How this instrument works
An equilateral triangle has all three sides equal and all three angles fixed at exactly 60°, so a single side length s determines every other measure at once: area = (√3 ⁄ 4)s², perimeter = 3s, and height = (√3 ⁄ 2)s. Each formula falls directly out of splitting the triangle in half along its own height, producing two identical 30-60-90 right triangles whose side ratios are already fixed by that special angle set.
The height formula in particular is worth deriving by eye: dropping a perpendicular from one corner to the midpoint of the opposite side creates a right triangle with hypotenuse s and one leg s⁄2 (half the base), and the Pythagorean theorem then gives the height as √(s² − (s⁄2)²), which simplifies to (√3 ⁄ 2)s.
Because every equilateral triangle is a scaled copy of every other one — same angles, same proportions, just a different size — all three measures scale together in fixed, predictable ways: doubling the side length doubles the perimeter and height (both linear in s) but quadruples the area (which grows with s²).
- Enter the triangle's side length into the Side length field.
- Read Area: the sheet applies (√3 ⁄ 4)s² directly.
- Read Perimeter: simply three times the side length.
- Read Height: the sheet applies (√3 ⁄ 2)s, all three outputs updating together as the side length changes.
Worked example — a side length of 6
An equilateral triangle has a side length of 6 units. Its area is (√3 ⁄ 4) × 36 = 9√3 ≈ 15.59 square units, its perimeter is 3 × 6 = 18 units, and its height is (√3 ⁄ 2) × 6 = 3√3 ≈ 5.20 units — three different measures returned from the single side length.
A larger triangle with side 10 gives area 25√3 ≈ 43.30, perimeter 30, and height 5√3 ≈ 8.66 — the perimeter and height each grew in direct proportion (5⁄3 times larger, matching the side's own growth), while the area grew by (5⁄3)² ≈ 2.78 times, the faster quadratic rate.
Questions
What is the formula for an equilateral triangle's area?
A = (√3 ⁄ 4)s², where s is the side length. It comes from splitting the triangle into two 30-60-90 right triangles along its own height and applying the ordinary ½·base·height area formula to the result.
How do you find the height of an equilateral triangle?
h = (√3 ⁄ 2)s, derived by dropping a perpendicular from one corner to the midpoint of the opposite side and applying the Pythagorean theorem to the resulting right triangle, whose hypotenuse is s and whose base leg is s⁄2.
Why does one side length determine every measure?
Because an equilateral triangle's angles (all 60°) never change regardless of its size — every equilateral triangle is simply a scaled copy of every other one, so once the scale (the side length) is fixed, every proportional measure follows automatically.
Does doubling the side length double the area?
No — area scales with the SQUARE of the side length, so doubling the side quadruples the area. Perimeter and height, by contrast, scale in direct proportion to the side, so doubling the side simply doubles each of them.
Is an equilateral triangle also isosceles?
Yes — an equilateral triangle is technically a special case of an isosceles triangle (which requires at least two equal sides), just with all three sides equal rather than only two. Every property that holds for an isosceles triangle holds for an equilateral one too.