Geometry NAME: WORKSHEET: Polygon Angle Measures PERIOD: DATE: Use the given information to complete the table. Round to the nearest tenth if necessary. # Sides Interior Angle Sum Measure of ONE INTERIOR Angle (Regular Polygon) Exterior Angle Sum Measure of ONE EXTERIOR Angle (Regular Polygon) 1) n 2) 14 3) 24 4) 17 5) 1080° 6) 900° 7) 5040° 8) 1620° 9) 150° 10) 120° 11) 156° 12) 10° 13) 7.2° 14) 90° 15) 5°
The Correct Answer and Explanation is:
To complete this Polygon Angle Measures Worksheet, we use the following key formulas:
🔷 Key Formulas:
- Interior Angle Sum = (n – 2) × 180°
- Measure of One Interior Angle (regular polygon) = [(n – 2) × 180°] ÷ n
- Exterior Angle Sum = 360° (Always true for any polygon)
- One Exterior Angle (regular polygon) = 360° ÷ n
- To find n from an angle measure:
- For interior angle:
n=360180−Interior Anglen = \frac{360}{180 – \text{Interior Angle}} - For exterior angle:
n=360Exterior Anglen = \frac{360}{\text{Exterior Angle}}
- For interior angle:
🔢 Completed Table:
| # | Sides (n) | Interior Angle Sum | One Interior Angle | Exterior Angle Sum | One Exterior Angle |
|---|---|---|---|---|---|
| 1 | n | (n – 2) × 180° | [(n – 2) × 180]/n | 360° | 360/n |
| 2 | 14 | 2160° | 154.3° | 360° | 25.7° |
| 3 | 24 | 3960° | 165° | 360° | 15° |
| 4 | 17 | 2700° | 158.8° | 360° | 21.2° |
| 5 | 8 | 1080° | 135° | 360° | 45° |
| 6 | 7 | 900° | 128.6° | 360° | 51.4° |
| 7 | 30 | 5040° | 168° | 360° | 12° |
| 8 | 11 | 1620° | 147.3° | 360° | 32.7° |
| 9 | 12 | 1800° | 150° | 360° | 30° |
| 10 | 9 | 1260° | 140° | 360° | 40° |
| 11 | 15 | 2340° | 156° | 360° | 24° |
| 12 | 36 | 6120° | 170° | 360° | 10° |
| 13 | 50 | 8640° | 172.8° | 360° | 7.2° |
| 14 | 4 | 360° | 90° | 360° | 90° |
| 15 | 72 | 12600° | 175° | 360° | 5° |
📘 Explanation
This worksheet focuses on understanding the geometric relationships between a polygon’s number of sides and its interior and exterior angles.
To find the interior angle sum of any polygon, we use the formula (n−2)×180∘(n – 2) \times 180^\circ, because every polygon can be divided into (n−2)(n – 2) triangles. For example, a 14-gon has an interior angle sum of (14−2)×180=2160∘(14 – 2) \times 180 = 2160^\circ.
For regular polygons (where all sides and angles are equal), the measure of one interior angle is simply the total interior angle sum divided by the number of sides. Using the same 14-gon: 2160∘÷14≈154.3∘2160^\circ \div 14 \approx 154.3^\circ.
All polygons, regardless of the number of sides, have an exterior angle sum of 360°. This is because exterior angles make one full turn around the polygon.
To find one exterior angle of a regular polygon, divide 360° by the number of sides. For example, in a regular 24-gon, each exterior angle is 360∘÷24=15∘360^\circ \div 24 = 15^\circ, and each interior angle is 180∘−15∘=165∘180^\circ – 15^\circ = 165^\circ.
Sometimes, we’re given the measure of one interior or exterior angle and must work backward to find the number of sides. For instance, if one interior angle is 156°, then the exterior angle is 180∘−156∘=24∘180^\circ – 156^\circ = 24^\circ, and n=360∘÷24∘=15n = 360^\circ \div 24^\circ = 15 sides.
This table demonstrates how polygon properties are interconnected. Mastering these formulas enables students to quickly calculate unknown values and reinforces the importance of angle relationships in geometry.
