Hypotenuse Calculator
Solve any right triangle in seconds. Enter two legs to find the hypotenuse with the Pythagorean theorem, use one leg and an acute angle, or select a special triangle (30-60-90 or 45-45-90). The calculator returns every missing side, both angles, the altitude to the hypotenuse, and the perimeter and area.
Formula
Worked example
Two legs a=3, b=4: c = sqrt(9+16) = 5. Angles: alpha = asin(3/5) = 36.87 deg, beta = 53.13 deg. Altitude h = (3*4)/5 = 2.4. Reversing: with c=13 and one leg=5, missing leg = sqrt(169-25) = 12 (the 5-12-13 triple). With a 30-60-90 triangle and short leg=6, the hypotenuse = 2*6 = 12 and long leg = 6*sqrt(3) = 10.392.
How to find the hypotenuse: four solve modes explained
This calculator covers every standard method for solving the hypotenuse of a right triangle. The most common is the Pythagorean theorem: c = sqrt(a squared + b squared). When you only know one leg and an acute angle, trig steps in: if the angle is opposite the known leg, use c = leg / sin(angle); if the angle is adjacent (next to the leg), use c = leg / cos(angle). The third option is the reverse: if you know the hypotenuse and one leg, rearrange the theorem to missing = sqrt(c squared - known squared). The fourth mode handles the two special right triangles that appear so often in geometry, construction and trigonometry that they deserve their own shortcut.
Angles, altitude and area: the full triangle picture
Beyond the hypotenuse length, this calculator returns the two acute angles (alpha, opposite leg a, and beta, opposite leg b), which always sum to exactly 90 degrees since the third angle is the right angle. It also computes the altitude from the right-angle vertex to the hypotenuse, given by h = (a times b) divided by c. This altitude is particularly useful in surveying and engineering, where the perpendicular distance from a vertex to the opposite side is needed. Area is one-half times the product of the two legs, and the perimeter is the simple sum of all three sides.
Special right triangles: 30-60-90 and 45-45-90
Two right triangles have such clean side ratios that every trig textbook covers them first. The 45-45-90 triangle is an isosceles right triangle: both legs are equal, and the hypotenuse is exactly leg times the square root of 2 (roughly 1.414). This arises naturally when you cut a square diagonally. The 30-60-90 triangle has sides in the ratio 1 : sqrt(3) : 2. The shortest leg is opposite the 30-degree angle, the long leg is opposite the 60-degree angle, and the hypotenuse is twice the short leg. These triangles appear in design, architecture, and physics problems, and knowing their ratios lets you skip the calculator entirely for mental arithmetic.
Pythagorean triples and real-world uses
Some right triangles have all three sides as whole numbers, called Pythagorean triples. The most famous is 3-4-5; scale it up and you get 6-8-10, 9-12-15, and so on. The 5-12-13 triple and 8-15-17 triple come up often in construction. The classic application is the builder's "3-4-5 rule": measure 3 units along one wall edge and 4 along the other; the diagonal between the marks should be exactly 5 if the corner is square. The formula also underlies the distance formula between two points on a coordinate grid (d = sqrt of delta-x squared plus delta-y squared), vector magnitudes in physics, and ramp, rafter, and staircase length calculations.
Common Pythagorean triples and special triangles
| Type | Leg a | Leg b | Hypotenuse c | Notes |
|---|---|---|---|---|
| Triple | 3 | 4 | 5 | Scale: 6-8-10, 9-12-15, ... |
| Triple | 5 | 12 | 13 | Common in surveying |
| Triple | 8 | 15 | 17 | Common in construction |
| Triple | 7 | 24 | 25 | |
| Triple | 20 | 21 | 29 | |
| Triple | 9 | 40 | 41 | |
| Special | a | a | a*sqrt(2) | 45-45-90, ratio 1:1:sqrt(2) |
| Special | a | a*sqrt(3) | 2a | 30-60-90, ratio 1:sqrt(3):2 |
Exact integer triples and the two key special right triangles. The calculator detects these automatically.
Frequently asked questions
How do you calculate the hypotenuse from two legs?
Use the Pythagorean theorem: c = sqrt(a squared + b squared). Square each leg, add the results, then take the square root. For legs 6 and 8, c = sqrt(36 + 64) = sqrt(100) = 10.
How do I find the hypotenuse from one leg and an angle?
Use trigonometry. If the angle alpha is opposite the known leg a, then c = a / sin(alpha). If the angle beta is adjacent (next to the known leg), then c = b / cos(beta). For example, leg = 5 and opposite angle = 30 deg gives c = 5 / sin(30 deg) = 5 / 0.5 = 10.
How do I find a missing leg when I know the hypotenuse?
Rearrange the theorem to missing = sqrt(c squared - known squared). With c = 13 and known leg = 5, missing = sqrt(169 - 25) = sqrt(144) = 12, confirming the 5-12-13 triple.
What is the hypotenuse of a 45-45-90 triangle?
In a 45-45-90 triangle both legs are equal, so the hypotenuse is leg times the square root of 2 (approximately 1.41421). If each leg is 7, the hypotenuse is 7 * sqrt(2) = 9.8995.
What is the hypotenuse of a 30-60-90 triangle?
The hypotenuse of a 30-60-90 triangle is exactly twice the short leg. If the short leg (opposite 30 deg) is 5, the hypotenuse is 10, and the long leg is 5 * sqrt(3) = 8.6603.
What is the altitude to the hypotenuse?
The altitude from the right-angle vertex to the hypotenuse equals the product of the two legs divided by the hypotenuse: h = (a * b) / c. For the 3-4-5 triangle, h = (3 * 4) / 5 = 2.4. This length is important in geometric mean and similarity proofs.
Can the hypotenuse ever be shorter than a leg?
No. The hypotenuse is opposite the right angle and is always the longest side of a right triangle. If a claimed hypotenuse is shorter than or equal to one of the legs, no valid right triangle exists with those dimensions.