Rectilinear Figures [Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium]

238. The Regular Polygon's Secret · Equal sides, equal angles, perfect symmetry

Interior + Exterior angle at any vertex always equals 180°.

60°60°60°60°60°60°60°60°60°60°60°60°60°60°60°60°Drag the slider to change sides: n = 6Drag the slider to change sides: n = 6Each exterior angle = 360° ÷ 6 = 60°Each exterior angle = 360° ÷ 6 = 60°Sum of exterior angles = 6 × 60° = 360°Sum of exterior angles = 6 × 60° = 360°Interior angle = 180° − 60° = 120°Interior angle = 180° − 60° = 120°P
A regular polygon has all sides equal and all angles equal. At each vertex the interior and exterior angles are supplementary (add to 180°), and the exterior angles of any polygon sum to 360°, so each exterior angle of a regular n-gon is 360°/n.

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Selina ICSE: Rectilinear Figures [Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium]

What this lesson covers

Try to break it

Drag P around the circle to rotate the polygon. The interior and exterior angles at vertex P always sum to 180° at every position — they're a linear pair on the side line, so the sum is fixed. Try to drag P to a place where they don't sum to 180°; impossible.

How you build it

Construct a regular hexagon with a compass.

  • Point tool: mark the centre point O.
  • Point tool: mark point A a little away from O. This will be the first vertex.
  • Arc tool: click O, then A. This draws the circumcircle and locks the compass to the radius OA — and a hexagon''s side equals its radius.
  • Arc tool: click A (same radius). The arc cuts the circle at the next vertex.
  • Point tool: mark B where the arc meets the circle. AB equals the radius.
  • Arc tool: click B (same radius).
  • Point tool: mark C where the arc meets the circle.
  • Arc tool: click C (same radius).
  • Point tool: mark D where the arc meets the circle.
  • Arc tool: click D (same radius).
  • Point tool: mark E where the arc meets the circle.
  • Arc tool: click E (same radius).
  • Point tool: mark F where the arc meets the circle. All six vertices are now an equal radius-step apart.
  • Segment tool: join A to B.
  • Segment tool: join B to C.
  • Segment tool: join C to D.
  • Segment tool: join D to E.
  • Segment tool: join E to F.
  • Segment tool: join F to A. All six sides equal the radius — a regular hexagon.

The proof, step by step

Prove that the interior and exterior angle at each vertex of a regular polygon add to 180°.

  • The sum of exterior angles of any convex polygon is always 360°.
  • For a regular polygon with n sides, each exterior angle = 360° / n.
  • Interior angle + Exterior angle = 180° (linear pair).
  • Therefore, Interior angle = 180° - 360°/n = (2n-4)×90°/n.

Worked example

In a regular polygon, each exterior angle measures 40°. How many sides does the polygon have?

Each exterior angle = 360°/n. Given 360°/n = 40°, solving for n gives n = 360/40 = 9. The polygon has 9 sides.

  • 6
  • 8
  • 9 — correct
  • 10
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