An area from three sides

A triangular field has sides 13, 14 and 15. No height is marked anywhere.

The rule

Semi-perimeter — half the perimeter of a shape.

Rule: Half the perimeter. Take each side from it. Multiply the four numbers. Then find the number that squares to the product.

That number is the square root of the product, written √. Call the sides a, b and c, and half the perimeter s. Then s = (a + b + c) ÷ 2. The area is √(s × (s − a) × (s − b) × (s − c)).

The perimeter is 13 + 14 + 15 = 42, so s = 21. Now take each side from 21. 21 − 13 = 8. 21 − 14 = 7. 21 − 15 = 6. We call 8, 7 and 6 the brackets. The four numbers are s and the three brackets. Multiply them: 21 × 8 × 7 × 6 = 7056. Since 84 × 84 = 7056, the area is 84 square units.

This is Heron's rule. Heron was a Greek mathematician in Alexandria.

Check it another way. Take 14 as the base. The height on that base is 12. The height splits the base into 5 and 9. That makes right triangles 5, 12, 13 and 9, 12, 15. And 14 × 12 ÷ 2 = 84. Both routes agree.