265. Area of an Equilateral Triangle · √3/4 a², every time

Try to break it

Drag B along the base to grow or shrink the equilateral triangle. The base a and the height a√3/2 grow together, and the area always equals (√3/4)·a². The √3/4 ratio is the same for every equilateral triangle — try to drag B to change it; impossible.

How you build it

Construct an equilateral triangle and find its area.

• Point tool: mark A — one end of the side.
• Point tool: mark B — the other end of the side.
• Segment tool: join A to B — the base side a.
• Arc tool: centre A, open out to B (radius AB) and swing an arc.
• Arc tool: keep the opening and click B — the arc crosses the first one at C.
• Point tool: mark C where the two arcs cross — CA = CB = AB, so the triangle is equilateral.
• Segment tool: join A to C.
• Segment tool: join B to C — all three sides equal a, area = (√3/4)·a².

The proof, step by step

Prove that the area of an equilateral triangle is (√3/4) × side².

• Draw altitude CM from C to AB. Since △ABC is equilateral, CM bisects AB, so AM = MB = a/2.
• In right △AMC, by Pythagoras: AC² = AM² + CM² ⇒ a² = (a/2)² + h² ⇒ h² = 3a²/4 ⇒ h = a√3/2.
• Area of △ABC = ½ × base × height = ½ × a × (a√3/2) = √3/4 a². Hence proved.

Worked example

An equilateral triangle has a perimeter of 60 cm. What is its area? (Take √3 = 1.73)

Perimeter = 3a = 60 ⇒ a = 20 cm. Area = √3/4 × a² = √3/4 × 400 = 100√3 ≈ 100 × 1.73 = 173.20 cm².

• 150.00 cm²
• 173.20 cm² — correct
• 180.50 cm²
• 200.00 cm²