How would you find the area of an equilateral triangle using the Pythagorean Theorem?

Question

Theodore Abad · Accepted Answer

We can see that if we split an equilateral triangle in half, we are left with two congruent equilateral triangles. Thus, one of the legs of the triangle is #1/2s#, and the hypotenuse is #s#.

If we want to find the height, we use the Pythagorean Theorem:

#(1/2s)^2+h^2=(s)^2#
#1/4s^2+h^2=s^2#
#h^2=3/4s^2#
#h=sqrt3/2s#

If we want to determine the area of the entire triangle, we know that #A=1/2bh#. We also know that the base is #s# and the height is #sqrt3/2s#, so we can plug those in to the area equation to see the following for an equilateral triangle:

#A=1/2bh=>1/2(s)(sqrt3/2s)=(s^2sqrt3)/4#

Violet Aldrich · Answer

undefined You cannot find the area of an equilateral triangle using the Pythagorean Theorem alone, as the Pythagorean Theorem is used to find the lengths of the sides of a right-angled triangle. Instead, you can use the formula for the area of an equilateral triangle, which is:

$$ \text{Area} = \frac{\sqrt{3}}{4} \times \text{side}^2 $$

Where the side represents the length of one side of the equilateral triangle.