Study guides › Level 7 · Geometry

Regular polygons: each angle

Level 7 · Geometry · for questions like “A regular octagon has 8 equal sides and angles. How big is each inside angle?”

💡 The idea

Walk around any polygon and you turn a full circle, 360° in all. In a regular polygon every turn is the same, and each turn (the outside angle) sits next to an inside angle on a straight line.

• each outside angle = 360° ÷ n
• each inside angle = 180° − outside angle

🪜 The steps

1. Outside angle: 360 ÷ number of sides.
2. Inside angle: 180 minus that.
3. Going backwards from the inside angle: outside = 180 − inside, then sides = 360 ÷ outside.

✏️ Worked examples

1. Regular octagon: 360 ÷ 8 = 45, so each inside angle is 180 − 45 = 135°.

2. Each inside angle is 150°. How many sides? Outside = 30, and 360 ÷ 30 = 12 sides.

⚠️ Watch out

• Inside angle or turn? Check which one the question asks for.

🎯 Try these

1. Regular hexagon: each inside angle?
2. Regular 10-sided polygon: each turn?
3. Each inside angle is 160°. How many sides?

Answers: 1) 120° 2) 36° 3) 18

This guide is also in the 10MathQ app: tap “📖 Study guide” after a question.