Is it a Right Triangle? Calculator
Enter the measurements you know and find out immediately whether your triangle has a 90-degree angle. Choose from three input modes: three side lengths (Pythagorean theorem check), two non-right angles (they must add to 90 degrees), or two sides and one angle (trigonometric check). The calculator shows every step of the working so you can see exactly why the answer is yes or no.
What makes a triangle a right triangle?
A right triangle is a triangle that has exactly one interior angle equal to 90 degrees. That 90-degree angle is called the right angle, the side opposite it is the hypotenuse (always the longest side), and the other two sides are called the legs. The three angles of any triangle must add to 180 degrees, so in a right triangle the two remaining angles must add to exactly 90 degrees - they are complementary. This simple relationship is the basis for all three tests this calculator can perform.
The Pythagorean theorem test (three sides)
When you know all three side lengths, the Pythagorean theorem provides the definitive test: if the square of the longest side equals the sum of the squares of the other two sides, the triangle is a right triangle. Written as a formula: c squared = a squared + b squared, where c is the longest side. Integer sets of side lengths that satisfy this are called Pythagorean triples; the most famous is 3-4-5 (because 9 + 16 = 25). Any multiple of a Pythagorean triple also works: 6-8-10, 9-12-15, 30-40-50. The table below lists several common triples for quick reference. When working with measured or computed values rather than exact integers, small rounding errors are normal - the Pythagorean deficit (a squared + b squared minus c squared) should be very close to zero, not necessarily exactly zero.
Special right triangles: 45-45-90 and 30-60-90
Two right triangles appear so often in geometry and trigonometry that they have their own names. The 45-45-90 triangle is an isosceles right triangle: both legs are equal, and the hypotenuse is the leg length multiplied by the square root of 2. For example, legs of 1 unit each give a hypotenuse of about 1.414 units. The 30-60-90 triangle has sides in the ratio 1 to the square root of 3 to 2: the shortest leg is opposite the 30-degree angle, the longer leg is opposite the 60-degree angle, and the hypotenuse is twice the shortest leg. These ratios make it easy to find unknown sides without the full Pythagorean calculation.
Testing with angles or mixed input
If you know two of the three angles rather than the sides, use the angle test: add the two known angles and check whether they sum to 90 degrees. If so, the third angle is 90 degrees and you have a right triangle. If you know two sides and one angle, the calculator applies the Law of Sines to derive all three angles, then checks whether any of them equals 90 degrees. This is the most flexible mode and handles any combination of leg-leg-angle or leg-hypotenuse-angle. Note that for the angle and mixed modes the calculator cannot compute an area unless the absolute side lengths are provided, so the area and perimeter outputs are shown only when all three side lengths are known or derivable.
Common Pythagorean triples
| Side a | Side b | Hypotenuse c | a² + b² | c² |
|---|---|---|---|---|
| 3 | 4 | 5 | 25 | 25 |
| 5 | 12 | 13 | 169 | 169 |
| 8 | 15 | 17 | 289 | 289 |
| 7 | 24 | 25 | 625 | 625 |
| 20 | 21 | 29 | 841 | 841 |
| 9 | 40 | 41 | 1681 | 1681 |
| 11 | 60 | 61 | 3721 | 3721 |
| 12 | 35 | 37 | 1369 | 1369 |
Integer-sided right triangles. Any multiple of these triples is also a right triangle (e.g. 6-8-10, 10-24-26).
Frequently asked questions
How do I know which side is the hypotenuse?
The hypotenuse is always the longest side of a right triangle and is always opposite the 90-degree angle. When you enter three side lengths, this calculator automatically sorts them and uses the largest as the candidate hypotenuse. If the triangle is a right triangle, the longest side is confirmed as the hypotenuse.
Can the calculator handle decimal or fractional side lengths?
Yes. Enter any positive decimal values. The check uses a floating-point tolerance so that rounded measurements like 3.000, 4.000, 5.000 still pass even if tiny rounding errors are present. Very large or very small values work too, as long as they are positive numbers.
What is a Pythagorean triple?
A Pythagorean triple is a set of three positive integers (a, b, c) where a squared + b squared = c squared exactly. The most common example is 3-4-5. Any whole-number multiple of a Pythagorean triple is also a Pythagorean triple, so 6-8-10, 9-12-15, and 15-20-25 are all right triangles. They are useful because they let you build or check right angles using just a tape measure and integer lengths.
What is a right isosceles triangle?
A right isosceles triangle is a 45-45-90 triangle: a right triangle where the two legs are equal in length. Because both legs are the same, the two acute angles are also equal (45 degrees each). If each leg has length l, the hypotenuse has length l times the square root of 2, approximately 1.41421 times l.
Why does my measurement almost pass but not quite?
Real-world measurements are rounded, and even a small error in one side length shifts the Pythagorean deficit away from zero. For example, if your sides are 3, 4, and 5.001, the deficit is 5.001 squared minus 25 = 0.010001, which fails a strict test. The calculator reports the actual deficit so you can judge how close you are. If the deficit is less than a few thousandths of a percent of c squared, you likely have a right triangle with measurement noise.