times180\), \(120n=180n - 360\), \(60n=360\), \(n = 6\)
- Interior angle sum: \((6 - 2)\times180^{\circ}=720^{\circ}\)
- Exterior angle sum: \(360^{\circ}\)
- Measure of one exterior angle: \(\frac{360^{\circ}}{6}=60^{\circ}\)

Step12: Solve for row 11 (\(I = 156^{\circ}\))

- Use \(I=\frac{(n - 2)\times180^{\circ}}{n}\), so \(156n=(n - 2)\times180\), \(156n=180n-360\), \(24n=360\), \(n = 15\)
- Interior angle sum: \((15 - 2)\times180^{\circ}=2340^{\circ}\)
- Exterior angle sum: \(360^{\circ}\)
- Measure of one exterior angle: \(\frac{360^{\circ}}{15}=24^{\circ}\)

Step13: Solve for row 12 (\(E = 10^{\circ}\))

- Use \(E=\frac{360^{\circ}}{n}\), so \(n=\frac{360^{\circ}}{10}=36\)
- Interior angle sum: \((36 - 2)\times180^{\circ}=6120^{\circ}\)
- Measure of one interior angle: \(\frac{6120^{\circ}}{36}=170^{\circ}\)
- Exterior angle sum: \(360^{\circ}\)

Step14: Solve for row 13 (\(E = 7.2^{\circ}\))

- Use \(E=\frac{360^{\circ}}{n}\), so \(n=\frac{360^{\circ}}{7.2}=50\)
- Interior angle sum: \((50 - 2)\times180^{\circ}=8640^{\circ}\)
- Measure of one interior angle: \(\frac{8640^{\circ}}{50}=172.8^{\circ}\)
- Exterior angle sum: \(360^{\circ}\)

Step15: Solve for row 14 (\(E = 90^{\circ}\))

- Use \(E=\frac{360^{\circ}}{n}\), so \(n=\frac{360^{\circ}}{90}=4\)
- Interior angle sum: \((4 - 2)\times180^{\circ}=360^{\circ}\)
- Measure of one interior angle: \(\frac{360^{\circ}}{4}=90^{\circ}\)
- Exterior angle sum: \(360^{\circ}\)

Step16: Solve for row 15 (\(E = 5^{\circ}\))

- Use \(E=\frac{360^{\circ}}{n}\), so \(n=\frac{360^{\circ}}{5}=72\)
- Interior angle sum: \((72 - 2)\times180^{\circ}=12600^{\circ}\)
- Measure of one interior angle: \(\frac{12600^{\circ}}{72}=175^{\circ}\)
- Exterior angle sum: \(360^{\circ}\)

Answer:

# Sides	Interior Angle Sum	Measure of ONE INTERIOR Angle (Regular Polygon)	Exterior Angle Sum	Measure of ONE EXTERIOR Angle (Regular Polygon)
\(14\)	\(2160^{\circ}\)	\(154.3^{\circ}\)	\(360^{\circ}\)	\(25.7^{\circ}\)
\(24\)	\(3960^{\circ}\)	\(165^{\circ}\)	\(360^{\circ}\)	\(15^{\circ}\)
\(17\)	\(2700^{\circ}\)	\(158.8^{\circ}\)	\(360^{\circ}\)	\(21.2^{\circ}\)
\(8\)	\(1080^{\circ}\)	\(135^{\circ}\)	\(360^{\circ}\)	\(45^{\circ}\)
\(7\)	\(900^{\circ}\)	\(128.6^{\circ}\)	\(360^{\circ}\)	\(51.4^{\circ}\)
\(30\)	\(5040^{\circ}\)	\(168^{\circ}\)	\(360^{\circ}\)	\(12^{\circ}\)
\(11\)	\(1620^{\circ}\)	\(147.3^{\circ}\)	\(360^{\circ}\)	\(32.7^{\circ}\)
\(12\)	\(1800^{\circ}\)	\(150^{\circ}\)	\(360^{\circ}\)	\(30^{\circ}\)
\(6\)	\(720^{\circ}\)	\(120^{\circ}\)	\(360^{\circ}\)	\(60^{\circ}\)
\(15\)	\(2340^{\circ}\)	\(156^{\circ}\)	\(360^{\circ}\)	\(24^{\circ}\)
\(36\)	\(6120^{\circ}\)	\(170^{\circ}\)	\(360^{\circ}\)	\(10^{\circ}\)
\(50\)	\(8640^{\circ}\)	\(172.8^{\circ}\)	\(360^{\circ}\)	\(7.2^{\circ}\)
\(4\)	\(360^{\circ}\)	\(90^{\circ}\)	\(360^{\circ}\)	\(90^{\circ}\)
\(72\)	\(12600^{\circ}\)	\(175^{\circ}\)	\(360^{\circ}\)	\(5^{\circ}\)