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Related Expert Tools
More precision tools in the same niche.
30-60-90 Triangle Calculator
The 30-60-90 Triangle Calculator computes all three side lengths of a 30-60-90 special right triangle when you enter any one known side. It applies the fixed 1 to square-root-of-3 to 2 ratio to find the remaining two sides instantly. Use it for geometry homework, architectural drafting, and trigonometry problems involving this widely used special triangle.
Area of a Right Triangle Calculator
Calculates the area of a right triangle using three methods: two legs (A = ab/2), base and height (A = bh/2), or hypotenuse and one angle. Also shows perimeter and hypotenuse where applicable.
Hypotenuse Calculator
Calculates the hypotenuse of a right triangle given the two legs using c = sqrt(a² + b²). Also outputs the two acute angles, area, and perimeter.
An equilateral triangle has three equal sides and three equal angles of 60 degrees each. Because every dimension flows from a single measurement, entering the side length is enough to work out the area, perimeter, height, inradius, and circumradius in one step. This calculator does exactly that. It also supports reverse calculations: if you know the height or area instead of the side, you can figure out the side length first and then compute everything else. Whether you are solving a geometry problem, planning a construction layout, or checking a design, this tool gives you every measurement without carrying out multiple manual calculations.
All Five Formulas Explained
Given that all three sides are equal (each has length a) and every angle is 60 degrees, a single variable drives all five key measurements. The area is A = (sqrt(3) / 4) x a squared. The height, which is the perpendicular distance from any vertex to the opposite side, is h = (sqrt(3) / 2) x a. The perimeter is simply P = 3a. The inradius (radius of the inscribed circle that touches all three sides from inside) is r = a / (2 x sqrt(3)), which simplifies to a x sqrt(3) / 6. The circumradius (radius of the circumscribed circle passing through all three vertices) is R = a / sqrt(3), or equivalently a x sqrt(3) / 3. On top of that, R is always exactly twice r for any equilateral triangle, a relationship that holds regardless of size. The Omni Calculator equilateral triangle tool provides an interactive version of these formulas with unit conversion support.
| Property | Formula (side = a) | Example: a = 10 | Approximate value |
|---|---|---|---|
| Area | (sqrt(3)/4) x a² | (sqrt(3)/4) x 100 | 43.301 |
| Perimeter | 3a | 30 | 30.000 |
| Height | (sqrt(3)/2) x a | (sqrt(3)/2) x 10 | 8.660 |
| Inradius | a x sqrt(3) / 6 | 10 x sqrt(3) / 6 | 2.887 |
| Circumradius | a x sqrt(3) / 3 | 10 x sqrt(3) / 3 | 5.774 |
How to Derive the Height Formula
Drawing the height from one vertex to the opposite side splits the equilateral triangle into two congruent right triangles. Each right triangle has a hypotenuse equal to a (the full side), a base of a/2 (half the opposite side), and a height h. Applying the Pythagorean theorem: h squared plus (a/2) squared equals a squared, so h squared equals a squared minus a squared over 4, giving h squared equals 3a squared over 4. Taking the square root: h = (sqrt(3)/2) x a. As a result, for a triangle with side 10, the height is 8.660. This derivation is based on the 30-60-90 right triangle that always appears when you build up the altitude of an equilateral triangle. Our Pythagorean Theorem Calculator handles the underlying right triangle calculation if you want to verify this step.
Working Backwards from Height or Area
A common question on Quora is how to find the area of an equilateral triangle when only the height is known, not the side. The reverse formula for side from height is a = 2h / sqrt(3). For example, if h = 8.66, then a = 2 x 8.66 / 1.732 = 10. Once you narrow down the side length from the height, enter it into this calculator to get area, perimeter, inradius, and circumradius instantly. Similarly, to find the side from the area: rearrange A = (sqrt(3)/4) x a squared to get a = sqrt(4A / sqrt(3)). For an area of 43.3, this gives a = sqrt(173.2 / 1.732) = sqrt(100) = 10. That said, these reverse formulas involve a square root, so rounding errors in the intermediate step can affect the final answer. Using the calculator lets you carry out the computation in full precision, eliminating any accumulated rounding error from manual intermediate steps.
Finding the Inradius and Circumradius
The inradius and circumradius locate the two circles associated with the triangle. The inscribed circle (inradius r) is the largest circle that fits entirely inside the triangle and touches all three sides. The circumscribed circle (circumradius R) passes through all three vertices. With that in mind, for an equilateral triangle the centres of both circles coincide at the centroid, which lies at one-third of the height from the base. The ratio R = 2r means that the circumscribed circle is always twice as large as the inscribed circle. Engineers use these values when designing triangular cross-sections, circular clamps, or hexagonal arrangements built from equilateral triangles. The CalculatorSoup equilateral triangle calculator and the Cuemath area of equilateral triangle guide both cover the inradius and circumradius with worked numerical examples. Our Area of Right Triangle Calculator is useful alongside this one when a problem involves a mix of equilateral and right triangles, such as when the altitude divides one equilateral triangle into two right triangles.