Area from the inner circle

The idea

A triangle with sides a, b, c whose incircle has radius r has area r(a + b + c)/2.

A circle inside

The second circle formula uses the circle that fits inside the triangle and touches all three sides. Call its radius r.

A triangle has sides 3, 4, 5, and its incircle has radius r = 1. What does r(a + b + c) ÷ 2 come to?

• 6
• 12
• 1

Test r(a + b + c) ÷ 2

Pick a triangle and tap Check the formula. The toy uses the radius r measured from the drawing and the perimeter. Then it compares with Heron's formula. Do all three triangles.

Half the perimeter times r

A triangle with sides a, b, c whose incircle has radius r has area r(a + b + c)/2.

Here a + b + c is the perimeter, so the area is r times half the perimeter. For the 13, 14, 15 triangle, r = 4. So r(a + b + c) ÷ 2 = 4 × 42 ÷ 2 = 84, the same as Heron's formula. The proof of this formula needs a result that you will study in Grade 10.

Notes

A triangle with sides a, b, c whose incircle has radius r has area = r(a + b + c)/2.

Check yourself

A triangle has sides 3, 4, 5 and inradius r = 1. Use area = r(a + b + c) ÷ 2. What is the area?

Answer: 6

Area = 1 × 12 ÷ 2 = 6.

A triangle has sides 5, 5, 6 and inradius r = 1.5. What is its area?

Answer: 12

Area = 1.5 × 16 ÷ 2 = 24 ÷ 2 = 12.

A triangle with sides 13, 14, 15 has area 84. Use area = r(a + b + c) ÷ 2 to find its inradius r.

Answer: 4

84 = r × 21, so r = 84 ÷ 21 = 4.

A triangle has perimeter 30 cm, and its incircle has radius 2 cm. What is its area?

• 60 cm². That is r × perimeter. The formula divides by 2.
• 30 cm² — correct. Yes. Area = 2 × 30 ÷ 2 = 30.
• 15 cm². That is perimeter ÷ 2. Multiply by r = 2 as well.