Triangles from the centre

The idea

The area enclosed by a regular polygon equals half its perimeter × the radius of the circle that fits tightly within the polygon.

A circle inside a polygon

Archimedes (c. 250 BCE) used a property of regular polygons. Draw the circle that fits tightly inside the polygon, touching every side. Its radius is r.

Join the centre to the corners of a regular polygon. It splits into triangles. What do all these triangles share?

• The same height r
• The same angle
• Nothing

Polygons around a circle

Pick a polygon. See its area as half the perimeter × r, and check it against the area measured from the drawing. Try all five polygons.

Half the perimeter times r

The area enclosed by a regular polygon equals half its perimeter × the radius of the circle that fits tightly within the polygon.

Each triangle has a side of the polygon as its base and the radius r as its height. Its area is 1/2 × side × r. Add all the triangles: 1/2 × (sum of the sides) × r = 1/2 × P × r. With more and more sides, the polygon becomes the circle, P becomes 2πr and the area becomes πr². This is how Archimedes got A = πr².

Notes

The area enclosed by a regular polygon is half its perimeter × the radius of the circle that fits tightly within it. With more sides, it becomes the circle and gives πr².

Check yourself

A square has side 8 cm, so its perimeter is 32 cm. The circle that fits tightly inside it has radius 4 cm. What is the area of the square?

Answer: 64 cm²

1/2 × 32 × 4 = 16 × 4 = 64 cm². This matches 8 × 8.

A regular polygon has perimeter 60 cm. The circle that fits tightly inside it has radius 4 cm. What is the area of the polygon?

Answer: 120 cm²

1/2 × 60 × 4 = 30 × 4 = 120 cm².

In the formula area = 1/2 × perimeter × r, what is r?

A regular polygon has area 150 cm². The circle that fits tightly inside it has radius 5 cm. What is its perimeter?

Answer: 60 cm

150 = 1/2 × P × 5, so P × 5 = 300 and P = 60 cm.

• The distance from the centre to a corner. That is the radius of the circle that passes through the corners. The formula uses the circle that touches the sides.
• The length of one side. The side is part of the perimeter. r is the height of the triangles.
• The radius of the circle that fits tightly inside the polygon — correct. Yes. It runs from the centre to the middle of a side, and it is the height of each triangle.