Equilateral Triangle Area Calculator
Calculate the area of an equilateral triangle from just its side length.
Returns area, height, perimeter, and inradius in any common unit.
An equilateral triangle has three equal sides and three 60° angles. Given just the side, every other measurement falls out of the formulas.
Area formula:
A = (√3 / 4) × s²
Numerically, that’s about 0.4330 × s². A 10 cm equilateral triangle has area 43.30 cm².
Where the √3 comes from: drop a perpendicular from one vertex to the opposite side. It bisects the side, and you get a 30-60-90 right triangle. The long leg of that right triangle is s√3/2 — that’s the height of the equilateral. Multiply ½ × base × height with base = s and height = s√3/2 and you get the formula.
Other measurements from the same side s:
- Height: h = s × √3 / 2 ≈ 0.866 × s
- Inradius (inscribed circle radius): r = s × √3 / 6 ≈ 0.289 × s
- Circumradius (circumscribed circle radius): R = s × √3 / 3 ≈ 0.577 × s
- Perimeter: P = 3s
Where equilateral triangles show up:
- Warning signs in many countries (yield, hazard) are equilateral triangles. A US “Yield” sign is 36 in on each side, area 561.2 sq in.
- Geodesic domes are built from equilateral triangles. A 4 ft side panel covers 6.93 sq ft.
- The triangular Toblerone bar cross-section is equilateral (approximately).
- The triangular face of a regular tetrahedron is equilateral, as is the face of an octahedron and an icosahedron.
- Bicycle frame triangles approximate equilateral for a balanced ride feel — though most actual frames are slightly stretched.
Worked example — banner for a kids’ party:
You’re cutting equilateral triangle pennants 6 in on a side. Area per pennant = 0.4330 × 36 = 15.59 sq in. For 20 pennants you need 312 sq in of fabric = about 2.17 sq ft. Buy a yard of 36-inch fabric (9 sq ft) and you have plenty of room for waste and seam allowance.
Why this shape is structurally efficient: the equilateral triangle is the only triangle with all three angles equal. It also has the smallest perimeter for a given area among all triangles — useful when minimizing the fence around a fixed triangular plot, or the seam length around a triangular cushion.