Hypotenuse tutorial
How to Find the Hypotenuse
A step-by-step guide to finding the longest side of a right triangle, with the formula, three worked examples, and a free calculator.
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What You Need to Know First
What is a right triangle?
A right triangle has exactly one 90-degree angle. The two sides that form the right angle are called legs, labeled a and b. The side opposite the right angle, always the longest side, is called the hypotenuse, labeled c.
The Pythagorean theorem
The relationship between the three sides is:
a² + b² = c²
This equation is always true for right triangles. To find c, the hypotenuse,
rearrange it to:
c = √(a² + b²)
If you want a deeper explanation of the theorem itself, see The Pythagorean Theorem.
Method
How to Find the Hypotenuse: 4 Steps
Identify the right angle and label the sides
Find the corner with the 90-degree symbol. The two sides meeting at that corner are the legs,
call them a and b. The remaining side, opposite the right angle, is the
hypotenuse c. You are solving for c.
Square each leg
Multiply each leg by itself.
a² = a × a
b² = b × b
3² = 9
4² = 16
Add the two squares
Add the results from Step 2 together.
a² + b² = sum
9 + 16 = 25
Take the square root
The hypotenuse is the square root of the sum from Step 3.
c = √(a² + b²)
c = √25 = 5
The hypotenuse is 5.
The formula in one line: c = √(a² + b²)
Substitute your values for a and b, and you have the hypotenuse.
Examples
Worked Examples
These three examples progress from a simple integer result to a decimal answer to a real-world scenario. Each one follows the same four steps.
Example 1: Find the hypotenuse of a 6-8-? triangle
Given: a = 6, b = 8. Find c.
Basic
| Square the legs | 6² = 36, 8² = 64 |
|---|---|
| Add | 36 + 64 = 100 |
| Square root | c = √100 = 10 |
The hypotenuse is 10.
6-8-10 is a multiple of the 3-4-5 Pythagorean triple.
Example 2: Find the hypotenuse of a 5-7-? triangle
Given: a = 5, b = 7. Find c.
Intermediate
| Square the legs | 5² = 25, 7² = 49 |
|---|---|
| Add | 25 + 49 = 74 |
| Square root | c = √74 ≈ 8.602 |
The hypotenuse is approximately 8.602.
Not every triangle produces a whole number. Use the calculator for precise decimal results.
Example 3: A ramp rises 1.2 m over a horizontal distance of 3.5 m. How long is the ramp?
Given: a = 1.2 m, b = 3.5 m. Find c.
Applied
| Square the legs | 1.2² = 1.44, 3.5² = 12.25 |
|---|---|
| Add | 1.44 + 12.25 = 13.69 |
| Square root | c = √13.69 = 3.7 m |
The ramp is 3.7 m long.
For more examples including word problems, see Pythagorean Theorem Examples and Word Problems with Solutions.
Special cases
Special Cases
When both legs are equal
When a = b, the triangle is a 45-45-90 isosceles right triangle. The hypotenuse
simplifies to:
c = a√2
If a = b = 5, then c = 5√2 ≈ 7.071.
When one angle is 30° or 60°
In a 30-60-90 triangle, the sides follow a fixed ratio. If the short leg is a,
then:
hypotenuse c = 2a
longer leg b = a√3
If a = 4, then c = 8 and b = 4√3 ≈ 6.928.
Pitfalls
Common Mistakes to Avoid
Adding the sides instead of their squares
Writing c = a + b instead of c = √(a² + b²). The hypotenuse is never simply the sum of the legs.
Always square first, then add, then take the square root.
Forgetting the square root
Stopping at a² + b² and reporting that as the answer. That value is c², not c.
The final step is always √(a² + b²) to get c.
Labeling the hypotenuse as a leg
Plugging the hypotenuse value into a or b instead of c. This breaks the setup.
The hypotenuse is always opposite the right angle and always the longest side. Assign it to c.
Rounding too early
Rounding a² or b² before adding them. Early rounding compounds errors, especially with decimal inputs.
Keep full precision through the intermediate steps. Round only the final answer.
Calculator
Find the Hypotenuse Instantly
Skip the manual steps and let the calculator do the arithmetic, with full working shown.
Hypotenuse Calculator
Enter two legs and get the hypotenuse with step-by-step working.
Calculate NowRight Triangle Calculator
Solve all sides, angles, and area of any right triangle.
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Frequently Asked Questions
Square both legs, add the results, and take the square root:
c = √(a² + b²). For example, if a = 3 and b = 4, then
c = √(9 + 16) = √25 = 5. This four-step process works for any right triangle.
Not with the Pythagorean theorem alone, you need both legs. If you have one leg and one angle, you can use trigonometry instead. The Right Triangle Calculator handles angle-based cases.
That is normal. Most right triangles do not produce integer hypotenuse values. Use a calculator or square-root function and round to the precision your problem requires, usually 2 or 3 decimal places.
No. Both legs must be in the same unit before you apply the formula. Convert one leg to match the other first, then calculate. The result will be in that same unit.
Finding the hypotenuse uses c = √(a² + b²), you add the squared legs. Finding a
missing leg uses a = √(c² − b²), you subtract one squared side from the other.
For missing-leg calculations, use the Missing Side Calculator.
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