Regular Polygons in a Circle സമബഹുഭുജം
The chapter measures circles by drawing regular polygons inside them. This page shows how to draw such polygons, how joining the centre cuts them into equal triangles, and how to find their area with ½ × base × height.
- A circle is all points at distance r (radius) from the centre; perimeter of a polygon = sum of its sides.
- The angles round a point add up to 360°.
1. What a regular polygon is
A regular polygon (സമബഹുഭുജം) has all its sides equal and all its angles equal: the equilateral triangle, the square, the regular pentagon, the regular hexagon, and so on.
To draw a regular polygon with n sides inside a circle, divide the full turn at the centre into n equal angles of 360° ÷ n, mark where the arms meet the circle, and join the points in order.
2. Any number of sides
Each triangle from the centre has two sides equal to the radius, so they are all congruent isosceles triangles. With more sides the polygon looks more and more like the circle.
3. Area of a triangle: ½ × base × height
4. Area of a regular polygon from its triangles
Cut it into n triangles from the centre. Each has base s (a side) and height h (the perpendicular from the centre to the side). Area = n × ½sh = ½ × (ns) × h = ½ × perimeter × h. Chapter 8 uses exactly this on page 159.
5. The regular hexagon is special
The angle at the centre of each hexagon triangle is 360° ÷ 6 = 60°, and the two other angles are equal, so all three are 60°: the triangles are equilateral. Hence side of the hexagon = radius. Each interior angle of the hexagon is 60° + 60° = 120°. In general the interior angle of a regular n-gon is (n − 2) × 180° ÷ n.
An equilateral triangle of side a has area (√3/4)a², so the hexagon has 6 × (√3/4)a².
Bee honeycombs, nuts and bolts, floor tiles and football panels are regular hexagons and pentagons. Engineers make round shapes on a computer screen by drawing regular polygons with hundreds of tiny sides.
6. Check yourself
- A regular n-gon in a circle: equal central angles of 360° ÷ n; joining the centre gives n congruent isosceles triangles.
- Triangle area = ½ × base × height; polygon area = n × ½sh = ½ × perimeter × h.
- Regular hexagon: six equilateral triangles, side = radius, interior angle 120°, area 6 × (√3/4)a².
- The more sides, the closer the polygon is to the circle.