Find Area of Equilateral Triangle given Height 6sqrt3?

Question

I am getting 2 different answers from 2 different sources. 27units Squared and 36units Squared.

Answer 1

Orient the equilateral triangle so that one side is the base.

 

Form a right triangle using half the base along with the height and one side of the triangle.  The Pythagorean Theorem applies to this right triangle; the sides are n*1, n*2 (by definition of what we just did) and n*√3 [PLZ remember these numbers] where n is some constant.  This is a 30-60-90 degree triangle and is half the area of the equilateral triangle.

 

The base of this right triangle is n*1, the height is n*√3, and the hypotenuse is n*2 [PLZ verify that this fits the Pythagorean Theorem so you will know which side is which].

 

Since the height of the right triangle is 6√3, then n=6 and the base of the right triangle (half the base of the equilateral triangle) is 6.  That makes the base (and two other sides) of the equilateral triangle 12 units.

 

Since area of a triangle is (1/2)*base*height, he area of the right triangle is (1/2)*6*6√3 = 18√3 square units.  That makes the area of the equilateral triangle 36√3 square units.

 

CHECK:  If you use s as the side length of an equilateral triangle, the formula is:

   A = base*height

   A = (s/2)*(s/2)√3)

   A = (1/4)s²√3

 

The height is 6√3, and s = 12.

   A = (1/4)144√3

   A = 36√3 square units;      Check!

Answer 2

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

the height divides one of the 60º angles into two 30º angles

the side opposite the 30º angle is half of the hypotenuse

now to solve the problem...

x√3/2=the height in an equilateral triangle where x=length of a side

x√3/2=6√3

x=6√3/(√3/2)

x=6√3(2/√3)

x=12 units (we could have gotten the x value from x√3/2=6√3; x/2=6 and x=2*6=12)

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

A=(1/2)(12)(6√3)

A=6(6√3)

A=36√3 square units

 

Answer 3

Hi Harry,

 

We are given the height so we need to find the length of the sides. Since this is an equilateral triangle, the triangles formed by height will be special triangles with 30, 60 and 90 angles. Then if we call the side length a, the side across from 30 degrees will be a/2 units long. Then we can write according to the Pythagorean Theorem

(a/2)^2 + (6√3)^2 = a^2

a^2/4 + 108 = a^2

108 = 3a^2/4 (subtracted a^2/4 from each side)

a = 12 units (side length)

 

Area of an equilateral triangle is

√3*a^2/4 where a side length

so we found that a=12

area = √3*12^2/4

= 36√3 unit^2