Hypotenuse Calculator with Steps

This hypotenuse calculator finds the length of the longest side of a right triangle when you enter the other two side lengths. It uses the Pythagorean theorem, shows the formula, and gives you a step-by-step solution with the correct units. You get both the exact answer and the rounded decimal value.

Formula

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides. In symbols:

c = √(a² + b²)

• c – hypotenuse length (the side you want to find), in the same unit as a and b
• a – length of one leg (the shorter sides adjacent to the right angle), in meters, feet, inches, etc.
• b – length of the other leg, in the same unit

The formula only applies to right triangles – one angle must be exactly 90°. If you don’t have a right angle, you need the law of cosines instead.

Worked Example

Using the default values: side a = 3 m, side b = 4 m.

1. Square side a: 3² = 9
2. Square side b: 4² = 16
3. Add the squares: 9 + 16 = 25
4. Take the square root: √25 = 5

The hypotenuse is 5 m. This is the classic 3-4-5 triangle, often used in construction to check right angles. The calculator also shows the acute angles: angle A = atan(3/4) ≈ 36.87°, angle B = atan(4/3) ≈ 53.13°.

How to Use the Calculator

• Enter the length of side a in the first box.
• Enter the length of side b in the second box.
• Make sure both sides are in the same unit – the calculator doesn’t convert between meters and feet.
• Press calculate (or the equivalent button) to get the hypotenuse and the two acute angles.

The result updates instantly. The first output (the hypotenuse) is the main answer, shown with two decimal places. The angles are given in degrees, rounded to two decimals.

When to Use the Hypotenuse Formula

You use the Pythagorean theorem whenever you need to find a missing side of a right triangle and you know the other two sides. Common real-world situations include:

• Finding the length of a ladder needed to reach a certain height on a wall (the ladder is the hypotenuse).
• Calculating the diagonal of a rectangular screen, table, or room from its width and height.
• Determining the shortest distance between two points when you know the horizontal and vertical distances.
• Checking if a corner is square – if a triangle with sides 3, 4, and 5 fits, the corner is 90°.

The formula also appears in physics (resultant of perpendicular forces) and in navigation (direct distance from east/north offsets).

Common Mistakes and Misconceptions

• Mixing units: If one side is in meters and the other in centimeters, the result is wrong. Convert first – e.g., 1 m = 100 cm.
• Using the wrong side as hypotenuse: The hypotenuse is always the longest side, opposite the right angle. If you enter the hypotenuse as one of the legs, you’ll get a nonsense answer.
• Forgetting to square root: Some people stop after adding the squares. The Pythagorean theorem gives c², so you must take the square root to get c.
• Applying to non-right triangles: The theorem only works when one angle is exactly 90°. For other triangles, use the law of cosines: c² = a² + b² − 2ab·cos(C).
• Rounding too early: If you round intermediate values, the final answer may be off. Keep full precision until the last step.

Limitations of the Formula

The Pythagorean theorem is exact for ideal right triangles in Euclidean geometry. It doesn’t apply to:

• Triangles on curved surfaces (like the Earth’s surface for large distances) – use spherical trigonometry.
• Non-right triangles – use the law of cosines or law of sines.
• Three-dimensional problems – you need the 3D version: diagonal of a box = √(l² + w² + h²).

Also, the formula only gives the length of the hypotenuse. If you need the area of the triangle, use (a × b) / 2. If you need the perimeter, add a + b + c.

How to Check Your Answer

You can verify the result by plugging the hypotenuse back into the theorem: a² + b² should equal c². For the example, 9 + 16 = 25, and 5² = 25. Also, the hypotenuse must be longer than either leg – if it isn’t, you’ve made an error. For a quick sanity check, remember that the hypotenuse is always the largest side in a right triangle.

Frequently Asked Questions

What if I only know one leg and the hypotenuse?

Then you can find the missing leg by rearranging the formula: a = √(c² − b²). Enter the hypotenuse as one of the sides and the known leg as the other – but be careful: the calculator expects the two legs, so if you enter the hypotenuse as side a, you’ll get a wrong value for the hypotenuse. Instead, use the rearranged formula manually or use a dedicated “leg calculator.�

Can I use this for any triangle?

No, only for right triangles. If you have an angle between two sides that isn’t 90°, use the law of cosines: c² = a² + b² − 2ab·cos(θ). For example, if two sides are 5 and 7 with an included angle of 60°, the third side is √(25 + 49 − 2×5×7×cos60°) = √(74 − 35) = √39 ≈ 6.24.

What units does the calculator use?

The calculator is unit-agnostic – it works with any consistent unit (meters, feet, inches, etc.). The output uses the same unit you entered. If you mix units, the result is meaningless. Convert all sides to the same unit before calculating.

Why is the hypotenuse always the longest side?

Because the square of the hypotenuse equals the sum of the squares of the other two sides. Since both a² and b² are positive, c² is larger than either a² or b² alone, so c must be longer than both a and b. This is a direct consequence of the Pythagorean theorem.

How do I find the hypotenuse if I only know the area and one leg?

From the area A = (a × b)/2, you can solve for the other leg: b = 2A/a. Then use the Pythagorean theorem to find c. For example, if area = 6 and a = 3, then b = 4, and c = 5.

Is the Pythagorean theorem always exact?

Yes, for ideal right triangles in flat (Euclidean) geometry, it is exact. In real-world measurements, there is always some measurement error, but the formula itself is mathematically precise. For very large distances on Earth, you’d need to account for curvature.

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in a | Side a | m | 3 | 0
in b | Side b | m | 4 | 0
out c | Hypotenuse | m | sqrt(a^2 + b^2) | 2
out angleA | Angle A | deg | atand(a/b) | 2
out angleB | Angle B | deg | atand(b/a) | 2