Only the three sides

The idea

A triangle with sides a, b, c and semi-perimeter s = (1/2)(a + b + c) has area √(s(s – a)(s – b)(s – c)).

No height given

You know that the area of a triangle is 1/2 × base × height. But what if you know only the three sides and no height? Heron of Alexandria found a formula that uses just the sides.

A triangle has sides 3, 4 and 5. What is its area?

• 6
• 7.5
• 12

Try some triangles

Choose the three sides. The toy finds s, then √(s(s − a)(s − b)(s − c)), and checks it against 1⁠/2 × base × height measured on the drawn triangle. Try three different triangles.

A strange-looking formula

A triangle with sides a, b, c and semi-perimeter s, which is half the perimeter, has area √(s(s − a)(s − b)(s − c)). Here s = (a + b + c) ÷ 2.

Test it on the 3, 4, 5 triangle. s = 6, so the area is √(6 × 3 × 2 × 1) = √36 = 6. Since 3² + 4² = 5², the triangle is right-angled, with base 3 and height 4. Then 1⁠/2 × 3 × 4 = 6 too. The same answer! The proof of Heron's formula comes in Grade 10.

Notes

A triangle with sides a, b, c and semi-perimeter s = (a + b + c) ÷ 2 has area √(s(s − a)(s − b)(s − c)) (Heron's formula).

Check yourself

A triangle has sides 5 cm, 12 cm and 13 cm. What is its semi-perimeter s?

Answer: 15 cm

s = (5 + 12 + 13) ÷ 2 = 30 ÷ 2 = 15 cm.

For the triangle with sides 5, 12 and 13, s = 15. Use Heron's formula. What is the area?

Answer: 30 cm²

Area = √(15 × 10 × 3 × 2) = √900 = 30 cm².

A triangle has sides 13 cm, 14 cm and 15 cm. What is its area?

Answer: 84 cm²

s = 21. Area = √(21 × 8 × 7 × 6) = √7056 = 84 cm².

When is Heron's formula especially useful?

• When we know the three sides of a triangle but not its height — correct. Yes. It needs only a, b and c.
• When we know only the perimeter. The perimeter alone is not enough. Different triangles can have the same perimeter and different areas.
• When we know the base and the height. Then 1/2 × base × height is quicker.