Equilateral Triangle Calculator
Our free triangle calculator solves equilateral triangle problems. Get worked examples, visual aids, and downloadable results.
Formula
Area = s^2 x sqrt(3) / 4, Height = s x sqrt(3) / 2
Where s is the side length. All sides equal s, all angles equal 60 degrees. The circumradius R = s / sqrt(3) and inradius r = s / (2 sqrt(3)), with R = 2r always.
Worked Examples
Example 1: Complete Properties from Side Length
Problem: An equilateral triangle has a side length of 12 cm. Find its height, area, perimeter, inradius, and circumradius.
Solution: Side = 12 cm\nHeight = 12 x sqrt(3) / 2 = 12 x 0.8660 = 10.3923 cm\nArea = 12^2 x sqrt(3) / 4 = 144 x 0.4330 = 62.3538 sq cm\nPerimeter = 3 x 12 = 36 cm\nInradius = 12 / (2 x sqrt(3)) = 12 / 3.4641 = 3.4641 cm\nCircumradius = 12 / sqrt(3) = 6.9282 cm
Result: Height = 10.3923 cm | Area = 62.3538 sq cm | Inradius = 3.4641 cm | Circumradius = 6.9282 cm
Example 2: Finding Side Length from Area
Problem: An equilateral triangle has an area of 100 square meters. Find the side length and all other properties.
Solution: Area = side^2 x sqrt(3) / 4 = 100\nside^2 = 400 / sqrt(3) = 400 / 1.7321 = 230.9401\nside = sqrt(230.9401) = 15.1967 m\nHeight = 15.1967 x sqrt(3) / 2 = 13.1607 m\nPerimeter = 3 x 15.1967 = 45.5901 m\nInradius = 15.1967 / 3.4641 = 4.3869 m\nCircumradius = 15.1967 / 1.7321 = 8.7738 m
Result: Side = 15.1967 m | Height = 13.1607 m | Perimeter = 45.5901 m
Frequently Asked Questions
What is an equilateral triangle and what are its key properties?
An equilateral triangle is a triangle in which all three sides have equal length and all three interior angles measure exactly 60 degrees. It is the most symmetric type of triangle, possessing three lines of symmetry and rotational symmetry of order three. In an equilateral triangle, every altitude is also a median, angle bisector, and perpendicular bisector, which means all four major triangle centers (centroid, circumcenter, incenter, orthocenter) coincide at the same point. The circumradius is exactly twice the inradius, the minimum possible ratio for any triangle.
How do you calculate the area of an equilateral triangle?
The area of an equilateral triangle with side length s is given by the formula: Area = (s squared times sqrt(3)) / 4. This formula is derived from the general triangle area formula (1/2 times base times height) where the height of an equilateral triangle is s times sqrt(3) / 2. Substituting: Area = (1/2) times s times (s times sqrt(3) / 2) = s squared times sqrt(3) / 4. For a side length of 10, the area equals 100 times sqrt(3) / 4 = 25 times sqrt(3), which is approximately 43.301 square units.
What is the height of an equilateral triangle?
The height (altitude) of an equilateral triangle with side s equals s times sqrt(3) / 2, which is approximately 0.866 times the side length. This can be derived by splitting the equilateral triangle into two congruent 30-60-90 right triangles along the altitude. The base of each right triangle is s/2, the hypotenuse is s, and the height is found using the Pythagorean theorem: h = sqrt(s squared - (s/2) squared) = sqrt(3s squared / 4) = s times sqrt(3) / 2. The altitude, median, angle bisector, and perpendicular bisector all coincide in an equilateral triangle.
What is the circumradius and inradius of an equilateral triangle?
For an equilateral triangle with side length s, the circumradius R (radius of the circumscribed circle) equals s / sqrt(3), or equivalently s times sqrt(3) / 3, which is approximately 0.5774 times s. The inradius r (radius of the inscribed circle) equals s / (2 times sqrt(3)), or equivalently s times sqrt(3) / 6, which is approximately 0.2887 times s. The circumradius is exactly twice the inradius (R = 2r), a unique property of equilateral triangles. The center of both circles is the same point, the centroid of the triangle.
How do equilateral triangles tile the plane?
Equilateral triangles are one of only three regular polygons that can tile (tessellate) the Euclidean plane without gaps or overlaps, the other two being squares and regular hexagons. Six equilateral triangles meet at each vertex, since 6 times 60 degrees = 360 degrees. This tiling has been used in art, architecture, and flooring since ancient times. Two equilateral triangles placed base-to-base form a rhombus, and six form a regular hexagon. The equilateral triangle tiling is the dual of the regular hexagonal tiling, meaning each generates the other by connecting centers of adjacent tiles.
How is an equilateral triangle related to other geometric shapes?
The equilateral triangle has deep connections to many other geometric shapes. Six equilateral triangles form a regular hexagon. The equilateral triangle is the face of a regular tetrahedron, octahedron, and icosahedron (three of the five Platonic solids). The Star of David (hexagram) consists of two overlapping equilateral triangles. In a regular hexagonal grid, connecting alternate vertices creates equilateral triangles. The Sierpinski triangle fractal is constructed from equilateral triangles. The Reuleaux triangle, formed from arcs centered at equilateral triangle vertices, is a curve of constant width used in drill bits.