How to find side of an equilateral triangle when area is given?

Let area = A units ² and side length = y units
1/2 x y x y sin 60° = A
(1/2) y² ( √3/2 ) = A
y² = 4 A / √3
y = 2 [ A/√3 ]^(1/2)...Show more

Side of equilateral triangle = square root of (area*2)/(sine 60 degrees)...Show more

In an equilateral triangle, if you draw a line down from one top down to the opposite base, you create two right triangles that are identical.

The smaller triangle's base is half that of the original, and the height of both are the same.

So if the hypotenuse is "s", then the base is "s/2". Knowing this is a right triangle we can solve for the height using Pythagorean Theorem:

a² + b² = c²
(s/2)² + b² = s²
s²/4 + b² = s²
s² + 4b² = 4s²
4b² = 3s²
b² = 3s²/4
b = √(3)s/2

That's the height of the triangle, with the base being "s" so we can now come up with an area equation:

A = bh/2
A = [s√(3)s/2] / 2
A = s²√(3)/4

If you know the area and want to solve for the side, we get:

4A = s²√(3)
4A / √3 = s²
√(4A / √3) = s

which simplifies to:

2√A / ⁴√3 = s

If we want to rationalize the denominator, multiply both halves by ⁴√(3³) to get:

2 ⁴√(3³) √A / ⁴√3⁴ = s
2 ⁴√(27) √A / 3 = s

And if we want to have the 27 and A under a single radical, we get:

2 ⁴√(27A²) / 3 = s

Kind of ugly. This would likely make more sense if we had the area to start with and just worked the numbers from there, but this will get you the side knowing the area of any equilateral triangle....Show more

Area = one quarter of √3 times the square of the side, so side = 2√(area / √3)

Rationalise the denominator if you must....Show more

Heron's formula tells us that the area of any triangle is:

sqrt(s * (s - a) * (s - b) * (s - c))

Where a , b , and c are side lengths and s is the semi-perimeter

a = b = c

sqrt(s * (s - a) * (s - a) * (s - a)) =>
sqrt(s * (s - a)^3) =>
(s - a) * sqrt(s * (s - a))

s = (a + b + c) / 2 = (a + a + a) / 2 = 3a/2

(3a/2 - a) * sqrt((3a/2) * (3a/2 - a)) =>
a * (3/2 - 1) * sqrt(a * (3/2) * a * (3/2 - 1)) =>
a * (1/2) * sqrt(a^2 * (3/2) * (1/2)) =>
a * (1/2) * a * (1/2) * sqrt(3) =>
(sqrt(3)/4) * a^2

A = (sqrt(3)/4) * a^2

If you have the area

A * 4 / sqrt(3) = a^2
A * 4 * sqrt(3) / 3 = a^2
2 * sqrt(A * sqrt(3) / 3) = a
(2/3) * sqrt(3 * sqrt(3) * A) = a

a is the side length.

(2/3) * sqrt(3^(3/2) * A)
(2/3) * 3^(3/4) * A^(1/2)

2 * 3^(-1/4) * A^(1/2)...Show more