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Equilateral Triangle Calculator

Calculate area, perimeter, height, side length, inradius, and circumradius of an equilateral triangle. Free online geometry calculator with formula display.

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What is the Equilateral Triangle Calculator?

The Equilateral Triangle Calculator is a free online tool for computing all key properties of an equilateral triangle. Given a single side length, it instantly calculates the area, perimeter, height, inradius, circumradius, and fixed 60-degree angles. It also supports inverse calculation to find the side length from a known area. This tool uses the standard formulas: A = (√3/4)s² for area, P = 3s for perimeter, h = s√3/2 for height, inradius r = s√3/6, and circumradius R = s√3/3.

This calculator is essential for students learning geometry, architects designing triangular trusses, engineers working on structural frameworks, and anyone dealing with regular triangular shapes in tiling, signage, or manufacturing.

How to Use the Equilateral Triangle Calculator

Using the calculator is straightforward:

  1. Enter the side length of the equilateral triangle in your preferred unit (meters, centimeters, millimeters, inches, feet, or yards)
  2. The tool instantly computes all properties including area, perimeter, height, inradius, and circumradius
  3. Results update in real-time as you adjust the side length

Understanding the Equilateral Triangle Formula

An equilateral triangle is a regular polygon with three equal sides and three 60-degree angles. All its properties derive from a single side length s:

  • Area: A = (√3/4)s² — this is the space enclosed by the triangle
  • Perimeter: P = 3s — the total boundary length
  • Height: h = s√3/2 — the perpendicular distance from any vertex to the opposite side
  • Inradius: r = s√3/6 — the radius of the inscribed circle (touches all three sides)
  • Circumradius: R = s√3/3 — the radius of the circumscribed circle (passes through all three vertices); note R = 2r

The 30-60-90 right triangle formed by dropping a perpendicular from any vertex gives the height formula. The inradius is exactly one-third of the height, and the circumradius is two-thirds of the height.

Applications of Equilateral Triangle Calculator

  • Architecture: Designing triangular trusses and roof structures
  • Tiling: Equilateral triangles tessellate the plane perfectly without gaps
  • Traffic signage: Warning signs use equilateral triangles by international convention
  • Education: Teaching geometry and trigonometric relationships
  • Manufacturing: Cutting triangular components and decorative panels

Frequently Asked Questions

How do you calculate the area of an equilateral triangle?

Use the formula A = (√3/4)s², where s is the side length. For example, with s = 6 cm, A = (√3/4) × 36 = 9√3 ≈ 15.588 cm².

What is the height of an equilateral triangle?

The height is h = s√3/2. For a triangle with side 6 cm, h = 3√3 ≈ 5.196 cm. This comes from the 30-60-90 right triangle formed by the altitude.

What are the angles of an equilateral triangle?

All three angles equal 60 degrees. The sum of any triangle's angles is 180 degrees, so 3 × 60° = 180°.

What is the difference between inradius and circumradius?

Inradius r = s√3/6 is the radius of the inscribed circle that 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.

How is an equilateral triangle related to a regular hexagon?

A regular hexagon decomposes into 6 equilateral triangles meeting at its center. This is why the hexagon's circumradius equals its side length.

Is every equilateral triangle also isosceles?

Yes, every equilateral triangle is isosceles because all isosceles triangles have at least two equal sides. However, not every isosceles triangle is equilateral.

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