Hypotenuse Calculator
Find the hypotenuse of a right triangle using two known sides, one side and an angle, or the triangle's area. Enter your values to get the missing length, a labeled triangle diagram, and a step-by-step solution.
Enter valid measurements to see a result.
Hypotenuse (c)
Triangle measurements
- Leg a
- 3 m
- Leg b
- 4 m
- Angle α
- 36.8699°
- Angle β
- 53.1301°
- Area
- 6 m²
- Perimeter
- 12 m
Diagram and precision notes
For very thin triangles the smaller leg may be below screen resolution; labels stay outside the shape.
≈ indicates a computed approximation. Values display up to 10 significant digits. Steps use meters and square meters internally, without rounding intermediate calculations. Angles display up to four decimal places.
Show calculation
- c = √(a² + b²)
- c = √((3)² + (4)²) = √(9 + 16) ≈ 5 m
- α = atan2(a, b) ≈ 36.86989765°; β = atan2(b, a) ≈ 53.13010235°
- A = ab / 2 ≈ 6 m²; P = a + b + c ≈ 12 m
How to Calculate the Hypotenuse of a Right Triangle
The hypotenuse c is opposite the right angle. The legs a and b meet at 90°. The Pythagorean theorem applies only to right triangles; measurements must use compatible units.
Using Two Legs: The Pythagorean Theorem
c = √(a² + b²)
Choose Two legs and enter both lengths. This online hypotenuse calculator updates as you type, then displays both acute angles, area and perimeter.
Using One Side and an Angle
c = opposite / sin(α) or c = adjacent / cos(α)
Choose the relationship to the known acute angle. Here a is opposite α, b is adjacent to α, and β = 90° − α.
Using the Area and One Leg
b = 2A / a; c = √(a² + b²)
Area uses squared units. In feet + inches mode, enter area in square feet. A measurement such as 2 ft 15 in is normalized to 3 ft 3 in; decimal feet is a separate unit choice.
Hypotenuse Formula and Step-by-Step Examples
These examples use sample values, not your current inputs.
Two legs: 3–4–5
c = √(3² + 4²) = √(9 + 16) = √25 = 5
The area is 3 × 4 / 2 = 6 square units, and the perimeter is 3 + 4 + 5 = 12 units.
A side and an angle
a = 5, α = 30°: c = 5 / sin(30°) = 5 / 0.5 = 10
The adjacent leg is b = 5√3 ≈ 8.660254, and β = 60°.
Area and a leg
A = 24, a = 6: b = 2 × 24 / 6 = 8; c = √(36 + 64) = 10
Use square units for area and the matching linear units for a.
Find a missing leg
c = 13, a = 5: b = √(13² − 5²) = √144 = 12
The reverse mode finds b, not c. The known hypotenuse must be strictly longer than the known leg.
Special Right Triangles: 30-60-90 and 45-45-90
In a 30-60-90 triangle, the short leg, long leg and hypotenuse have ratio 1 : √3 : 2. A short leg of 5 gives a hypotenuse of 10. In a 45-45-90 triangle, the legs are equal: 1 : 1 : √2. Two legs of 7 give c = 7√2 ≈ 9.899495.
Frequently Asked Questions
How do you calculate the hypotenuse?
If you know both legs of a right triangle, square each length, add the squares, and take the square root: c = √(a² + b²). For legs of 6 and 8, the hypotenuse is 10.
Can I find the hypotenuse with only one side?
Not from one side alone. You also need additional information, such as an acute angle or the triangle’s area.
How do I find the hypotenuse using an angle?
Divide the opposite side by the sine of the known acute angle, or divide the adjacent side by its cosine. Select the relationship that matches your measurements.
Is the hypotenuse always the longest side?
Yes. In a right triangle, the hypotenuse is opposite the 90° angle and is longer than either leg.
How do I calculate the hypotenuse in feet and inches?
Convert both leg measurements to a common unit, apply the Pythagorean theorem, and convert the result back to feet and inches. For example, legs of 3 ft and 4 ft give a hypotenuse of 5 ft.
How can I find a missing leg if I know the hypotenuse?
Use the rearranged Pythagorean theorem: a = √(c² − b²). The hypotenuse must be longer than the known leg. In the missing-leg mode here, the known leg is a and the result is b.
Does the Pythagorean theorem work for every triangle?
No. It applies to right triangles. For other triangles, a more general method such as the law of cosines may be required.
Sources and assumptions
OpenStax — Right Triangle Trigonometry. All calculations assume a right triangle and positive measurements; input measurement uncertainty is not modeled.
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