CalConverters

Equilateral Triangle Calculator: Area, Height, Perimeter

Find an equilateral triangle's area, height, and perimeter from any one known measure — side, perimeter, semi-perimeter, area, or height — with exact results.

Loading calculator…

Equilateral triangle formulas

An equilateral triangle has all three sides equal to a single length aa, and all three interior angles equal to 60°60°. Every other measure follows directly from aa:

P=3as=3a2h=32aK=34a2P = 3a \qquad s = \frac{3a}{2} \qquad h = \frac{\sqrt{3}}{2}a \qquad K = \frac{\sqrt{3}}{4}a^2

Tip: The height comes straight from the Pythagorean theorem: it splits the triangle into two right triangles with legs a/2a/2 and hh, hypotenuse aa, so h=a2−(a/2)2=3a24=32ah = \sqrt{a^2 - (a/2)^2} = \sqrt{\frac{3a^2}{4}} = \frac{\sqrt{3}}{2}a.

Worked example: side a = 6

P=3(6)=18s=3(6)2=9P = 3(6) = 18 \qquad s = \frac{3(6)}{2} = 9

h=32(6)=33≈5.1962K=34(36)=93≈15.5885h = \frac{\sqrt{3}}{2}(6) = 3\sqrt{3} \approx 5.1962 \qquad K = \frac{\sqrt{3}}{4}(36) = 9\sqrt{3} \approx 15.5885

Since 6 is even, both the height and area coefficients reduce to whole numbers (33 and 99). For an odd side, like a=5a=5, the coefficient itself is a fraction: h=523≈4.3301h = \frac{5}{2}\sqrt{3} \approx 4.3301 and K=2543≈10.8253K = \frac{25}{4}\sqrt{3} \approx 10.8253 — still an exact closed form, just not a whole-number multiple of 3\sqrt{3}.

Solving in the other direction

This calculator also works backward: give it any one of the five measures and it solves for the side first, then derives the rest.

KnownSolve for sideExact?
Side aa— (already known)Yes
Perimeter PPa=P/3a = P/3Yes — a plain rational division
Semi-perimeter ssa=2s/3a = 2s/3Yes — a plain rational division
Area KKa=4K/3a = \sqrt{4K/\sqrt{3}}No — dividing by the irrational 3\sqrt{3} almost never lands back on a whole number
Height hha=2h/3a = 2h/\sqrt{3}No — same irrational division
Tip: Perimeter and semi-perimeter reverse EXACTLY (no irrational step involved), but area and height reverse only approximately — expect a decimal side even when the area or height you typed looks like a clean number.

Common mistakes

  • Using K=12a⋅aK = \frac{1}{2}a \cdot a. That formula only works when the two sides meeting at a vertex are perpendicular. In an equilateral triangle the angle is 60°60°, not 90°90°, so the correct factor is 34\frac{\sqrt{3}}{4}, not 12\frac{1}{2}.
  • Confusing the height with a side. The height is always shorter — about 0.866a0.866a — never equal to or longer than a side.
  • Forgetting every angle is 60°60°. Unlike a general isosceles or right triangle, there's nothing to solve for the angles here; they're fixed by the equal-sides property alone.

Where this shows up

  • Structural engineering: the equilateral triangle is the most rigid basic shape used in trusses and frameworks, since none of its angles can flex without changing a side length.
  • Signage and tiling: road-hazard signs, warning triangles, and equilateral-triangle floor or wall tiles all rely on the same fixed 60°60° geometry.
  • Trigonometry reference: the 3030-6060-9090 right triangle formed by an equilateral triangle's height is one of the two standard "special" right triangles used throughout trigonometry.

An equilateral triangle is the most special case of both an isosceles triangle (all three sides equal, not just two) and a regular polygon (a regular 3-gon). Its height is found by solving a right triangle formed by half the base, the height itself, and a full side as the hypotenuse.

Frequently asked questions

What is an equilateral triangle?
A triangle with all three sides equal (aa) and all three angles equal to 60°60°. It is the most symmetric triangle — both a special isosceles triangle and a special case of a regular polygon (the regular 3-gon).
What is the formula for the area of an equilateral triangle?
K=34a2K = \frac{\sqrt{3}}{4}a^2. For side a=6a=6: K=34(36)=93≈15.5885K = \frac{\sqrt{3}}{4}(36) = 9\sqrt{3} \approx 15.5885. Using K=12a⋅aK = \frac{1}{2}a \cdot a instead is a common mistake — the two sides meeting at a vertex are not perpendicular, so that shortcut only works for right triangles.
What is the formula for the height of an equilateral triangle?
h=32ah = \frac{\sqrt{3}}{2}a. It comes from the Pythagorean theorem: the height splits the triangle into two right triangles with legs a/2a/2 and hh and hypotenuse aa, so h=a2−(a/2)2=32ah = \sqrt{a^2 - (a/2)^2} = \frac{\sqrt{3}}{2}a.
How do I find the side if I only know the perimeter or area?
From the perimeter, a=P/3a = P/3 — always exact. From the area, a=4K/3a = \sqrt{4K/\sqrt{3}} — this involves dividing by the irrational 3\sqrt{3}, so it almost always comes back as a decimal approximation rather than a clean whole number, even when the area itself looks tidy.
Why are all three angles exactly 60°?
A triangle's interior angles always sum to 180°. Since an equilateral triangle's three angles must also be equal to each other, each one is 180°/3 = 60° — this follows from the equal-sides property alone, with no extra assumption needed.
Is the height the same as any of the sides?
No — the height is always shorter than a side. Since h=32a≈0.866ah = \frac{\sqrt{3}}{2}a \approx 0.866a, the height is about 86.6% of the side length, never equal to it and never longer.
How is this related to the isosceles and right triangle?
An equilateral triangle is the special case of an isosceles triangle where all three sides (not just two) are equal. Its own height, in turn, is found by solving a right triangle formed by half the base, the height, and a full side as the hypotenuse.

Related calculators