Area of an Equilateral Triangle

Calculates the area of an equilateral triangle as (√3 ÷ 4) × side². Its height is (√3 ÷ 2) × side.

Explanation

An equilateral triangle has three equal sides and three angles of 60 degrees. Because its shape is completely determined, the side length alone gives you both the area and the height.

S = \dfrac{\sqrt{3}}{4} a^2 \qquad h = \dfrac{\sqrt{3}}{2} a

$S$ — the area
$h$ — the height
$a$ — the side length

Example

With the default side $a = 6$ , the height is

h = \dfrac{\sqrt{3}}{2} \times 6 = 3\sqrt{3} \approx 5.1962

and the area is

S = \dfrac{\sqrt{3}}{4} \times 6^2 = 9\sqrt{3} \approx 15.5885

Where the formula comes from

Drop a perpendicular from the top vertex to the base. It cuts the base in half and leaves a right triangle with hypotenuse $a$ and legs $\dfrac{a}{2}$ and $h$ . The Pythagorean theorem gives $h^2 = a^2 - \dfrac{a^2}{4} = \dfrac{3a^2}{4}$ , so $h = \dfrac{\sqrt{3}}{2} a$ . Substituting that into $S = \dfrac{1}{2} a h$ yields $S = \dfrac{\sqrt{3}}{4} a^2$ .

Watch out

The height is always shorter than the side, by a factor of $\dfrac{\sqrt{3}}{2} \approx 0.8660$ . Using the side where the height belongs inflates the area. Note too that the area grows with the square of the side: double the side and the area becomes four times as large. The side must be a positive number.