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Equilateral Triangle Calculator

Area equals root three over four s squared

Solution

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Calculate Equilateral Triangle Area from Side

Use this form when the side length is known. All three sides are equal, so a single dimension fully determines the area.

A = (√3 / 4) s²

Calculate Equilateral Triangle Perimeter

Use this form for the boundary length — three equal sides times the side length.

P = 3s

Calculate Equilateral Triangle Height

Use this form for the altitude (perpendicular from any vertex to the opposite side).

h = s√3 / 2

Calculate Equilateral Triangle Side from Area

Use this rearrangement when the area is known and you need the side length.

s = √(4A / √3)

How It Works

This equilateral triangle calculator solves the formulas for a triangle with three equal sides and three 60° angles. From the side length s, every other property follows: area A = (√3/4)s², perimeter P = 3s, height h = s√3/2, inradius r = s√3/6, circumradius R = s√3/3. The circumradius is exactly twice the inradius for any equilateral triangle.

Example Problem

An equilateral triangle has a side length of 6 cm. Compute its area, perimeter, height, and radii.

  1. Knowns: s = 6 cm, all angles = 60°
  2. Area: A = (√3/4) · 36 = 9√3 ≈ 15.588 cm²
  3. Perimeter: P = 3s = 18 cm
  4. Height (altitude): h = 6√3/2 = 3√3 ≈ 5.196 cm
  5. Inradius (apothem): r = 6√3/6 = √3 ≈ 1.732 cm
  6. Circumradius: R = 6√3/3 = 2√3 ≈ 3.464 cm; note R = 2r exactly.

Every regular hexagon decomposes into 6 equilateral triangles meeting at the center, which is why the hexagon's apothem equals its inscribed equilateral triangle's height.

When to Use Each Variable

  • Solve for Area — when the side length is known — equilateral tiles, billiard rack, traffic-warning signs.
  • Solve for Perimeter — when you need the total boundary length.
  • Solve for Height — when you need the altitude for inscribed circles, stacking, or vertical clearance.
  • Solve for Side — when the area is known and you need the side length.

Key Concepts

An equilateral triangle is a special triangle with all three sides equal and all three angles equal to 60°. It's the only triangle that's both equilateral and equiangular. The height equals s√3/2 by the 30-60-90 right triangle formed when you drop a perpendicular from any vertex to the opposite side. The inradius (apothem) is exactly one-third the height; the circumradius is two-thirds. Equilateral triangles tile the plane perfectly (along with squares and regular hexagons) — they're one of the three regular polygon tilings.

Applications

  • Tiling: equilateral triangles tessellate the plane perfectly without gaps
  • Trusses: triangular frames provide structural rigidity (3 sides locked = no degrees of freedom)
  • Traffic warnings: caution signs are equilateral triangles by international convention
  • Geometry instruction: foundational shape; the 30-60-90 right triangle comes from cutting an equilateral in half

Common Mistakes

  • Using base × height / 2 with the wrong height — for equilateral, height is s√3/2, not s
  • Forgetting the √3 in the area formula — A = s² is wrong (that's a square)
  • Confusing inradius with circumradius — inradius = s√3/6, circumradius = s√3/3, so R = 2r
  • Treating the apothem and altitude as the same thing — for equilateral they're related (apothem = h/3), not equal

Frequently Asked Questions

How do you calculate the area of an equilateral triangle?

Use A = (√3/4) s². For s = 6 cm, A = 9√3 ≈ 15.588 cm².

What is the height of an equilateral triangle?

h = s√3/2. For s = 6 cm, h = 3√3 ≈ 5.196 cm. This comes from the 30-60-90 right triangle formed by dropping a perpendicular from any vertex.

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

P = 3s. For s = 6 cm, P = 18 cm.

What are the angles of an equilateral triangle?

All three angles equal 60°. This is the only triangle where all angles are equal. The sum of any triangle's angles is 180°, so 3 × 60° = 180° ✓.

How is an equilateral triangle related to a hexagon?

A regular hexagon decomposes into 6 equilateral triangles meeting at its center. This is why a regular hexagon's circumradius equals its side length and its apothem equals an equilateral triangle's height.

What is the inradius vs circumradius of an equilateral triangle?

Inradius r = s√3/6 (inscribed circle, touches all three sides). Circumradius R = s√3/3 (passes through all three vertices). R = 2r — the circumradius is exactly twice the inradius for an equilateral triangle.

Is an equilateral triangle the same as an isosceles triangle?

Every equilateral triangle is isosceles (because all isosceles triangles have at least two equal sides — equilateral has three). But not every isosceles triangle is equilateral.

Can an equilateral triangle be a right triangle?

No. A right triangle has one 90° angle, but every angle in an equilateral triangle is 60°. The two categories are disjoint.

Reference: Weisstein, Eric W. "Equilateral Triangle." MathWorld — A Wolfram Web Resource. https://mathworld.wolfram.com/EquilateralTriangle.html

Worked Examples

Tiling

How much surface does an equilateral 6 cm tile cover?

An equilateral triangular tile has 6 cm sides. Compute its area.

  • Knowns: s = 6 cm
  • Formula: A = (√3/4) s²
  • A = (√3/4) · 36 = 9√3 ≈ 15.588 cm²

Area ≈ 15.59 cm² per tile

Equilateral triangles tile the plane perfectly — they're one of three regular polygons that do.

Truss Design

What is the height of a 12 m equilateral truss?

An equilateral truss has 12 m sides. Find its height (altitude from base to apex).

  • Knowns: s = 12 m
  • Formula: h = s√3/2
  • h = 12 · √3 / 2 = 6√3 ≈ 10.392 m

Height ≈ 10.39 m

Equilateral truss heights follow the 30-60-90 right triangle: half the base is s/2, the height is s√3/2, and the slant (full side) is s.

Inverse Solve

What side does a 50 cm² equilateral triangle need?

Find the side length of an equilateral triangle with area exactly 50 cm².

  • Knowns: A = 50 cm²
  • Formula: s = √(4A / √3)
  • s = √(200 / √3) ≈ √115.47 ≈ 10.745 cm

Side ≈ 10.74 cm

Useful for sizing equilateral cutouts in fabrication or decorative panels.

Equilateral Triangle Formulas

A single side length s fully determines an equilateral triangle. All three angles are 60°.

Area equals root three over four s squaredPerimeter equals three sHeight equals s root three over twoSide equals the square root of 4A over root 3
Equilateral triangle with side s and height hhsss

Where:

  • A — area (m², ft², in²)
  • P — perimeter = 3s
  • h — altitude (height) = s√3/2
  • s — side length (all sides equal)
  • r — inradius (apothem) = s√3/6 (one-third of h)
  • R — circumradius = s√3/3 (two-thirds of h, and R = 2r)

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