Question

what is the area of an equilateral triangle of side 'a' units?

Answer 1

(1) Let ABC be the equilateral triangle with base BC and all sides AB , BC, CA each equal to a. Draw perpendicular to base BC from A to meet BC at point D. Then D bisects  BC such that BD = DC.                                Now to find height of triangle ABC:-AD² = AC² - DC² = a² - (a/2)² = (3/4)a².      Or height AD = √[(3/4)a²] = [(√3)/2)] x a.                                      Area of triangle ABC = (1/2)xHeight x Base                                                            = (1/2)xADxBC = (1/2)[(√3/2)a]xa = (√3/4)a²                                                     (2) using formula Area of a triangle = √[s(s-a)(s-b)(s-c)] where a, b, c are three sides of a triangle and s = (a+b+c)/2.                                               For equilateral triangle, s = (3a/2), s-a = s-b = s-c = (3a/2) - a = a/2.                Hence area = √[(3a/2)(a/2)³] = [(√3)/4]xa²