Equilateral Triangle Calculator
Enter any one of side, perimeter, height, area, circumradius or inradius and get every other measurement — with a to-scale diagram, unit conversion table and step-by-step working.
What is an equilateral triangle?
An equilateral triangle is the simplest possible regular polygon: three sides, all cut to the same length. That one constraint on the sides quietly fixes everything else about the shape. The three interior angles have to match too, and because the angles in any triangle add to 180°, each corner works out to exactly 60°. It has three axes of mirror symmetry and a 120° rotational symmetry — properties no other triangle shares.
A practical consequence of all that symmetry is that one number is enough. Give this calculator any single measurement — the side, the perimeter around the outside, the vertical height, the area of the interior, the inscribed circle's radius, or the circumscribed circle's radius — and the remaining five follow from a short chain of algebra. That is why the input box below only asks for one value at a time.
Equilateral triangle, case by case
Everything from the side length
The side is the natural input — every other quantity is a fixed multiple of it. Perimeter is three times the side; height is √3/2 times the side; area is √3/4 times the side squared. If you already know a, no other input is needed.
Where the area formula comes from
The altitude bisects the base, so the half-triangle is a right triangle with hypotenuse a and one leg a/2. Pythagoras gives the other leg (the height) as (√3/2)·a. Substitute that back into base × height ÷ 2 and the a's collect: A = ½·a·(√3/2)·a = (√3/4)·a².
Reverse: side from area, perimeter or height
Because the ratios between quantities are fixed, any single measurement locks the triangle down. Solve backwards to get the side, then the rest follows automatically.
The two circles — inradius and circumradius
The incircle is the largest circle that fits inside the triangle; the circumcircle passes through all three vertices. For an equilateral triangle both circles share the same centre (the centroid), and the circumradius is exactly twice the inradius — a relationship that is unique to this shape.
Starting from the perimeter
If the fence around the triangle is what you know, divide by three to recover the side and then let every other quantity fall out of it. Perimeter is the friendliest reverse input because no square roots appear until you go for the height or area.
Starting from the circumradius R
The circumcircle passes through all three vertices; its radius fixes the scale of the triangle. Multiply R by √3 to get the side, then use R = 2r to read off the inradius for free — no separate calculation needed.
Starting from the inradius r
The incircle is the largest circle that fits snugly inside the triangle, touching each side once. Its radius is exactly half the circumradius, so doubling r and multiplying by √3 gives the side — everything else follows the standard chain.
Formulas at a glance
| Quantity | Formula | Reverse |
|---|---|---|
| Perimeter | P = 3a | a = P / 3 |
| Height | h = (√3 / 2)·a | a = 2h / √3 |
| Area | A = (√3 / 4)·a² | a = √(4A / √3) |
| Circumradius | R = a / √3 | a = R·√3 |
| Inradius | r = a / (2√3) | a = 2r·√3 |
| R vs r | R = 2r | Unique to equilateral triangles. |
What this tool does for you
- Accepts any one of six inputs — side, perimeter, height, area, circumradius or inradius — and back-solves the other five in a single click.
- Lets you pick the length unit up front (mm, cm, m, km, in, ft, yd) so both the input you type and every output stay in the same system.
- Renders a conversion table alongside the answer so a side of 10 cm is instantly visible as mm, m, inches and feet without a second calculation.
- Draws a to-scale SVG of the triangle with the height dashed, the inscribed and circumscribed circles optional, and all three 60° corners labelled.
- Ships with one-tap presets — a real-world yield sign, a tetrahedron face, a roof gable, and reverse-from-area / perimeter / height starting points.
- Prints a personalised working-out that plugs your actual number into each formula, so the algebra is easy to copy into homework or a report.
- Includes a copy-to-clipboard action plus a diagram snapshot download, both respecting the significant-figures setting you choose.
- Runs entirely in your browser — nothing you type is uploaded, and the page works offline once loaded.
Frequently asked questions
+What is an equilateral triangle?
An equilateral triangle is a triangle in which all three sides have the same length. Because equal sides face equal angles, all three interior angles are also equal — and because they must add to 180°, each angle is exactly 60°. That single symmetry makes it the most regular of all triangles: it's both isosceles (twice over) and equiangular.
+How do you find the area of an equilateral triangle?
Use A = (√3 / 4)·a², where a is the side length. It comes from the standard base × height ÷ 2 formula with height h = (√3/2)a substituted in. For a = 6, A = (√3/4)·36 = 9√3 ≈ 15.588.
+How do you find the height of an equilateral triangle?
Drop an altitude from any vertex to the opposite side — it bisects that side. The resulting right triangle has hypotenuse a and one leg a/2, so by Pythagoras the height is h = √(a² − (a/2)²) = (√3/2)·a. For a = 10, h ≈ 8.660.
+How do you find the side length from the area?
Invert A = (√3/4)·a² to get a = √(4A / √3). For A = 25, a = √(100/√3) ≈ 7.6.
+What are the circumradius and inradius of an equilateral triangle?
R (circumradius, the radius of the circle through all three vertices) = a / √3. r (inradius, the radius of the largest inscribed circle) = a / (2√3). Note R = 2r exactly — a property unique to equilateral triangles.
+Are all equilateral triangles similar?
Yes. Since every equilateral triangle has three 60° angles, any two equilateral triangles are similar by AAA. They only differ in scale, so ratios like h/a = √3/2, R/a = 1/√3 and area/side² = √3/4 are the same for every one of them.
+How is this calculator different from the general Triangle Calculator?
This page only needs one measurement — a side, the perimeter, the height, the area, R, or r — because equilateral triangles are fully determined by scale. The general Triangle Calculator needs three inputs (SSS, SAS, ASA, AAS, SSA) because a scalene triangle has three independent degrees of freedom.
+Where do equilateral triangles show up in real life?
Traffic yield signs, the faces of a regular tetrahedron, the fundamental tile of a triangular tiling, roof-truss webs, geodesic domes, the outline of many warning symbols, and the isogonic (Fermat) point of certain networks. Anywhere maximum symmetry or rigidity matters, the equilateral triangle shows up.