Equilateral Triangle Calculator
An equilateral triangle has three equal sides and three 60 degree angles, so knowing any one property fixes the entire shape. Choose what you know, enter the value, and the calculator solves for all six properties, including the inscribed circle radius and the circumscribed circle radius.
Formula
Worked example
For a side a = 6: height = (sqrt(3)/2) x 6 = 5.1962, area = (sqrt(3)/4) x 36 = 15.5885, perimeter = 18, semiperimeter = 9, inradius = (sqrt(3)/6) x 6 = 1.7321, circumradius = (sqrt(3)/3) x 6 = 3.4641. Note R = 2r always holds.
How the equilateral triangle formulas work
An equilateral triangle is the most symmetric triangle: all three sides share the same length a, and all three interior angles equal exactly 60 degrees. Drop a perpendicular from any vertex to the opposite side and it splits the triangle into two identical 30-60-90 right triangles, each with a short leg of a/2. The Pythagorean theorem gives the height as h = sqrt(a^2 - (a/2)^2) = sqrt(3a^2/4) = (sqrt(3)/2) x a. Area follows from the standard formula half base times height: (1/2) x a x (sqrt(3)/2) x a = (sqrt(3)/4) x a^2. The perimeter is simply 3a, and the semiperimeter is half of that. The constant sqrt(3)/4 = 0.4330 is worth memorizing as the area-per-unit-side-squared for any equilateral triangle.
Inradius, circumradius, and why one value is enough
Every equilateral triangle has two notable circles. The incircle is the largest circle that fits inside and touches all three sides; its radius r = (sqrt(3)/6) x a = a / (2 x sqrt(3)). The circumcircle passes through all three vertices; its radius R = (sqrt(3)/3) x a = a / sqrt(3). A clean relationship always holds: R = 2r, meaning the circumscribed circle is exactly twice the size of the inscribed one. Because all equilateral triangles are similar to each other, knowing any single property, whether that is the side, height, area, perimeter, inradius, or circumradius, is enough to compute all the others. The calculator supports all six starting points so you can work with whatever measurement is most convenient.
Scaling relationships worth knowing
The side enters the area as a square but the perimeter linearly, so the two scale at different rates. Triple the side and the perimeter also triples, but the area becomes nine times larger. The height, inradius, and circumradius all scale linearly with the side, so they too triple. These scaling laws are useful for estimation: if you know an equilateral triangle with side 1 cm has an area of about 0.433 cm^2, then one with side 10 cm has an area of 43.3 cm^2 without any re-calculation.
Equilateral triangle quick reference
| Side (a) | Height | Area | Perimeter | Inradius (r) | Circumradius (R) |
|---|---|---|---|---|---|
| 1 | 0.8660 | 0.4330 | 3 | 0.2887 | 0.5774 |
| 4 | 3.4641 | 6.9282 | 12 | 1.1547 | 2.3094 |
| 6 | 5.1962 | 15.5885 | 18 | 1.7321 | 3.4641 |
| 10 | 8.6603 | 43.3013 | 30 | 2.8868 | 5.7735 |
| 20 | 17.3205 | 173.2051 | 60 | 5.7735 | 11.5470 |
| 100 | 86.6025 | 4330.1270 | 300 | 28.8675 | 57.7350 |
All values rounded to 4 decimal places. R = circumradius, r = inradius.
Frequently asked questions
How do you find the area of an equilateral triangle from one side?
Use A = (sqrt(3)/4) x a^2, where a is the side length. The factor sqrt(3)/4 = 0.4330 is a fixed constant for any equilateral triangle. For a side of 6, the area is 0.4330 x 36 = 15.59 square units. If you know a different property, this calculator can work backwards to find the side first, then the area.
What is the height of an equilateral triangle?
The height (altitude from any vertex to the opposite side) is h = (sqrt(3)/2) x a, where a is the side. Splitting the triangle down the middle creates a 30-60-90 right triangle, and the Pythagorean theorem gives this height. For a side of 6, h = 0.8660 x 6 = 5.196. To reverse-solve for the side from the height, use a = 2h / sqrt(3).
What are the inradius and circumradius of an equilateral triangle?
The inradius r is the radius of the largest circle that fits inside the triangle and touches all three sides: r = a x sqrt(3)/6 = a / (2 x sqrt(3)). The circumradius R is the radius of the circle passing through all three vertices: R = a x sqrt(3)/3 = a / sqrt(3). In an equilateral triangle, R is always exactly twice r. For a side of 6: r = 1.732, R = 3.464.
Can I calculate the side from the area or perimeter?
Yes. From the perimeter P, the side is simply a = P / 3. From the area A, the side is a = sqrt(4A / sqrt(3)). From the height h, the side is a = 2h / sqrt(3). From the inradius r, the side is a = 6r / sqrt(3). From the circumradius R, the side is a = 3R / sqrt(3). Use the "I know the" dropdown to pick your starting measurement and this calculator handles the algebra for you.
Are all the angles in an equilateral triangle the same?
Yes. An equilateral triangle has three equal angles, each exactly 60 degrees, summing to the 180 degrees required of any triangle. Equal sides force equal angles, which is why a single measurement is enough to determine the entire geometry.