Find the area of an equilateral triangle (regular 3-gon) with the given measurement. 6-inch apothem A = sq. in.

Question

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

6-inch apothem

A = sq. in.

Answer 1

draw an equilateral triangle

draw the altitude

the apothem of an equilateral triangle is 1/3 the height

therefore the height is 18 in

the height divides the triangle into two 30-60-90 right triangles

take one of the triangles

the longest leg is 18, the shortest leg is "x", and the hypotenuse is "2x"  (the height bisects the 60º angle and the side opposite the 30º angle is half the hypotenuse)

use the Pythagorean Theorem to get...

(2x)^2=18^2+x^2

4x^2=324+x^2

3x^2=324

x^2=108

take the square root of both sides

x=√108   (108=36*3)

x=6√3

therefore each side of the equilateral triangle is 2*6√3=12√3

A=(1/2)(b)(h)

A=(1/2)(12√3)(18)

A=108√3 sq in or ≈187.06149 sq in

 

another approach is to use the formula A=absinC/2 where a and b are both 12√3 from above and angle C=60º

A=(12√3)(12√3)sin60º/2

A=(432*[√3/2])/2

A=216√3/2

A=108√3 sq in

Answer 2

The area of such triangle is calculated as 

A=(√3/4)l^2;

apothem=(√3/6)l