Interior and Exterior Angles of Polygons Worksheet Answers: Understanding the Concepts Behind Polygon Angles and Their Applications

Understanding the interior and exterior angles of polygons is fundamental in geometry. Whether you’re working with triangles, quadrilaterals, pentagons, or more complex shapes, having a strong grasp of these concepts allows you to solve for unknown angles and check the accuracy of your design layouts. Here are answers and explanations commonly found on an "Interior and Exterior Angles of Polygons Worksheet."

Key Formulas

1. Sum of Interior Angles:
2. For a polygon with n sides:
3. Sum = (n - 2) × 180°
4. Each Interior Angle (Regular Polygon):
5. =(n - 2) × 180° / n
6. Each Exterior Angle (Regular Polygon):
7. = 360° / n
8. The sum of all exterior angles, for any convex polygon, is always 360°.

Example Answers

1. Find the sum of the interior angles of a hexagon (6 sides):
2. Sum = (6 - 2) × 180° = 4 × 180° = 720°
3. What is each interior angle of a regular pentagon (5 sides)?
4. Sum = (5 - 2) × 180° = 3 × 180° = 540°
5. Each angle = 540° / 5 = 108°
6. What is each exterior angle of a regular octagon (8 sides)?
7. Each exterior angle = 360° / 8 = 45°
8. If the sum of the interior angles of a polygon is 1,260°, how many sides does it have?
9. (n - 2) × 180° = 1,260°
10. n - 2 = 1,260 ÷ 180 = 7
11. n = 9
12. The polygon is a nonagon (9 sides).
13. A regular decagon (10 sides): Find each interior and exterior angle.
14. Interior angle = (10 - 2) × 180° ÷ 10 = 8 × 180° ÷ 10 = 1,440° ÷ 10 = 144°
15. Exterior angle = 360° ÷ 10 = 36°

As a designer, I often calculate these angles when planning wall placements, ceiling details, or designing custom furniture that follows polygonal geometry. The precise knowledge of interior and exterior angles enhances the flow and efficiency of the space. If you want to visualize and lay out various polygonal structures before building, a room planner can streamline your workflow, ensuring accuracy and creativity in your interior designs.

Tips 1:

When working on polygon-based designs—such as tiling, joinery, or even open-plan zones—knowing how to quickly calculate angles saves time and reduces costly mistakes. Digital tools can help you experiment with different layouts in a risk-free environment.

FAQ

Q: What is the sum of the exterior angles of any polygon?

A: It is always 360°, regardless of the number of sides.

Q: How do you find the measure of each interior angle in a regular polygon?

A: Use (n – 2) × 180° divided by the number of sides (n).

Q: Can a polygon have an interior angle greater than 180°?

A: No; in a convex polygon, all interior angles are less than 180°.

Q: How does understanding angles help in interior design?

A: Accurate angle measurements are crucial for fitting furniture, tiling, and custom woodworking, affecting both aesthetics and function.

Q: What tool can I use to design rooms with precise angles?

A: Room planner tools allow you to visualize and calculate polygonal spaces efficiently before construction.