Answer key: Polygon Angles Worksheet

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Goal: Find the interior, exterior and total angles of regular polygons, discover the formulas behind them, and use them to explain which shapes tile a floor.

Completed results table

Values are the readouts of the tool (Regular polygon explorer) for each setting.

Answers

1. By how much does the angle sum grow each time you add one side? Write a formula for the angle sum of an n-sided polygon.

   It grows by 180° for each extra side, because each extra side adds one more triangle inside the polygon. Angle sum = (n − 2) × 180°. For a hexagon: (6 − 2) × 180° = 720°.

2. Multiply each polygon’s exterior angle by its number of sides. What do you notice?

   The answer is always 360°: 3 × 120° = 4 × 90° = 6 × 60° = 12 × 30° = 360°. Walking once around any convex polygon, you turn through one full turn.

3. Find the interior angle of a regular octagon by hand.

   The angle sum is (8 − 2) × 180° = 1,080°, and 1,080° ÷ 8 = 135°.

4. Which regular polygons in your table can tile a floor with no gaps or overlaps? Use the interior angles to explain.

   Triangles (60°), squares (90°) and hexagons (120°). Their interior angles divide 360° exactly (6 × 60°, 4 × 90° and 3 × 120°), so copies fit around a point. A pentagon’s 108° does not: 3 × 108° = 324° leaves a gap, and 4 would overlap.

5. Use the diagonal formula n(n − 3) ÷ 2 to check your table for 12 and 20 sides.

   12 × 9 ÷ 2 = 54 and 20 × 17 ÷ 2 = 170, matching the explorer.

6. Challenge: a regular polygon has interior angles of 150°. How many sides does it have?

   Each exterior angle is 180° − 150° = 30°. The exterior angles add up to 360°, so n = 360° ÷ 30° = 12 sides.