Equilateral Triangle Area Calculator

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, which is 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 frames are built around triangles for rigidity, though the front triangle is nowhere near equilateral on any real bike.

Worked example: bunting for a 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, and of every triangle enclosing a given area it is the one with the shortest perimeter. That matters when you are fencing a triangular plot or binding the seam around a triangular cushion: any other proportion costs you more edge for the same ground.