Area from the outer circle

The idea

A triangle with sides a, b, c whose circumcircle has radius R has area abc/(4R).

A circle and an area

Heron's formula uses only the sides. There are two more formulas for the area of a triangle, and both have a connection with circles. The first uses the circle through the three vertices. Call its radius R.

A triangle has sides 3, 4, 5, and its circumcircle has radius R = 2.5. What does abc ÷ 4R come to?

• 6
• 10
• 12

Test abc ÷ 4R

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

A beautiful symmetry

A triangle with sides a, b, c whose circumcircle has radius R has area abc/(4R).

For the 13, 14, 15 triangle, R = 8.125. So abc ÷ 4R = 2730 ÷ 32.5 = 84, the same as Heron's formula. The formula has a beautiful symmetry: the three sides a, b and c appear in exactly the same way.

Notes

A triangle with sides a, b, c whose circumcircle has radius R has area = abc/(4R).

Check yourself

A triangle has sides 3, 4, 5 and circumradius R = 2.5. Use area = abc ÷ 4R. What is the area?

Answer: 6

Area = 60 ÷ (4 × 2.5) = 60 ÷ 10 = 6.

A triangle has sides 6, 8, 10 and circumradius R = 5. What is its area?

Answer: 24

Area = 480 ÷ (4 × 5) = 480 ÷ 20 = 24.

A triangle with sides 5, 5, 6 has area 12. Use area = abc ÷ 4R to find its circumradius R.

Answer: 3.125

4R = 150 ÷ 12 = 12.5, so R = 3.125.

What happens to abc ÷ 4R if you swap the names of the sides a and b?

• The area changes. a × b × c does not depend on the order of the sides. The area of the triangle is the same.
• Nothing: the value stays the same — correct. Yes. abc is the same product either way. This is the symmetry of the formula.
• The formula stops working. It works the same: the sides a, b and c appear in exactly the same way.