Hypotenuse Calculator
Find the hypotenuse of a right triangle from two legs, one leg and an angle, one leg and the area, or a 45-45-90 or 30-60-90 triangle — or find a missing leg from the hypotenuse. Shows the exact radical form, both angles, area, perimeter and the worked steps.
How it works
c = √(a² + b²) · a = √(c² − b²) · c = a ÷ sin α = a ÷ cos β · b = 2 × Area ÷ a · 45-45-90: c = a√2 · 30-60-90: c = 2 × short leg · α = arctan(a ÷ b)
The hypotenuse is the longest side of a right triangle — the one opposite the right angle. The Pythagorean theorem says the square on it equals the sum of the squares on the two legs, so c = √(a² + b²). Turned around, a missing leg is √(c² − other leg²). If you know one leg and an acute angle, trigonometry gives the hypotenuse: divide the leg by the sine of the angle across from it, or by the cosine of the angle next to it. If you know one leg and the area, the other leg is 2 × area ÷ leg. Two special triangles have fixed ratios: in a 45-45-90 triangle the hypotenuse is a leg times √2, and in a 30-60-90 triangle it is twice the short leg. With whole-number or short-decimal sides, the answer is also shown as an exact simplified radical, such as 5√2. You also get both acute angles, the area, perimeter, height to the hypotenuse and the radii of the inscribed and circumscribed circles. The steps show each calculation with your numbers.
Worked example
For legs a = 3 and b = 4: c = √(3² + 4²) = √(9 + 16) = √25 = 5, the classic 3-4-5 triple. The angles are arctan(3 ÷ 4) ≈ 36.87° and 53.13°, the area is 6 and the perimeter 12. For legs 5 and 5, c = √50 = 5√2 ≈ 7.0711. To find a missing leg from c = 13 and a = 5: b = √(169 − 25) = √144 = 12. With a = 10 and a 30° angle opposite it, c = 10 ÷ sin 30° = 20.
Sources: Pythagorean theorem — Wikipedia · Special right triangle — Wikipedia
Frequently asked questions
Which side is the hypotenuse?
It is the side opposite the 90° angle, and it is always the longest side of a right triangle. The other two sides, which form the right angle, are the legs.
How do I find a leg if I know the hypotenuse?
Choose “Hypotenuse and one leg”. The calculator uses leg = √(c² − other leg²). For c = 10 and a = 6, b = √(100 − 36) = 8. The hypotenuse must be longer than the known leg, or there is no real triangle.
How do I find the hypotenuse from one side and an angle?
If the angle is opposite the known leg, c = leg ÷ sin(angle). If it is next to the leg, c = leg ÷ cos(angle). For a leg of 7 next to a 40° angle, c = 7 ÷ cos 40° ≈ 9.1378.
What is a Pythagorean triple?
Three whole numbers that satisfy a² + b² = c², such as 3-4-5, 5-12-13 and 8-15-17. They make right triangles with all-integer sides. The calculator tells you when your sides form one, and whether it is a multiple of a smaller triple, such as 6-8-10 = 2 × 3-4-5.
What does the exact form like 5√2 mean?
When a² + b² is not a perfect square, the hypotenuse is irrational. The exact form keeps the square root and takes out any perfect-square factor: √50 = √(25 × 2) = 5√2. Use it when your teacher asks for an exact answer instead of a rounded decimal.
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This calculator is for educational purposes. Double-check important results before acting on them.