Calculate all sides, area, perimeter, altitudes, and radii of a 30-60-90 special right triangle. Enter any one side and get every property instantly.
The 30-60-90 triangle is one of the two "special" right triangles in geometry (the other being the 45-45-90). Its fixed side ratios of 1 : √3 : 2 let you compute every dimension from a single known side — no trigonometry tables required. The short leg (opposite 30°) is half the hypotenuse, the long leg (opposite 60°) is √3 times the short leg, and the hypotenuse is twice the short leg.
This triangle appears everywhere: it is exactly half of an equilateral triangle split along its altitude. Engineers encounter it in structural trusses, roof pitches, and hexagonal geometry. Surveyors use the ratio to compute inaccessible heights, and physicists apply it when resolving forces at 30° or 60° angles.
Beyond the three sides, this calculator derives the full suite of properties: area, perimeter, semi-perimeter, all three altitudes, the circumradius (R), the inradius (r), and the medians. A visual ratio comparison bar chart and a reference table of common 30-60-90 triangles make exploring the relationships intuitive. Choose a preset or enter any single side — short leg, long leg, or hypotenuse — and see every measurement update instantly.
Knowing the 30-60-90 ratios by heart is a powerful shortcut for standardized tests (SAT, GRE, ACT) as well as practical tasks in carpentry, machining, and graphic design.
The 30-60-90 triangle appears so often that it is worth treating as a geometry shortcut instead of a full trigonometry problem. This calculator is useful when you want to move quickly from one known side to the entire triangle, including the area, altitudes, medians, and circle radii. It is especially practical for test prep, drafting, surveying setups, and any design work involving equilateral triangles, hexagons, or 60-degree layouts where the fixed 1 : √3 : 2 ratio saves time.
Ratios — short : long : hypotenuse = 1 : √3 : 2 Short leg: a Long leg: b = a√3 Hypotenuse: c = 2a Area: A = (a × b) / 2 = (a² √3) / 2 Perimeter: P = a + a√3 + 2a = a(3 + √3) Altitude to hypotenuse: h_c = (a × b) / c = (a√3) / 2 Circumradius: R = c / 2 = a Inradius: r = (a + b − c) / 2
Result: Short ≈ 6.93 cm, Long = 12 cm, Hyp ≈ 13.86 cm, Area ≈ 41.57 cm²
If the known side is the long leg and value = 12 cm, the short leg is 12 / √3 ≈ 6.93 cm. The hypotenuse is 2 × 6.93 ≈ 13.86 cm. Area = (6.93 × 12) / 2 ≈ 41.57 cm², and the perimeter is about 6.93 + 12 + 13.86 = 32.78 cm.
A 30-60-90 triangle is completely determined by a single side because the other two sides are locked into the ratio 1 : √3 : 2. That means every property downstream, including area, perimeter, inradius, circumradius, and the altitude to the hypotenuse, can be written in terms of one measurement. This is why the shape is such a standard shortcut in geometry courses and technical drawing: once you identify the triangle type, most of the work is already done.
Splitting an equilateral triangle down the middle creates two congruent 30-60-90 triangles. The same structure appears repeatedly in regular hexagons, bolt-circle layouts, triangular trusses, and honeycomb-style designs. If you know the side of a hexagon or the altitude of an equilateral panel, you can often reduce the problem to this special triangle and solve it with simple multiplication instead of full trigonometric tables.
A quick consistency check is to compare the known values against the special-triangle pattern. The hypotenuse should be exactly twice the short leg, and the long leg should be about 1.732 times the short leg. If those relationships do not match your manual work, either the triangle is not 30-60-90 or one of the numbers was entered incorrectly. That makes this calculator useful both for solving new problems and for verifying hand calculations before you move on to a larger design or proof.
The sides are always in the ratio 1 : √3 : 2. The shortest side is opposite the 30° angle, the medium side is opposite 60°, and the longest side (hypotenuse) is opposite the 90° angle.
Multiply the short leg by 2. If the short leg is 7, the hypotenuse is 14. If you know the long leg, divide it by √3 to get the short leg first.
Area = (short leg² × √3) / 2. For example, if the short leg is 6, area = (36 × 1.732) / 2 ≈ 31.18 square units.
If you draw an altitude in an equilateral triangle, it splits into two congruent 30-60-90 triangles. The equilateral side becomes the hypotenuse, and the altitude becomes the long leg.
They appear in hexagonal geometry (bolts, tiles, honeycombs), roof trusses, surveying, force resolution at 30° or 60° angles, and standardized math tests. Use this as a practical reminder before finalizing the result.
The circumradius R = hypotenuse / 2 = short leg. So if the short leg is 5, R = 5 and the hypotenuse is 10.