30-60-90 Triangle Calculator
A 30-60-90 triangle locks its sides into the ratio 1 : √3 : 2. Enter any one known value, select which measurement it is, and this calculator returns all three sides, the altitude, area, perimeter, inradius, and circumradius instantly, with a full show-your-work breakdown.
Formula
Worked example
Given a short leg a = 5 cm: b = 5 x sqrt(3) = 8.6603 cm, c = 10 cm, height h = 8.6603 / 2 = 4.3301 cm, area A = (5 x 8.6603) / 2 = 21.6506 cm^2, perimeter P = 5 + 8.6603 + 10 = 23.6603 cm, inradius r = 5 x (1.7321 - 1) / 2 = 1.8301 cm, circumradius R = 5 cm.
How the 30-60-90 ratio locks all measurements
A 30-60-90 triangle has angles of 30 degrees, 60 degrees, and 90 degrees, which force its three sides into the exact ratio 1 : sqrt(3) : 2, where the short leg a sits opposite 30 degrees, the long leg b = a times sqrt(3) sits opposite 60 degrees, and the hypotenuse c = 2a sits opposite the right angle. Because the shape is determined entirely by its angles, a single measured value determines every other length. This calculator accepts any of six starting values: the short leg, long leg, hypotenuse, altitude to the hypotenuse, the area, or the perimeter. Each formula is inverted to recover the short leg first, then the full triangle falls out from the standard ratios.
Altitude, inradius, and circumradius explained
The altitude h (also called the height) is the perpendicular segment from the right-angle vertex to the hypotenuse. In a 30-60-90 triangle it equals a times sqrt(3) divided by 2, which is also the geometric mean altitude because h squared equals the product of the two hypotenuse segments created by the foot of the altitude. The inradius r is the radius of the largest circle that fits inside the triangle; for a 30-60-90 it equals a times (sqrt(3) minus 1) divided by 2. The circumradius R is the radius of the circle passing through all three vertices; by Thales's theorem, R equals the hypotenuse divided by 2, which simplifies to R = a for a 30-60-90 triangle. These two circle radii appear frequently in geometry problems and in engineering applications where you need to fit a triangle inside or around a circular boundary.
Solving from area or perimeter
When neither a side nor the altitude is known but the area or perimeter is, the short leg can still be recovered algebraically. Because the area is A = (sqrt(3)/2) times a squared (derived from A = a x b / 2 with b = a x sqrt(3)), solving for a gives a = sqrt(2A / sqrt(3)). Because the perimeter is P = a times (3 plus sqrt(3)), solving for a gives a = P divided by (3 plus sqrt(3)). Once a is known all other measurements follow. This makes the calculator useful in design situations where the available area or the total boundary length is the constraint, not a linear measurement.
Where 30-60-90 triangles appear in practice
The 30-60-90 triangle is half of an equilateral triangle cut along its altitude, which is why it appears in equilateral-triangle problems and in regular hexagon geometry. In architecture and woodworking the 30-60-90 set square is one of the two standard drafting triangles. In trigonometry it provides exact values: sin 30 = 0.5, cos 30 = sqrt(3)/2, tan 30 = 1/sqrt(3); sin 60 = sqrt(3)/2, cos 60 = 0.5, tan 60 = sqrt(3). In physics it is the natural shape of a 30-degree inclined plane, and in electronics the phasor diagram of a balanced three-phase system contains 30-60-90 triangles.
30-60-90 triangle reference values
| Short leg a | Long leg b | Hypotenuse c | Height h | Area A | Perimeter P | Inradius r | Circumradius R |
|---|---|---|---|---|---|---|---|
| 1 | 1.7321 | 2 | 0.8660 | 0.8660 | 4.7321 | 0.3660 | 1 |
| 2 | 3.4641 | 4 | 1.7321 | 3.4641 | 9.4641 | 0.7321 | 2 |
| 5 | 8.6603 | 10 | 4.3301 | 21.6506 | 23.6603 | 1.8301 | 5 |
| 10 | 17.3205 | 20 | 8.6603 | 86.6025 | 47.3205 | 3.6603 | 10 |
| 12 | 20.7846 | 24 | 10.3923 | 124.7077 | 56.7846 | 4.3923 | 12 |
| 15 | 25.9808 | 30 | 12.9904 | 194.8557 | 70.9808 | 5.4904 | 15 |
All measurements from the short leg a using exact ratios (4 decimal places). Area = (sqrt(3)/2) x a^2 = a x b / 2.
Frequently asked questions
What is the side ratio of a 30-60-90 triangle?
The sides are always in the ratio 1 : sqrt(3) : 2, written as short leg : long leg : hypotenuse. The long leg equals the short leg times sqrt(3) (about 1.732), and the hypotenuse equals the short leg times 2. This ratio holds for every 30-60-90 triangle regardless of size.
How do you find the other sides if you only know the hypotenuse?
Divide the hypotenuse by 2 to get the short leg, then multiply that short leg by sqrt(3) to get the long leg. For example, a hypotenuse of 10 gives a short leg of 5 and a long leg of 5 x sqrt(3) = 8.66. Select "Hypotenuse c" from the dropdown and this calculator handles it automatically.
What is the area of a 30-60-90 triangle?
Area = half times the two legs = (a x b) / 2. Because b = a x sqrt(3), this simplifies to A = (sqrt(3)/2) x a^2. For a short leg of 5, the area is (sqrt(3)/2) x 25 = 21.65 square units.
What is the altitude (height) of a 30-60-90 triangle?
The altitude from the right-angle vertex to the hypotenuse is h = a x sqrt(3) / 2, where a is the short leg. For a short leg of 5, the height is 5 x 1.732 / 2 = 4.330. You can also enter the height directly and solve for all other measurements.
What are the inradius and circumradius of a 30-60-90 triangle?
The inradius (radius of the inscribed circle) is r = a x (sqrt(3) - 1) / 2, which equals about 0.366 times the short leg. The circumradius (radius of the circumscribed circle) is R = c / 2 = a, exactly equal to the short leg. For a short leg of 5, r = 1.830 and R = 5.
Can you solve a 30-60-90 triangle if you only know the area or perimeter?
Yes. If you know the area A, the short leg is a = sqrt(2A / sqrt(3)). If you know the perimeter P, the short leg is a = P / (3 + sqrt(3)). Select either option from the "Which value do you know?" dropdown and enter the value to get all side lengths and other measurements.