All three sides equal

The idea

An equilateral triangle of side a has height (√3/2)a and area (√3/4)a².

Three equal sides

Put the same length a for all three sides into Heron's formula. Then find the area again with 1/2 × base × height. Do the two ways agree?

An equilateral triangle has side 4. Is its area a whole number?

• Yes, 8
• Yes, 16
• No, it is not a whole number

Three ways, one answer

Change the side a. Check that Heron's formula, the height method and the formula for the area all give the same answer. Try three different sides.

The same formula

An equilateral triangle of side a has height (√3⁠/2)a and area (√3⁠/4)a².

Heron: s = 3a⁠/2 and s − a = a⁠/2, so the area is √((3a⁠/2)(a⁠/2)(a⁠/2)(a⁠/2)) = (√3⁠/4)a². Height: the height cuts the base into two halves of length a⁠/2, so by the Baudhāyana–Pythagoras theorem h² = a² − a²⁠/4 = 3a²⁠/4 and h = (√3⁠/2)a. Then 1⁠/2 × a × (√3⁠/2)a = (√3⁠/4)a². Same formula!

Notes

An equilateral triangle of side a has height (√3⁠/2)a and area (√3⁠/4)a².

Check yourself

An equilateral triangle has side 10 cm. Its height is (√3⁠/2)a. Use √3 ≈ 1.73. What is the height?

Answer: 8.65 cm

h = 1.73 × 10 ÷ 2 = 8.65 cm.

An equilateral triangle has side 2 cm. Use √3 ≈ 1.73. What is its area?

Answer: 1.73 cm²

Area = (√3⁠/4) × 2² = √3 ≈ 1.73 cm².

An equilateral triangle has side 8 cm. Use √3 ≈ 1.73. What is its area?

Answer: 27.68 cm²

Area = (√3⁠/4) × 64 = 16√3 ≈ 16 × 1.73 = 27.68 cm².

The side of an equilateral triangle is doubled. What happens to its area?

• It becomes 4 times as big — correct. Yes. The area is (√3⁠/4)a², and (2a)² = 4a².
• It doubles. The area has a², so doubling a multiplies the area by 2 × 2 = 4.
• It becomes 8 times as big. The area has a², not a³. Doubling a multiplies it by 4.