Special Right Triangles Calculator
Enter one side length and choose the triangle type to solve every property of any special right triangle. The calculator handles 45-45-90, 30-60-90, the 3-4-5 Pythagorean triple, and two more classic types. It returns all three sides, both acute angles, the area, and the perimeter, with a shape diagram and a full worked-step panel showing the arithmetic.
What are special right triangles?
A special right triangle is a right triangle whose angles or sides follow a predictable pattern, so you can find all sides from just one known value without solving the Pythagorean theorem from scratch. The two most famous are the 45-45-90 (the isosceles right triangle) and the 30-60-90 (half of an equilateral triangle). A third classic family is Pythagorean triples, where all three sides are whole numbers: 3-4-5 is the simplest example. Because their ratios are fixed, these triangles appear constantly in geometry, trigonometry, architecture, and engineering wherever precise angles matter.
45-45-90 triangle: the isosceles right triangle
When both acute angles are 45 degrees the two legs are equal, and the hypotenuse is exactly leg x sqrt(2). Cut a square along a diagonal and each half is a 45-45-90 triangle. If a leg has length a, the ratio is a : a : a*sqrt(2). Common uses include roof framing at 45 degrees, tile layouts, and square-diagonal distance problems in design. Worked example: if leg a = 7 cm, then leg b = 7 cm and hypotenuse c = 7 * sqrt(2) = 9.8995 cm. Area = (1/2) * 7 * 7 = 24.5 cm^2.
30-60-90 triangle: half an equilateral triangle
Slice an equilateral triangle with an altitude and each half is a 30-60-90 triangle. The angles are 30, 60, and 90 degrees. The short leg (opposite 30 deg) is a, the long leg (opposite 60 deg) is a*sqrt(3), and the hypotenuse (opposite 90 deg) is 2a. Ratios: 1 : sqrt(3) : 2. This triangle appears in the unit circle at 30 deg and 60 deg, making it essential for trigonometry. Worked example: if the short leg a = 5 m, then long leg b = 5*sqrt(3) = 8.6603 m, hypotenuse c = 10 m. Area = (1/2)*5*8.6603 = 21.6506 m^2.
Pythagorean triples and other special types
A Pythagorean triple is a set of three positive integers (a, b, c) satisfying a^2 + b^2 = c^2. The simplest is 3-4-5: check 9 + 16 = 25. You can scale it by any factor k to get sides 3k, 4k, 5k. Other triples include 5-12-13, 8-15-17, and 7-24-25. Two irrational-hypotenuse types are also common in geometry: 1-2-sqrt(5) (legs in ratio 1 : 2, useful for slope problems) and 1-3-sqrt(10) (appears in constructions related to the square root of 10). For these types this calculator applies the same ratio logic: enter any one side and all other dimensions are derived instantly.
Special right triangle side ratios
| Type | Short leg (a) | Long leg (b) | Hypotenuse (c) | Angle A | Angle B |
|---|---|---|---|---|---|
| 45-45-90 | 1 | 1 | sqrt(2) ~ 1.4142 | 45 deg | 45 deg |
| 30-60-90 | 1 | sqrt(3) ~ 1.7321 | 2 | 30 deg | 60 deg |
| 3-4-5 | 3 | 4 | 5 | ~36.87 deg | ~53.13 deg |
| 1-2-sqrt(5) | 1 | 2 | sqrt(5) ~ 2.2361 | ~26.57 deg | ~63.43 deg |
| 1-3-sqrt(10) | 1 | 3 | sqrt(10) ~ 3.1623 | ~18.43 deg | ~71.57 deg |
Quick-reference for all five triangle types. Multiply the ratio by any scale factor k to get the actual side lengths.
Frequently asked questions
How do I find the hypotenuse of a 45-45-90 triangle?
Multiply either leg by sqrt(2). If each leg is length a, the hypotenuse is a * sqrt(2) = a * 1.41421. Example: leg = 10 in gives hypotenuse = 10 * 1.41421 = 14.1421 in. You can also enter the leg in this calculator and select "45-45-90" to get the result instantly.
What are the side ratios of a 30-60-90 triangle?
The sides are in ratio 1 : sqrt(3) : 2 (short leg : long leg : hypotenuse). If the short leg is a, then the long leg is a * sqrt(3) = a * 1.7321, and the hypotenuse is 2a. You can enter any of the three sides and this calculator derives the other two.
Can I enter the hypotenuse and solve backwards?
Yes. In the "Known side" drop-down, choose "Hypotenuse (c)", enter its length, and the calculator solves for both legs along with the area and perimeter. This works for all five triangle types.
What is a Pythagorean triple?
A Pythagorean triple is a set of three positive integers that satisfy the Pythagorean theorem: a^2 + b^2 = c^2. The classic example is 3-4-5 because 9 + 16 = 25. Multiples also work: 6-8-10, 9-12-15, and so on. The 3-4-5 option in this calculator scales the triple by any factor you choose.
How is the area of a special right triangle calculated?
Area = (1/2) x base x height. For a right triangle, the two legs serve as the base and height, so Area = (1/2) x a x b, where a and b are the two legs. For a 45-45-90 triangle with leg a: Area = (1/2) x a x a = a^2 / 2. For a 30-60-90 triangle with short leg a: Area = (1/2) x a x a*sqrt(3) = (a^2 * sqrt(3)) / 2.
What units does this calculator support?
The calculator supports meters, centimeters, millimeters, feet, inches, and yards. All outputs (sides, area, perimeter) use the same unit you select. Area is shown in square units of the chosen length unit.