How do you find the area of an equilateral triangle?

Area = (sqrt(3) / 4) x side squared. For side 6: area = (sqrt(3) / 4) x 36 = 9 x sqrt(3) = approximately 15.59 square units. Every interior angle of an equilateral triangle is 60 degrees.

What is the height of an equilateral triangle?

Height = (sqrt(3) / 2) x side. For side 6: height = 3 x sqrt(3) = approximately 5.20 units. The height bisects the base and creates two 30-60-90 right triangles.

What are the inradius and circumradius of an equilateral triangle?

The inradius (largest circle fitting inside) = side / (2 x sqrt(3)). The circumradius (circle through all three corners) = side / sqrt(3). For side 6: inradius = sqrt(3) = 1.73, circumradius = 2 x sqrt(3) = 3.46. The circumradius is always exactly twice the inradius.

Equilateral Triangle Calculator

An equilateral triangle is the most symmetrical triangle possible: all three sides are the same length, all three interior angles are exactly 60 degrees, and every line of symmetry maps one vertex onto another. Because of this perfect regularity, a single number, the side length, determines every other measurement. This calculator takes the side length and returns the area, height, perimeter, circumradius, and inradius, all updating as you type. The area formula, square root of three over four times the side squared, is a classic result in geometry worth knowing by heart. The height is the perpendicular distance from any side to the opposite vertex, equal to the side times root three over two. This height splits the equilateral triangle into two 30-60-90 right triangles, which is the geometric origin of all those root-three terms. The perimeter is simply three times the side. The circumradius is the radius of the circle that passes through all three vertices, and the inradius is the radius of the largest circle that fits inside touching all three sides. For an equilateral triangle the circumradius is exactly twice the inradius, a clean relationship not shared by other triangles. The tool is useful for students learning plane geometry and checking homework, for engineers and designers working with triangular bracing, trusses, or floor tiles, and for anyone who needs to quickly scale a triangular shape. Enter any positive side length and all five results appear instantly.

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Side length (s)

units

15.59

area (square units)

Height5.20

Perimeter18

Circumradius3.46

Inradius1.73

How it works

For an equilateral triangle with side s: area = (√3 / 4) × s²; height = (√3 / 2) × s; perimeter = 3s; circumradius = s / √3; inradius = s / (2√3). Every interior angle is 60 degrees. The circumradius is twice the inradius for any equilateral triangle.

Worked example

With side 6: area = (√3 / 4) × 36 = 9√3 ≈ 15.59 square units. Height = (√3 / 2) × 6 = 3√3 ≈ 5.20 units. Perimeter = 3 × 6 = 18 units. Circumradius = 6 / √3 = 2√3 ≈ 3.46 units. Inradius = 6 / (2√3) = √3 ≈ 1.73 units. These match the pre-filled defaults above.

Related calculators

Isosceles Triangle Calculator: area, height, angles and perimeter of an isosceles triangle.
Square Calculator: area, perimeter and diagonal of a square.
Triangular Prism Calculator: volume and surface area of a prism with any triangular cross-section.
Law of Sines and Cosines: solve any triangle from sides and angles.

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