Question

Find the area of an equilateral triangle of height h cm.

Answer 1

Since an equilateral triangle has (by definition) 3 equal sides, if I draw a perpendicular line from the base to the opposite vertex, I have not only drawn an angle bisector, but I have bisected the the base. This means that I have created two new triangles which are a 30/60/90 (degree) triangles. If my original equilateral triangle has sides x, then each 30/60/90 triangle have sides x, 1/2 x and h (in terms of x, h = √3/2 x). using the Pythagorean Theorem, we can find x in terms of h {h2 + (1/2 x)2 = x2}. Solving for x we would get x = h/√3 (or rationalizing the denominator, x = h√3/3). Now using the formula for the area of a triangle (A = 1/2 bh) we get A = 1/2 (h/√3)(h) = h2/√3 (or h2√3/3)

Answer 2

Answer:

Step-by-step explanation:

Altitude of an equilateral triangle , bisects the base. If each side = a unit

By Pythagiras law, a² = (a/2)² + h²

=> a² - a²/4 = h²

=> h = √(3a²/4) = √3a/2

=> a = 2h/√3

Area( triangle) = 1/2 * a * h

=> area = 1/2 * 2h/√3 * h = h²/√3 = √3h² /3

Area = √3h² /3