c = √(a² + b²)
Hypotenuse Calculator
Find the hypotenuse from any two known values: both legs, a leg and an angle, the area, the altitude, or the perimeter.
Find the hypotenuse of a right triangle
The hypotenuse is the longest side of a right triangle, opposite the right angle. If you know both legs, it's
c = √(a² + b²). If you know one leg and an angle, it's c = a / sin(α) for the opposite
leg or c = b / cos(α) for the adjacent leg.
Enter any two values below and the calculator finds c, then every other side and angle. It's
pre-loaded with a ladder that reaches 12 ft up a wall at 75° to the ground.
Results
Scalene right triangle- Leg a
- 3.2154 ft
- Leg b
- 12 ft
- Hypotenuse c
- 12.4233 ft
- Angle α
- 15°
- Angle β
- 75°
- Altitude h
- 3.1058 ft
- Area
- 19.2923 ft²
- Perimeter
- 27.6387 ft
- Inradius
- 1.396 ft
- Circumradius
- 6.2117 ft
Step-by-step solution
The working keeps extra decimal places so you can check it by hand. Answers are rounded to your chosen decimal places.
- Find angle αα = 90° − βα = 90° − 75°= 15°
- Find the hypotenusec = b / cos(α)c = 12 / cos(15°)≈ 12.4233 ft
- Find leg aa = c·sin(α)a = 12.423314·
sin(15°) ≈ 3.2154 ft - Compute the areaArea = (1/2)·a·bArea = (1/2)·
3.21539· 12 ≈ 19.2923 ft² - Compute the perimeterP = a + b + cP = 3.21539 + 12 + 12.423314≈ 27.6387 ft
- Compute the altitude to the hypotenuseh = (a·b) / ch = (3.21539·
12) / 12.423314 ≈ 3.1058 ft - Inradius (radius of the inscribed circle)r = (a + b − c) / 2r = (3.21539 + 12 − 12.423314) / 2≈ 1.396 ft
- Circumradius (radius of the circumscribed circle)R = c / 2R = 12.423314 / 2≈ 6.2117 ft
Hypotenuse formulas for every pair of known values
You don't need a second side to find the hypotenuse. Any two values that pin down the triangle will do. In the
table, α is the angle opposite leg a, and h is the altitude from the right
angle to the hypotenuse. These are the same formulas the calculator shows in its step-by-step working.
| What you know | Hypotenuse formula |
|---|---|
| Both legs a and b | c = √(a² + b²) |
| Leg a and its opposite angle α | c = a / sin(α) |
| Leg b and the adjacent angle α | c = b / cos(α) |
| Altitude h and an angle α | c = h / (sin(α)·cos(α)) |
| Area and an angle α | c = √(2·Area / (sin(α)·cos(α))) |
| Perimeter P and an angle α | c = P / (sin(α) + cos(α) + 1) |
| Leg a and the area | b = 2·Area / a, then c = √(a² + b²) |
| Leg a and the altitude h | b = (h·a) / √(a² − h²), then c = √(a² + b²) |
| Leg a and the perimeter P | b = P(P − 2·a) / (2(P − a)), then c = P − a − b |
| Altitude h and the area | c = 2·Area / h |
| Altitude h and the perimeter P | c = P² / (2(P + h)) |
| Area and the perimeter P | c = (P² − 4·Area) / (2P) |
For the last two pairs, the altitude or the area with the perimeter, the calculator finds c
numerically rather than with the formula shown, and gets the same answer.
Worked example: how long a ladder do you need?
A ladder has to reach a gutter 12 ft up a wall, and safety guides say a leaning ladder should stand at about 75°
to the ground. The wall, the ground, and the ladder make a right triangle, drawn the same way in the diagram:
the height on the wall is leg b, the 75° angle at the foot of the ladder is β, and the
ladder is the hypotenuse. The height is opposite β and you want the hypotenuse, so the ratio
to use is sine, the "SOH" in SOH CAH TOA: sin(β) = b / c. Rearranged,
c = b / sin(β) = 12 / sin(75°) ≈ 12 / 0.965926 ≈ 12.4233 ft, about 12 ft 5 in of ladder between the
ground and the gutter. For answers read straight off a tape, to the nearest 1/16 in, use the
right triangle calculator in feet and inches.
The calculator's steps take the other route to the same answer: first the angle at the top of the ladder,
α = 90° − 75° = 15°, then c = b / cos(α) = 12 / cos(15°), using the leg next to that
angle. The foot of the ladder stands a ≈ 3.215 ft from the wall, close to the rule of thumb of 1 ft
out for every 4 ft up (12 / 4 = 3). If both legs had been known instead, say 8 and 15, you'd skip
the trigonometry: c = √(8² + 15²) = √289 = 17.
Why the hypotenuse is always the longest side
In any triangle, the longest side sits opposite the largest angle. A right triangle's largest angle is the 90°
one, so the side across from it, the hypotenuse, is always longer than either leg. You can see it in the
formula too: c² = a² + b² is bigger than a² or b² alone.
That's a quick sanity check on any answer. If a hypotenuse comes out shorter than a leg, a value was entered in the wrong box. To check whether three lengths you've measured form a right triangle, use the Pythagorean theorem calculator.
Hypotenuse FAQ
What is the formula for the hypotenuse?
c = √(a² + b²). It comes from the Pythagorean theorem, a² + b² = c²: square both legs, add them, and take the square root. For legs of 6 and 8, c = √(36 + 64) = √100 = 10. If you know an angle instead of the second leg, use c = a / sin(α) or c = b / cos(α).
Can you find the hypotenuse with only one side?
Not on its own. One leg fits infinitely many right triangles, so you need a second value: the other leg, one of the acute angles, the area, the altitude, or the perimeter. The only exception is a special right triangle, where the angles are already known: in a 45-45-90 triangle c = leg × √2, and in a 30-60-90 triangle c = 2 × the short leg.
How do I find the hypotenuse with an angle?
Use SOH CAH TOA. If you know the leg opposite the angle, divide it by the sine of the angle: c = a / sin(α). If you know the leg next to the angle, divide it by the cosine: c = b / cos(α). For a ladder that reaches 12 ft up a wall at 75° to the ground, the height is opposite the 75° angle, so c = 12 / sin(75°) ≈ 12.4233 ft.
How do I find a leg if I know the hypotenuse?
Rearrange the Pythagorean theorem: a = √(c² − b²). Subtract the square of the known leg from the square of the hypotenuse, then take the square root. For a hypotenuse of 10 and a leg of 6, the other leg is √(100 − 36) = √64 = 8. With the hypotenuse and an angle instead, a = c·sin(α) and b = c·cos(α).
Can I find the hypotenuse from the area or the perimeter?
Yes, with one more value. From the area and an angle, c = √(2·Area / (sin(α)·cos(α))). From the perimeter and an angle, c = P / (sin(α) + cos(α) + 1). The area and the altitude give c = 2·Area / h, and the area and the perimeter give c = (P² − 4·Area) / (2P). The formula table above lists every pair, and the calculator accepts all of them.
Is the hypotenuse always opposite the right angle?
Yes. The hypotenuse is defined as the side opposite the 90° angle, and it is always the longest side of a right triangle. Because the other two angles are each less than 90°, the sides opposite them (the legs) are always shorter than the hypotenuse.
What's the hypotenuse of a 45-45-90 or 30-60-90 triangle?
In a 45-45-90 triangle the hypotenuse is a leg times √2 (about 1.4142). In a 30-60-90 triangle it is exactly twice the short leg. The 45-45-90 and 30-60-90 calculators on this site solve these from a single side.
How long a ladder do I need to reach a certain height?
Safety guides put a leaning ladder at about 75°, roughly 1 ft out from the wall for every 4 ft up. The ladder is the hypotenuse of the triangle it makes with the wall and the ground. To reach 16 ft up, set the foot 4 ft out: the ladder must be at least √(16² + 4²) = √272 ≈ 16.49 ft long between those two points. If you will step off at the top, the ladder should also extend at least 3 ft above the landing.
How do I work out a TV or monitor screen size?
A screen size is its diagonal, which is the hypotenuse of the width and the height. A 16:9 screen 48 in wide and 27 in tall measures √(48² + 27²) = √3033 ≈ 55.07 in, so it is sold as a 55-inch TV. Measure the visible picture only, not the frame around it.
How long does a wheelchair ramp need to be?
The ramp surface is the hypotenuse of its rise and its run. In the US, the ADA allows a slope of at most 1:12 for wheelchair ramps: 1 inch of rise for every 12 inches of run. A ramp that climbs 2 ft needs a 24 ft run, and its surface is √(24² + 2²) = √580 ≈ 24.08 ft long. One ramp run may rise at most 30 in before it needs a level landing.
For a single known side, try the 45-45-90 or 30-60-90 triangle calculator. See the full right triangle FAQ for more general questions.
Cite this page
To cite this calculator in schoolwork or a report, you can use this reference (APA style):
Right Triangle Calculators. (2026, October 10). Hypotenuse calculator. https://righttrianglecalculators.com/hypotenuse-calculator
References
The formulas and definitions on this page match these standard references.
- Weisstein, Eric W. “Hypotenuse.” MathWorld, Wolfram Research.
- Weisstein, Eric W. “Pythagorean Theorem.” MathWorld, Wolfram Research.
- Weisstein, Eric W. “SOHCAHTOA.” MathWorld, Wolfram Research.
Pythagorean theorem calculator
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