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Height and area of an equilateral triangle

2026-09-22Plane GeometryMs Tam

Side a → height (√3/2)a and area (√3/4)a².

Key formulas

h = (√3/2)a
area = (√3/4)a²

Split in half

The altitude bisects the base. Pythagoras on half the triangle: h² + (a/2)² = a² → h=(√3/2)a.

Example

a=6 → h=3√3, area=9√3.

Angles

Every angle is 60°. The altitude also bisects the apex angle into two 30° angles.

Worked example

Worked example

Problem. An equilateral triangle has side 6 cm. Find its height and area.

Solution. h = (√3/2)×6 = 3√3 cm. Area = (√3/4)×36 = 9√3 cm².

30-60-90

Half an equilateral triangle is the standard 30-60-90 triangle.

Review quiz

Equilateral side 10. Area is:

Practice questions

  1. a=10. Find area in surd form.
  2. If height is 4√3, find the side.

Study notes

Is every 60° triangle equilateral?

No — you need all three sides equal, or all angles 60°.

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