Equilateral Triangle Calculator

Find all properties of an equilateral triangle from its side, height, or area.

△ What is an Equilateral Triangle?

An equilateral triangle is a triangle in which all three sides are equal in length and all three interior angles are exactly 60°. It is the most symmetric of all triangles — it has three lines of symmetry, one through each vertex and the midpoint of the opposite side. Because all angles are equal, it is also called an equiangular triangle. For triangles specifically, equilateral and equiangular are equivalent: any triangle with all sides equal must have all angles equal, and vice versa.

The key formulas all involve √3 because the height of an equilateral triangle — found by the Pythagorean theorem — is h = (s√3)/2. The area then follows from A = ½ × base × height = ½ × s × s√3/2 = (s²√3)/4. The approximate value √3 ≈ 1.7321, so the area is about 0.433 × s².

Equilateral triangles have a special relationship with regular hexagons: a regular hexagon is made up of six equilateral triangles. They also tile the plane (tessellate) along with squares and regular hexagons — the only three regular polygons that can fill a flat surface without gaps. This property makes equilateral triangles appear in honeycomb structures, crystal lattices, triangular grids, and architectural truss designs.

All four classical triangle centers (centroid, circumcenter, incenter, orthocenter) coincide at the same point in an equilateral triangle. The inradius (r = s√3/6) and circumradius (R = s√3/3) satisfy R = 2r — the circumradius is exactly twice the inradius, a relationship unique to equilateral triangles. These properties make the equilateral triangle the fundamental building block of many geometric and physical structures.

📐 Formulas

Area = (s²√3) ÷ 4 Height = (s√3) ÷ 2

s = side length · Perimeter = 3s

Inradius r = s/(2√3) = s√3/6 · Circumradius R = s/√3 = s√3/3

From height h: s = 2h/√3 · From area A: s = √(4A/√3)

Example (s=6): Area = 36√3/4 = 9√3 ≈ 15.59 · Height = 3√3 ≈ 5.196 · Perimeter = 18

📖 How to Use This Calculator

Steps

Select Side, Height, or Area tab depending on which value you know.

Enter the known value. The calculator accepts any positive number; units are implied (cm, m, inches, etc.).

Click Calculate to see side, height, area, perimeter, inradius, and circumradius.

💡 Example Calculations

Example 1 — Side = 6 cm

Given: side = 6 cm

Height = (6 × √3)/2 = 3√3 ≈ 5.196 cm

Area = (6² × √3)/4 = 9√3 ≈ 15.59 cm²

Perimeter = 18 cm. Inradius = √3 ≈ 1.732 cm. Circumradius = 2√3 ≈ 3.464 cm.

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Example 2 — Height = 10 m (find the side)

Given: height = 10 m

Side = 2 × 10 / √3 = 20/√3 = 20√3/3 ≈ 11.547 m

Area = (11.547² × √3)/4 ≈ 57.74 m²

Side ≈ 11.547 m. Perimeter ≈ 34.64 m. Area ≈ 57.74 m².

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Example 3 — Area = 100 cm² (find the side)

Given: area = 100 cm²

s = √(4 × 100 / √3) = √(400/1.7321) = √230.94 ≈ 15.197 cm

Height = 15.197 × √3/2 ≈ 13.161 cm. Perimeter ≈ 45.59 cm.

Side ≈ 15.197 cm. This is the equilateral triangle with minimum perimeter enclosing area = 100 cm².

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Example 4 — Traffic Sign (stop sign triangle side = 30 cm)

Equilateral warning sign with side = 30 cm

Height = (30 × √3)/2 = 15√3 ≈ 25.98 cm

Area = (900 × √3)/4 = 225√3 ≈ 389.7 cm². Perimeter = 90 cm.

A 30 cm equilateral warning sign has area ≈ 389.7 cm² and height ≈ 25.98 cm.

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Frequently Asked Questions

What is the formula for the area of an equilateral triangle?

Area of an equilateral triangle = (√3/4) × s², where s is the side length. For example, an equilateral triangle with side 6 cm: Area = (√3/4) × 36 = 9√3 ≈ 15.59 cm². This formula comes from using the Pythagorean theorem to find the height h = s√3/2, then Area = ½ × base × height = ½ × s × s√3/2 = s²√3/4.

How do you find the height of an equilateral triangle?

Height (altitude) = (√3/2) × s, where s is the side length. For a triangle with side 8 cm: height = (√3/2) × 8 = 4√3 ≈ 6.928 cm. The height creates a right angle with the base and bisects it, forming two 30-60-90 right triangles. To reverse: given height h, side = 2h/√3 = 2h√3/3.

What is the perimeter of an equilateral triangle?

Perimeter = 3 × s (three equal sides). For side = 7 m: Perimeter = 21 m. Simple. The equilateral triangle has the smallest perimeter of all triangles with the same area — it is the most 'efficient' triangle shape, analogous to how the circle is the most efficient 2D shape.

What is the inradius of an equilateral triangle?

Inradius r = s/(2√3) = s√3/6. For side 6: r = 6/(2√3) = 3/√3 = √3 ≈ 1.732. The inradius is the radius of the largest circle that fits inside the triangle, touching all three sides. For an equilateral triangle, r = h/3 (one-third of the height), where h is the altitude.

What is the circumradius of an equilateral triangle?

Circumradius R = s/√3 = s√3/3. For side 6: R = 6/√3 = 2√3 ≈ 3.464. The circumradius is the radius of the circle that passes through all three vertices. Notably, R = 2r (circumradius is exactly twice the inradius) — this 2:1 ratio is unique to equilateral triangles.

How do you find the side of an equilateral triangle given the area?

From Area = (√3/4)s², solve for s: s = √(4A/√3) = 2√(A/√3) = 2(A/√3)^(1/2). For Area = 25 cm²: s = 2 × √(25/√3) = 2 × √(25/1.732) = 2 × √14.434 = 2 × 3.799 ≈ 7.598 cm. Verify: (√3/4) × 7.598² = 0.433 × 57.73 ≈ 25 ✓.

What angles does an equilateral triangle have?

All three interior angles of an equilateral triangle are exactly 60°. This is the only triangle where all angles are equal (equiangular). Since equilateral (all sides equal) implies equiangular (all angles equal) and vice versa for triangles, the terms are interchangeable.

Is an equilateral triangle the same as a regular triangle?

Yes. An equilateral triangle is the only regular triangle — the only triangle that is both equilateral (all sides equal) and equiangular (all angles equal). It is the simplest regular polygon. All regular polygons are equilateral and equiangular; for triangles, these properties imply each other.

How does an equilateral triangle relate to the 30-60-90 triangle?

Dropping an altitude in an equilateral triangle with side s creates two congruent 30-60-90 right triangles, each with: hypotenuse = s, short leg = s/2, long leg = altitude = s√3/2. This is how the 30-60-90 ratio 1:√3:2 is derived and why the equilateral triangle formula involves √3.

What is the centroid of an equilateral triangle?

The centroid is at the intersection of the medians, at height h/3 = s√3/6 from the base. In an equilateral triangle, the centroid coincides with the circumcenter, incenter, and orthocenter — all four triangle centers are at the same point. The centroid divides each median in ratio 2:1 from vertex to midpoint.

🔗 Related Calculators

📌 Quick Tips

💡All three sides of an equilateral triangle are equal. All three angles are exactly 60°. It has three lines of symmetry (one through each vertex and the midpoint of the opposite side).

💡The altitude (height) of an equilateral triangle bisects the base at 90° and also bisects the apex angle into two 30° angles, creating two 30-60-90 triangles.

💡The inradius (radius of inscribed circle) is exactly half the circumradius: r = R/2 = s/(2√3). The centroid, circumcenter, incenter, and orthocenter all coincide at the same point in an equilateral triangle.

💡For an equilateral triangle with side s: area = s²√3/4 ≈ 0.433s². This is always about 43.3% of s² — useful for quick mental estimation.

💡An equilateral triangle can tile the plane (tessellate) perfectly. It is the only regular polygon besides the square and regular hexagon that can do so.