UNDERSTANDING SHAPES (Including Polygons)

165. The Regular Polygon's Secret · Angles, sides, and the magic of n

Exterior + Interior = 180°, and sum of all exterior angles = 360°.

OVVₙ₋₁V₁Exterior Angle = 360°/nInterior Angle = 180° − 360°/n312← sides (n) →360° ÷ 6 = 60°360° ÷ 6 = 60°180° − 360° ÷ 6 = 120°180° − 360° ÷ 6 = 120°n = 6n = 6Vn
This diagram shows a regular n-sided polygon inscribed in a circle. • Drag vertex V to change the polygon's size and rotation • Drag the green slider to change the number of sides (n) • Each exterior angle = 360° ÷ n (purple arc) • Each interior angle = 180° − 360°/n (yellow arc) • Together they form a linear pair — always adding to 180°. For a regular hexagon (n = 6): exterior = 60°, interior = 120°. For a regular decagon (n = 10): exterior = 36°, interior = 144°.

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

What this lesson covers

Try to break it

The purple arc shows the exterior angle = 360°/n. The yellow arc shows the interior angle = 180° − 360°/n. Together they form a linear pair (180°).

How you build it

Construct a regular hexagon (6 sides, each interior angle = 120°).

  • Mark the centre O anywhere on the canvas. Every vertex of the regular hexagon will be the same distance R from O.
  • Mark vertex A at any distance from O. The length OA becomes the circumradius R. Key fact: in a regular hexagon, side length equals R.
  • With centre O, draw the guide circle through A. Click O first, then click A. This circle will pass through all 6 vertices.
  • With centre A and the same radius, draw an arc. Click A. The arc crosses the guide circle at vertex B.
  • Mark B where the arc from A crosses the guide circle. Click the intersection — the canvas snaps to the exact crossing.
  • With centre B and the same radius, draw another arc. Click B. Its intersection with the guide circle gives vertex C.
  • Mark C where the arc from B crosses the guide circle. Keep going the same direction as A→B — click the next intersection.
  • With centre C and the same radius, draw an arc. Click C.
  • Mark D where the arc from C crosses the guide circle — halfway around, D is opposite to A.
  • With centre D and the same radius, draw an arc. Click D.
  • Mark E where the arc from D crosses the guide circle.
  • With centre E and the same radius, draw an arc. Click E.
  • Mark F where the arc from E meets the guide circle. If the construction is correct, F will also be exactly one radius away from A, closing the hexagon perfectly.
  • Draw side AB — connect A to B.
  • Draw side BC — connect B to C.
  • Draw side CD — connect C to D.
  • Draw side DE — connect D to E.
  • Draw side EF — connect E to F.
  • Draw side FA — connect F back to A. All 6 sides equal R, all 6 interior angles are 120°. A perfect regular hexagon!

The proof, step by step

Prove that each interior-exterior angle pair adds to 180° and the exterior angles sum to 360°.

  • The sum of exterior angles of any convex polygon is 360°.
  • In a regular polygon, all exterior angles are equal.
  • Therefore, each exterior angle = 360° / n.
  • Interior and exterior angles form a linear pair, so they sum to 180°.
  • Thus, each interior angle = 180° - (360°/n) = (2n-4)90°/n.

Worked example

A regular polygon has an exterior angle of 45°. How many sides does it have?

Exterior angle = 360°/n. So, 45° = 360°/n. Solving for n gives n = 360/45 = 8 sides.

  • 6
  • 8 — correct
  • 10
  • 12
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