Equilateral Triangle Calculator
Solve an equilateral triangle from any single value: side, height, area, or perimeter. Enter it in any unit and get the other measurements in the same unit, including the circumradius and inradius.
Example: with What do you know? Side length (a) · Value 10 → Height h = (√3/2)a: 8.66 units.
- Side (a)10 units
- Area A = (√3/4)a²43.301 square units
- Perimeter (3a)30 units
Computed by the calculator below using its default values. Change any input to see your own numbers.
All three sides equal, all angles 60°. Height h = (√3/2)a ≈ 0.866a — dropping the height splits it into two 30-60-90 triangles.
Where the √3 comes from
Drop a line from one vertex to the midpoint of the opposite side and an equilateral triangle splits into two mirror-image right triangles with legs a/2 and h, and hypotenuse a. The Pythagorean theorem gives h² = a² − (a/2)² = (3/4)a², so h = (√3/2)a ≈ 0.866a. The height is always about 86.6% of the side — never equal to it.
Area follows immediately: A = ½ × base × height = ½ × a × (√3/2)a = (√3/4)a² ≈ 0.433a². The center of the triangle sits ⅓ of the way up each height line, which is why the inradius a/(2√3) is exactly half the circumradius a/√3 — a ratio unique among triangles to the equilateral case.
How it’s calculated
Height h = (√3/2)a ≈ 0.866025a; area A = (√3/4)a² ≈ 0.433013a²; perimeter P = 3a; circumradius R = a/√3 ≈ 0.57735a; inradius r = a/(2√3) ≈ 0.288675a. Reverse solving: a = 2h/√3, a = √(4A/√3), a = P/3. √3 ≈ 1.7320508.
Equilateral triangles only — all sides equal, all angles 60°. For merely isosceles or scalene triangles these shortcuts do not hold.
Equilateral triangle dimensions
| Side a | Height (√3/2)a | Area (√3/4)a² | Perimeter |
|---|---|---|---|
| 1 | 0.87 | 0.43 | 3 |
| 2 | 1.73 | 1.73 | 6 |
| 5 | 4.33 | 10.83 | 15 |
| 10 | 8.66 | 43.30 | 30 |
| 12 | 10.39 | 62.35 | 36 |
Computed with h = (√3/2)a and A = (√3/4)a²; rounded to 2 decimals.
Common mistakes
- Using the side as the height — the height is 0.866a, so a "10-inch" triangle stands only 8.66 inches tall.
- Plugging a known height into A = (√3/4)a² as if it were the side; convert first with a = 2h/√3.
- Computing area as ½ a² (base × side instead of base × height) — that overstates area by about 15%.
- Doubling the side and expecting double the area; area grows with the square, so it quadruples.
Frequently asked questions
What is the height of an equilateral triangle?
h = (√3/2) × side ≈ 0.866 × side. A triangle with 10-unit sides is 8.66 units tall. It comes from the Pythagorean theorem applied to half the triangle.
What is the area formula?
A = (√3/4)a² ≈ 0.433a², where a is the side. Equivalently ½ × base × height once you know h = 0.866a.
How do I find the side from the height?
Multiply the height by 2/√3 ≈ 1.1547. A 6-unit height means sides of 6.93 units. People often wrongly assume side = height; the side is always about 15% longer.
What are the circumradius and inradius?
The circumradius (center to a corner) is a/√3 ≈ 0.577a, and the inradius (center to a side) is a/(2√3) ≈ 0.289a — exactly half. Both share the same center point, ⅓ of the way up each median.
Is an equilateral triangle related to the 30-60-90 triangle?
Yes — cutting an equilateral triangle along its height produces two 30-60-90 right triangles with sides in the ratio 1 : √3 : 2. That is where those famous ratios come from.