Equilateral Triangle Calculator

An equilateral triangle is a triangle in which all three sides have the same length. As a direct consequence of equal sides, all three interior angles are also equal, each measuring exactly 60 degrees. This combination of equal sides and equal angles gives the equilateral triangle the highest degree of symmetry among all triangles, and it is the only triangle that is also a regular polygon. Because all properties of an equilateral triangle derive from a single measurement — the side length — every dimension can be computed from one input. The area is (√3/4) × s², where s is the side length. This formula can be derived by applying the general triangle area formula (½ × base × height) after first finding the height. The perimeter is simply 3s, since all three sides are equal. The height (also called the altitude) of an equilateral triangle is the perpendicular distance from one vertex to the opposite side. It equals (√3/2) × s. This value comes directly from the Pythagorean theorem: the altitude bisects the base into two segments of length s/2, and the altitude h satisfies h² + (s/2)² = s², giving h = s√3/2 ≈ 0.866s. The inradius is the radius of the largest circle that fits inside the triangle (the inscribed circle). For an equilateral triangle, the inradius equals s√3/6 ≈ 0.289s. The circumradius is the radius of the smallest circle that passes through all three vertices (the circumscribed circle). It equals s√3/3 ≈ 0.577s. An important relationship: the circumradius is exactly twice the inradius for any equilateral triangle, and the centroid, incenter, circumcenter, and orthocenter all coincide at the same point. The √3 constant that appears throughout equilateral triangle formulas is the tangent of 60° and the square root of 3 (approximately 1.7321). Its prevalence is a consequence of all angles being 60°, making sine and cosine of 60° equal to √3/2 and 1/2 respectively. Equilateral triangles appear extensively in nature and human design. In chemistry, many molecules adopt trigonal planar geometry with 120° bond angles, which corresponds to a regular arrangement around a central atom. In engineering, triangular frameworks are the basis of structural trusses because a triangle is the only polygon that cannot change shape without changing side lengths. Equilateral triangles in particular provide maximum structural efficiency. In art and design, the perfect symmetry of equilateral triangles makes them staples of tessellation patterns, logos, and decorative motifs. Geodesic domes use networks of equilateral triangles to create self-supporting curved structures with minimal material. For practical applications, the calculator handles any positive side length — whether an integer like 6, a decimal like 4.5, or a large value like 100 — and returns results accurate to ten significant digits. All five output values update simultaneously so you can compare them at a glance.