Regular polygon

Properties

• The interior angles add up to (n − 2) × 180°.
• Each exterior angle is 360° ÷ n.
• It has n lines of symmetry.
• All its corners lie on one circle.
• The apothem is the distance from the centre to the middle of a side.

Why the formulas work

Join the centre to every corner. The polygon splits into n equal triangles, each with base a and height equal to the apothem. Its area is therefore n × ½ × a × apothem, or half the perimeter times the apothem. The apothem comes from the right-angled triangle formed by half a side and the angle 180°/n at the centre.

Worked example

Take a regular polygon with number of sides 6 and side 10 cm.

1. Area. Area = na² ⁄ (4 tan(180° ⁄ n)). Here: 6 × 10² ⁄ (4 × tan(180° ⁄ 6)) = 259.81 cm².
2. Perimeter. Perimeter = na. Here: 6 × 10 = 60 cm.
3. Each interior angle (degrees). Interior angle = (n − 2) × 180° ⁄ n. Here: (6 − 2) × 180 ⁄ 6 = 120.
4. Apothem. Apothem = a ⁄ (2 tan(180° ⁄ n)). Here: 10 ⁄ (2 × tan(180° ⁄ 6)) = 8.66 cm.

Finding a missing measurement

From the interior angle, find the number of sides through the exterior angle. An interior angle of 135° means an exterior angle of 45°, and 360 ÷ 45 = 8 sides: an octagon. From the perimeter, the side is the perimeter ÷ n.

Mistakes to avoid

• Using the formula for a polygon whose sides are not all equal. It works only for regular polygons.
• Mixing up interior and exterior angles. For a hexagon they are 120° and 60°.

Where you see it

Honeycomb cells and many floor tiles are regular hexagons, a STOP sign is a regular octagon, and old 2 rupee coins in India had eleven sides.

Try these

Work each one out, then tap to check.

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