Equilateral Triangle Area: The √3⁄4 Shortcut
The area of an equilateral triangle with side a is A = (√3 ⁄ 4)a². It is a shortcut for ½ × base × height.
Equilateral Triangle
All three sides are equal and every angle is 60°. Area, height and perimeter depend only on the side length.
- Area
- A = (√3 ⁄ 4) a2
- Height
- h = (√3 ⁄ 2) a
- Perimeter
- P = 3a
- a side length
Where √3 ⁄ 4 comes from
Drop a line from the top to the middle of the base. It makes two right triangles with hypotenuse a and base a ⁄ 2. By Pythagoras, the height is h = √(a² − (a ⁄ 2)²) = (√3 ⁄ 2)a. Then A = ½ × a × (√3 ⁄ 2)a = (√3 ⁄ 4)a².
Worked example
An equilateral triangle has sides of 8 cm.
- A = (√3 ⁄ 4) × 8² = (√3 ⁄ 4) × 64
- A = 27.71 cm²
- Height = (√3 ⁄ 2) × 8 = 6.93 cm
- Perimeter = 3 × 8 = 24 cm
Work back to the side
An equilateral triangle has an area of 50 cm². Find the side.
- a² = 50 × 4 ⁄ √3 = 115.47
- a = 10.75 cm
Try the equilateral triangle calculator
Try it yourself
Find the area of an equilateral triangle with sides of 10 cm.
Show the answer
A = (√3 ⁄ 4) × 100 = 43.3 cm²
More on the equilateral triangle formulas page.
Equilateral triangle: FAQ
What is the area of an equilateral triangle?
A = (√3 ⁄ 4)a², where a is the side length.
What is the height of an equilateral triangle?
h = (√3 ⁄ 2)a, about 0.866 × a.
Can I use ½ × base × height instead?
Yes. The shortcut gives exactly the same answer.