UNDERSTANDING SHAPES (Including Polygons)

164. The 360° Rule · Exterior angles always sum to a full circle

The sum of exterior angles is always 360°.

∠1 = 72°∠1 = 72°∠2 = 72°∠2 = 72°∠3 = 72°∠3 = 72°∠4 = 72°∠4 = 72°∠5 = 72°∠5 = 72°72° + 72° + 72° + 72° + 72° = 360°72° + 72° + 72° + 72° + 72° = 360°ABCDE
The sum of exterior angles of any convex polygon is always 360°, regardless of the number of sides. As you walk around the polygon, you turn through the exterior angles and complete one full revolution.

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For subscribers · Selina ICSE: UNDERSTANDING SHAPES (Including Polygons)

What this lesson covers

Try to break it

Drag any vertex. Each exterior angle changes, but the five of them always add to exactly 360°. Even when you drag a vertex inward to make the polygon concave, the right-turn exterior angles count as negative — and the total snaps back to 360°.

How you build it

Construct a pentagon with extended sides.

  • Draw a pentagon using the Polygon tool.
  • Extend side AB.
  • Extend side BC.
  • Extend side CD.
  • Extend side DE.
  • Extend side EA.

The proof, step by step

Prove that the exterior angles of a polygon add up to 360°.

  • A pentagon has 5 vertices. At each vertex, the interior angle and exterior angle form a linear pair, so Interior Angle + Exterior Angle = 180°.
  • Sum of all 5 (Interior + Exterior) angle pairs = 5 × 180° = 900°.
  • We know the sum of the 5 interior angles of a pentagon = (5 − 2) × 180° = 540°.
  • So, Sum of the 5 exterior angles = 900° − 540° = 360°.

Worked example

A regular polygon has 10 sides. What is the sum of its exterior angles?

The sum of exterior angles of any polygon is always 360°, regardless of the number of sides.

  • 180°
  • 360° — correct
  • 540°
  • 720°
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