Equilateral Triangle Calculator
Enter the length of one side and instantly get the area, perimeter, and height of your equilateral triangle. All three sides are equal, so one measurement is all you need.
Understanding Equilateral Triangle Geometry
Equilateral triangles are unique. All sides measure the same, all angles equal 60 degrees, and every property follows from symmetry. This makes calculations simpler than for general triangles because you only need one measurement to determine everything else.
The area formula A = (√3 / 4) × s² emerges from dropping a height from any vertex. That height bisects the base, creating two 30-60-90 right triangles. The height equals (√3 / 2) × s, and the base equals s, so the area becomes ½ × s × (√3 / 2) × s, which simplifies to the standard formula.
Because of this symmetry, the height, median, angle bisector, and perpendicular bisector from any vertex are all the same line. This triple coincidence makes equilateral triangles especially easy to analyze and construct.
Practical Applications of Equilateral Triangles
Equilateral triangles appear throughout engineering and design because they distribute forces evenly. Truss bridges use equilateral triangles for strength. Geodesic domes tessellate them to create stable curved surfaces. Even molecules like benzene have equilateral triangle symmetry.
In manufacturing, equilateral triangle patterns maximize packing efficiency for certain materials. Hexagonal tilings, which are built from six equilateral triangles meeting at a point, appear in honeycombs, floor tiles, and graphene lattices.
For construction, equilateral triangles are easy to lay out. Set a compass to the desired side length, draw a circle from one endpoint, then another circle from the other endpoint. The intersection points give you the third vertex. This compass-and-straightedge method has been known since ancient Greece.
Equilateral vs. Isosceles vs. Scalene
Equilateral triangles are a special case of isosceles triangles, which have at least two equal sides. An equilateral triangle has all three sides equal, making it maximally symmetric. Isosceles triangles have two equal sides but the third can differ. Scalene triangles have all three sides of different lengths.
The formulas differ accordingly. For equilateral triangles, you can derive area, perimeter, and height from a single side length. Isosceles triangles require the base and one equal side, or the base and height. Scalene triangles generally need all three sides or a combination of sides and angles.
This calculator exploits the simplicity of equilateral triangles. If your triangle has unequal sides, the formulas here will not apply. Always verify that all three sides are truly equal before using equilateral triangle formulas.
Frequently Asked Questions
What is an equilateral triangle?
An equilateral triangle has all three sides equal in length and all three angles equal to 60 degrees. It is the most symmetric type of triangle.
What is the area formula for an equilateral triangle?
The area equals (√3 / 4) × s², where s is the side length. This comes from the general triangle formula with the special 60-degree angles.
How do you find the height of an equilateral triangle?
The height is (√3 / 2) × s. Drop a perpendicular from any vertex to the opposite side, and it bisects that side while forming two 30-60-90 triangles.
Is the perimeter just three times the side?
Yes. Since all three sides are equal, the perimeter P = 3s.
Can I use this calculator for other triangle types?
No. This calculator applies only to equilateral triangles. For isosceles or scalene triangles, use the appropriate specialized calculator.