Question

Find the area of an equilateral triangle (regular 3-gon) with the given measurement.

6-inch apothem

A = sq. in.

Answer 1

what is the area of an equilateral triangle (regular 3-gon) with the given measurement?
6-inch apothem, apothem a = 6
an equilateral triangle has 6 30-60-90 triangles each with the shortest leg being the apothem, length = a = 6
find hypotenuse of each of the 6 triangles:
angle opposite to a is 30 degrees,
sin 30 = opposite/hypotenuse = a/hyp = 6/hyp
0.5 = 6/hyp
0.5hyp = 6
hyp = 12 inches
find adjacent of each of the 6 triangles:
cos 30 = adjacent/hypotenuse = adj/12
sqrt(3)/2 = adj/12
12sqrt(3)/2 = adj
6sqrt(3) = adj
find side of the equilateral triangle:
adj is 1/2 the length of a side of the equilateral triangle
side s = 2 * adj = 2 * 6sqrt(3) = 12sqrt(3)

find perimeter p of the equilateral triangle:
p = 3 * s = 3 * 12sqrt(3) = 36sqrt(3)
find area of one of the 6 30-60-90 triangles:
area of one of the 6 triangles = 1/2 * adj * a
area of one of the 6 triangles = 1/2 * 6sqrt(3) * 6 = 18sqrt(3)
find area of equilateral triangle:
area, A, of equilateral triangle = 6 * 18sqrt(3)
A = 108sqrt(3) square inches
A = 187.1 square inches rounded to nearest tenth
find area equation for any equilateral triangle given apothem:
A = 108sqrt(3) = 3 * 36sqrt(3) = 3 * perimeter of the equilateral triangle = a/2 * p
A = 1/2 * a * p
(p = perimeter, A = area, a = apothem, s = side)
(p = 3 * s)
A = 1/2 * a * 3 * s
s = 12sqrt(3) = 6 * 2sqrt(3) = 2a * sqrt(3)
A = 3/2 * a * 2a * sqrt(3)
A = 3 * a^2 * sqrt(3)