Equilateral Triangle Calculator
Our free triangle calculator solves equilateral triangle problems. Get worked examples, visual aids, and downloadable results.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
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 Height = 12 x sqrt(3) / 2 = 12 x 0.8660 = 10.3923 cm Area = 12^2 x sqrt(3) / 4 = 144 x 0.4330 = 62.3538 sq cm Perimeter = 3 x 12 = 36 cm Inradius = 12 / (2 x sqrt(3)) = 12 / 3.4641 = 3.4641 cm Circumradius = 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 side^2 = 400 / sqrt(3) = 400 / 1.7321 = 230.9401 side = sqrt(230.9401) = 15.1967 m Height = 15.1967 x sqrt(3) / 2 = 13.1607 m Perimeter = 3 x 15.1967 = 45.5901 m Inradius = 15.1967 / 3.4641 = 4.3869 m Circumradius = 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.
What are the real-world applications of equilateral triangles?
Equilateral triangles appear extensively in engineering, architecture, and design due to their structural stability and aesthetic appeal. The Warren truss bridge uses equilateral triangles for maximum load distribution. Geodesic domes, popularized by Buckminster Fuller, decompose spherical surfaces into equilateral triangles for extreme strength-to-weight ratios. Traffic yield signs are equilateral triangles in most countries. Crystal structures in materials science often exhibit equilateral triangular geometry. In telecommunications, cell tower placement follows equilateral triangle patterns for optimal coverage. Musical instruments like the triangle percussion instrument are shaped as equilateral triangles.
How do you construct an equilateral triangle with a compass and straightedge?
Constructing an equilateral triangle with compass and straightedge is one of the most basic geometric constructions. Start by drawing a line segment AB of the desired side length. Set the compass to the length AB. Place the compass point at A and draw an arc above the line. Without changing the compass width, place the point at B and draw another arc intersecting the first arc at point C. Connect A to C and B to C to complete the equilateral triangle. This works because AC = AB = BC, since all three distances equal the compass radius, ensuring all sides are equal.
What is the relationship between an equilateral triangle and the number sqrt(3)?
The number sqrt(3) (approximately 1.7321) is intimately connected to equilateral triangle geometry. The height equals side times sqrt(3)/2. The area equals side squared times sqrt(3)/4. The circumradius equals side times sqrt(3)/3. The inradius equals side times sqrt(3)/6. The tangent of 60 degrees equals sqrt(3), and the sine of 60 degrees equals sqrt(3)/2. The ratio of the diagonal to the side in a regular hexagon (composed of equilateral triangles) is sqrt(3). This constant appears because the 30-60-90 triangle, which forms half of any equilateral triangle, inherently involves sqrt(3) in its side ratios.
How does the equilateral triangle optimize certain geometric problems?
The equilateral triangle is the solution to several important optimization problems in geometry. Among all triangles with a given perimeter, the equilateral triangle has the maximum area, a result of the isoperimetric inequality for triangles. Among all triangles with a given area, the equilateral triangle has the minimum perimeter. It has the minimum circumradius-to-inradius ratio (exactly 2) among all triangles. The equilateral triangle also minimizes the ratio of circumscribed to inscribed circle areas. For the Fermat point problem (minimizing total distance to three points), the equilateral triangle gives the simplest solution with the center as the optimal point.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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