📏 Enter known values

Formula Reference

Worked example

Data

Leg

\(b = {\color{#6c5ce7}{5\,\text{cm}}}\)

Angle

\(c = {\color{#6c5ce7}{30^{\circ}}}\)

Hypotenuse

\(a = \,?\)

Formula

\[a = \frac{{b}}{\sin\left(\frac{{c}\cdot\pi}{180}\right)}\]

Steps

1Substitute the values

\[a = \frac{{\color{#6c5ce7}{5\,\text{cm}}}}{\sin\left(\frac{{\color{#6c5ce7}{30}}\cdot\pi}{180}\right)}\]

Each letter is replaced by its value.

2Multiply

\[a \approx \frac{{\color{#6c5ce7}{5\,\text{cm}}}}{\sin\left(\frac{94.2478}{180}\right)}\]

\(30\cdot\pi \approx 94.2478\)

3Divide

\[a \approx \frac{{\color{#6c5ce7}{5\,\text{cm}}}}{\sin\left(0.5236\right)}\]

\(\frac{94.2478}{180} \approx 0.5236\)

4Calculate the sine

\[a = \frac{{\color{#6c5ce7}{5\,\text{cm}}}}{0.5}\]

\(\sin\left(0.5236\right) \approx 0.5\)

5Divide

\[a = \frac{{\color{#6c5ce7}{5\,\text{cm}}}}{0.5} = 10\,\text{cm}\]

\(\frac{5}{0.5} = 10\)

Hypotenuse: 10 cm

Check

If we use the result to calculate Leg, we should get the value you entered:

\[b = {a}\cdot\sin\left(\frac{{c}\cdot\pi}{180}\right)\]

\[b = 10\,\text{cm}\cdot\sin\left(\frac{30\cdot\pi}{180}\right) = 5\,\text{cm}\]

It gives 5 cm, the same as the value entered.

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Calculate the Hypotenuse

Please fill in the fields:

Leg Angle

And leave empty

Hypotenuse

Calculate the Leg

Please fill in the fields:

Hypotenuse Angle

And leave empty

Leg

Calculate the Angle

Please fill in the fields:

Hypotenuse Leg

And leave empty

Angle

How to find the hypotenuse with a leg and an angle

In a right triangle, the sine of an acute angle is the leg opposite that angle divided by the hypotenuse. If you know one leg and the angle in front of it, the hypotenuse is:

\[ \text{Hypotenuse} = \frac{\text{Leg}}{\sin(\text{Angle})} \]

The same relation gives the other two values:

\[ \text{Leg} = \text{Hypotenuse} \times \sin(\text{Angle}) \]

\[ \text{Angle} = \arcsin\left(\frac{\text{Leg}}{\text{Hypotenuse}}\right) \]

The angle is in degrees and must be between 0° and 90°. The hypotenuse is always the longest side, so the leg has to be shorter than it.

How to use the calculator

Enter two of the three values: the hypotenuse, a leg and the angle opposite that leg.
Leave empty the one you want to find.
Press Calculate. If you know both legs instead of an angle, use the Pythagorean theorem calculator.

Worked examples

1. Leg of 5 cm with an opposite angle of 30°

\[ \text{Hypotenuse} = \frac{5}{\sin 30°} = \frac{5}{0.5} = 10\ \text{cm} \]

2. Length of a ladder

A ladder must reach a window 4 m high and stand at 75° to the ground. The height is the leg opposite the 75° angle, so the ladder measures \( 4 \div \sin 75° \approx 4 \div 0.9659 \approx 4.14 \) m.

3. Angle of a ramp

A ramp 6 m long rises 0.5 m. The rise is the leg opposite the angle of the ramp: \( \arcsin(0.5 \div 6) \approx 4.78° \).

If the angle is next to the leg

The calculator uses the angle opposite the leg. If you know the angle between the leg and the hypotenuse (the adjacent angle), use the cosine instead, or subtract the angle from 90° first:

\[ \text{Hypotenuse} = \frac{\text{Leg}}{\cos(\text{Adjacent angle})} \]

Example: a leg of 8 and an adjacent angle of 60°. The opposite angle is 90° − 60° = 30°, so the hypotenuse is 8 ÷ sin 30° = 16.

Hypotenuse for a leg of 1

Multiply the last column by your leg to get the hypotenuse. For example, with a leg of 7 and an angle of 45°, the hypotenuse is 7 × 1.414 ≈ 9.9.

Opposite angle	sin	Hypotenuse (leg = 1)
10°	0.1736	5.759
15°	0.2588	3.864
20°	0.342	2.924
30°	0.5	2
45°	0.7071	1.414
60°	0.866	1.155
75°	0.9659	1.035
80°	0.9848	1.015

When Do You Need to Calculate the Hypotenuse?

🪜 Ladders and Ramps

When you know the height a ladder or ramp must reach and the safe angle, the hypotenuse is the length you need. Ladder safety guides recommend about 75° to the ground.

Choose a ladder or ramp of the right length

🏠 Roof Rafters

A rafter is the hypotenuse of the right triangle formed by the rise and the run of the roof. With the rise and the roof angle you get the length to cut.

Cut lists for rafters and trusses

🧭 Navigation and Surveying

If you know how far north you moved and the angle of your course, the hypotenuse is the distance actually traveled. Surveyors use the same idea to measure slopes and distances they cannot walk.

Distances from a known side and an angle

📶 Cables and Guy Wires

Antenna masts and tents are held by guy wires anchored at an angle. The hypotenuse gives the cable length from the anchor height and the angle.

Order cable with the right length

⚡ Physics and Vectors

A force or a velocity at an angle has components along the axes. If you know one component and the angle, the hypotenuse is the magnitude of the vector.

Resolve forces and velocities

📚 Trigonometry Homework

Sine, cosine and tangent are first practiced on right triangles. Many exercises give one side and one angle and ask for the hypotenuse.

Check SOH-CAH-TOA calculations

Common Mistakes

⚠️ Using the Adjacent Angle

Common Error: The formula uses the angle opposite the leg. With the angle next to the leg, the result is wrong unless you use the cosine.

✅ Solution: Use the opposite angle, or 90° minus the adjacent angle

⚠️ Calculator in Radians

Common Error: In radian mode sin(30) is −0.988, not 0.5, and the hypotenuse comes out negative or absurd.

✅ Solution: Make sure the angle is in degrees

⚠️ Leg Longer Than the Hypotenuse

Common Error: When finding the angle, a leg longer than the hypotenuse has no solution: a sine cannot be greater than 1.

✅ Solution: Check the measurements: the hypotenuse is always the longest side

⚠️ Multiplying Instead of Dividing

Common Error: Writing leg × sin(angle) to get the hypotenuse. That gives a value smaller than the leg, which is impossible.

✅ Solution: Hypotenuse = leg ÷ sin(angle); leg = hypotenuse × sin(angle)

⚠️ Angles of 0° or 90°

Common Error: With 0° the triangle collapses, and 90° is the right angle itself, not one of the acute angles.

✅ Solution: Use an acute angle strictly between 0° and 90°

⚠️ Rounding the Sine

Common Error: Rounding sin 75° to 0.97 or to 1 changes the result noticeably for long sides.

✅ Solution: Keep at least four decimals, or let the calculator do it

Applications by Industry

Construction & Carpentry

Rafters: Rafter length from the rise and the roof angle
Stairs: Stringer length from the total rise and the stair angle
Bracing: Diagonal braces in frames and scaffolding

Telecommunications

Guy wires: Cable length for masts and towers
Line of sight: Distance to an antenna from its height and the elevation angle
Installation: Cable runs along sloped roofs

Navigation & Surveying

Courses: Distance traveled from the north or east displacement and the course angle
Slopes: Slope length from a height difference and an angle
Heights: Distance to the top of a building from its height and the viewing angle