The Equilateral Triangle

Part of the Triangles section of Coddy's Geometry journey. Lesson 64 of 71.

An equilateral triangle has three equal sides. Its height cuts it into two 30-60-90 triangles. In each one, the short leg is half the side, s/2, and the long leg is the height. The long leg of a 30-60-90 triangle is the short leg times √(3), so the height is s/2 × √(3). With side 4, the short leg is 2 and the height is 2√(3).

$An equilateral triangle of side 4 with a dashed height h splitting the base into halves of 2$

Side 4: half the base is 2, and the height is 2 root 3

A = 1/2 × 4 × 2√(3) = 4√(3)

The area keeps the root of 3, because the height is exact and the root does not simplify away.

For an equilateral triangle with side s, put the height s/2 × √(3) into the area formula. Half the base times the height gives:

A = √(3)/4 × s^2

With s = 4, the formula gives √(3)/4 × 16 = 4√(3), the same answer as before. The root of 3 comes from the height of the 30-60-90 triangle, and it stays in the answer.