What is the area of an equilateral triangle that has a perimeter of 36 units?

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Accepted Answer

Every side of an equilateral triangle is equal in length, so the length of one side of this triangle is 36 ⁄ 3 = 12 units. If a line is drawn from the vertex of an angle of the triangle to its opposite base, this line represents the height of the triangle. This line cuts the triangle into two identical 30-60-90 right triangles, since this line is perpendicular to the base, and every angle of an equilateral triangle is 60°. The length of this line is equal to the length of the longer base of a 30-60-90 right triangle, which is √3 times the length of the shorter base. The shorter base is equal to half the length of the hypotenuse. Since the hypotenuse is 12 units, the length of the shorter base is 6 units and the height is 6√3 units. Therefore, the area of the equilateral triangle is 1 ⁄ 2 (12)(6√3) = (6)(6√3) = 36√3 square units.

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