If the area of an equilateral triangle is a and its height is b, then b²…

For an equilateral triangle of side length s, two standard identities apply: the height is h = (√3/2)s, and the area is A = (√3/4)s². These follow from splitting the equilateral triangle into two 30-60-90 right triangles using the perpendicular from a vertex to the opposite side.

Apply these identities to this triangle, where the height is called b and the area is called a:

1. Let the side length be s. From b = (√3/2)s, solve for s: s = 2b/√3.

2. Square the height: b² = (3/4)s².

3. Write the area using the standard formula: a = (√3/4)s².

4. Divide b² by a: b²/a = (3/4)s² ÷ (√3/4)s² = 3/√3.

5. Simplify: 3/√3 = √3.

Cross-check with a specific value: take b = 1. Then s = 2/√3, and a = (√3/4)(2/√3)² = (√3/4)(4/3) = 1/√3. So b²/a = 1 ÷ (1/√3) = √3, matching the general result.

Therefore, b²/a = √3.