Equilateral Triangle Calculator
Our free triangle calculator solves equilateral triangle problems. Get worked examples, visual aids, and downloadable results.
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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.
Last reviewed: December 2025
Worked Examples
Example 1: Complete Properties from Side Length
Example 2: Finding Side Length from Area
Background & Theory
The Equilateral Triangle Calculator applies the following established principles and formulas. Mathematics rests on a hierarchy of number systems, each extending the previous. The natural numbers (1, 2, 3, ...) support counting and ordering. The integers add negative values and zero, enabling subtraction without restriction. The rational numbers, expressible as p/q where p and q are integers and q is nonzero, close the system under division. The real numbers fill the gaps left by irrationals such as the square root of 2 or pi, forming a complete ordered field. The complex numbers, written as a + bi where i is the square root of negative one, complete the algebraic closure of the reals and allow every polynomial to have a root. Prime factorization states that every integer greater than one is uniquely expressible as a product of primes, a result known as the Fundamental Theorem of Arithmetic. Computing the greatest common divisor (GCD) of two integers relies most efficiently on the Euclidean algorithm: repeatedly replace the larger number with the remainder when it is divided by the smaller, until the remainder is zero. The last nonzero remainder is the GCD. The least common multiple (LCM) follows from the identity LCM(a, b) = |a * b| / GCD(a, b). Modular arithmetic defines equivalence classes of integers that share the same remainder under division by a modulus n. Fermat's Little Theorem and Euler's Theorem arise from this structure and underpin modern cryptography. Logarithms are the inverses of exponential functions. If b raised to the power x equals y, then the logarithm base b of y equals x. The natural logarithm uses base e, approximately 2.71828. Combinatorics counts arrangements and selections. The number of ordered arrangements (permutations) of r objects from n distinct objects is nPr = n! / (n - r)!. The number of unordered selections (combinations) is nCr = n! / (r! * (n - r)!). Pascal's triangle arranges these binomial coefficients so that each entry equals the sum of the two entries directly above it. The Fibonacci sequence, defined by F(1) = 1, F(2) = 1, and F(n) = F(n-1) + F(n-2), appears throughout nature and connects deeply to the golden ratio via Binet's formula.
History
The history behind the Equilateral Triangle Calculator traces back through the following developments. Mathematics as a systematic discipline traces to ancient Mesopotamia. Babylonian clay tablets dating to around 1800 BCE demonstrate knowledge of quadratic equations, Pythagorean triples, and base-60 arithmetic, suggesting a practical mathematical tradition far preceding Greek formalism. Euclid of Alexandria compiled the Elements around 300 BCE, establishing the axiomatic method that would define rigorous mathematics for over two thousand years. His work organized plane geometry, number theory, and proportion into logically chained propositions derived from a small set of postulates. The algorithm bearing his name for computing GCDs appears in Book VII and remains in use today. In the 9th century, the Persian scholar Muhammad ibn Musa Al-Khwarizmi wrote Al-Kitab al-mukhtasar fi hisab al-jabr wal-muqabala, the treatise whose title gave algebra its name. He systematized the solution of linear and quadratic equations and described procedures that operated on unknowns as objects, a conceptual leap away from purely numerical calculation. Rene Descartes introduced coordinate geometry in 1637 by uniting algebra and Euclidean geometry, allowing curves to be studied through equations. This synthesis set the stage for calculus. Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus during the 1660s and 1670s, triggering a priority dispute that lasted decades and divided British and Continental mathematicians. Carl Friedrich Gauss proved the Fundamental Theorem of Algebra in 1799, showing that every nonconstant polynomial has at least one complex root. His Disquisitiones Arithmeticae of 1801 established modern number theory. David Hilbert's formalist program at the turn of the 20th century sought to place all of mathematics on an explicit axiomatic foundation, a project that Kurt Godel's incompleteness theorems of 1931 showed to be fundamentally limited. Alan Turing's work in the 1930s on computability introduced the theoretical model of the stored-program computer and linked mathematical logic directly to the limits of algorithmic calculation. His proof that no algorithm can decide in general whether an arbitrary program will halt or run forever placed fundamental boundaries on what mathematics can mechanically determine, and it opened the discipline now known as theoretical computer science.
Frequently Asked Questions
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.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy