A Pythagorean theorem calculator finds the missing side of a right triangle from the rule a squared plus b squared equals c squared. Enter any two sides and leave the unknown blank. With legs of 3 and 4, the hypotenuse is 5.
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How to Use the Pythagorean Theorem Calculator
- Identify the two sides you know on your right triangle.
- Enter those two, and leave the unknown side blank.
- Read the missing side, whether a leg or the hypotenuse.
- See the perimeter and area of the triangle.
Here is what each result means:
| Result | What it means |
|---|---|
| Missing side | The side you left blank, solved from the other two. |
| Perimeter | The three sides added together. |
| Area | Half of one leg times the other leg. |
What Is the Pythagorean Theorem?
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a^2 + b^2 = c^2. Here a and b are the legs that meet at the right angle, and c is the hypotenuse, the longest side opposite that angle.
It is one of the most useful results in geometry. Given any two sides of a right triangle, you can always find the third. That single relation underlies distance on a grid, diagonal measurements, navigation and countless construction tasks.
How Does the Calculator Work?
It rearranges the theorem to solve for whichever side you leave blank.
- If a leg is missing, subtract the known leg squared from the hypotenuse squared, then take the square root.
- If the hypotenuse is missing, add the two legs squared and take the square root.
- Use the sides for the perimeter and area.
On a coordinate grid this same idea gives the distance between two points; see the midpoint calculator for related coordinate geometry.
Pythagorean Theorem Example
A right triangle has legs of 3 and 4. Find the hypotenuse.
Calculation: c = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5. The perimeter is 3 plus 4 plus 5, which is 12, and the area is half of 3 times 4, which is 6. This 3-4-5 triangle is the best-known Pythagorean triple.
Common Pythagorean Triples
These are whole-number side sets that satisfy the theorem exactly.
| Legs a, b | Hypotenuse c |
|---|---|
| 3, 4 | 5 |
| 5, 12 | 13 |
| 8, 15 | 17 |
| 7, 24 | 25 |
Any multiple of a triple is also a triple, so 6, 8, 10 works just like 3, 4, 5. They are handy checks in construction and geometry.
Leg vs Hypotenuse: What You Are Solving For
Which side is missing changes whether you add or subtract.
| Missing side | What to do |
|---|---|
| Hypotenuse c | Add the legs squared, then take the square root |
| A leg (a or b) | Subtract the known leg squared from c squared, then take the square root |
The hypotenuse is always the longest side, so a leg you solve for must come out shorter than c.
What Affects the Result
Which Side Is Unknown
Solving for the hypotenuse adds the squares; solving for a leg subtracts, and needs c to be the largest side.
A True Right Angle
The theorem holds only when the triangle has a 90-degree angle. Without it, the result is wrong.
Unit Consistency
All sides must use the same units. Mixing feet and inches gives a meaningless answer.
When to Use a Pythagorean Theorem Calculator
Construction and DIY
Check that a corner is square, or find a diagonal brace length.
Geometry Homework
Solve for a missing side and confirm your working.
Distance and Navigation
Find straight-line distance across a grid from horizontal and vertical gaps.
Common Mistakes
1. Using It on a Non-right Triangle
The theorem needs a right angle. For other triangles, use the law of cosines instead.
2. Mixing Up Leg and Hypotenuse
The hypotenuse is the longest side, opposite the right angle. Subtract when solving for a leg.
3. Forgetting the Square Root
a squared plus b squared gives c squared. Take the square root to get c.
4. Inconsistent Units
Convert all sides to the same unit before calculating.
5. a Leg Longer than the Hypotenuse
If a known leg is longer than c, the triangle is impossible. Check your inputs.
Accuracy and Limitations
The theorem is exact; only the displayed decimals are rounded.
What it calculates accurately
- A missing leg or hypotenuse
- The perimeter of the triangle
- The area of the right triangle
What it does not do
- Work on non-right triangles
- Find angles without more steps
- Return exact surd forms
- Draw the triangle to scale
How We Solve Your Triangle
Frequently Asked Questions
What is the Pythagorean theorem?
It states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a squared plus b squared equals c squared. It lets you find any side from the other two.
How do I find the hypotenuse?
Square both legs, add them, and take the square root. For legs of 3 and 4, the hypotenuse is the square root of 9 plus 16, which is the square root of 25, or 5.
How do I find a missing leg?
Subtract the known leg squared from the hypotenuse squared, then take the square root. If c is 10 and one leg is 6, the other leg is the square root of 100 minus 36, which is 8.
Does it work for any triangle?
No. The Pythagorean theorem holds only for right-angled triangles. For triangles without a right angle, use the law of cosines to relate the sides and angles.
What is a Pythagorean triple?
It is a set of three whole numbers that fit the theorem, such as 3, 4, 5 or 5, 12, 13. Any multiple of a triple is also a triple, like 6, 8, 10.
Which side is the hypotenuse?
The hypotenuse is the longest side, opposite the right angle. The other two sides, which meet at the right angle, are the legs.
Can I use it to find distance between two points?
Yes. The straight-line distance on a grid is the hypotenuse of a right triangle whose legs are the horizontal and vertical gaps between the points.
Why does my leg answer seem too big?
A leg must be shorter than the hypotenuse. If a known leg is longer than c, the triangle is impossible, so check that c is really the longest side.
Is my information saved?
No. The calculation runs in your browser and nothing you enter is stored or sent anywhere, unless you choose Save, which keeps the result only on this device.
Sources
- Pythagorean theorem (Wikipedia).
- Pythagoras theorem explained (Maths Is Fun).
- Pythagorean triples (Maths Is Fun).
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Explore all math calculatorsThis calculator uses the Pythagorean theorem, which holds only for right-angled triangles. The hypotenuse c is always the longest side, opposite the right angle. Results are rounded for display. Spotted an error? Let us know.
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Shakeel Muzaffar is the Founder and Editor-in-Chief of MultiCalculators.com, bringing over 15 years of experience in digital publishing, product strategy, and online tool development. He leads the platform's editorial vision, ensuring every calculator meets strict standards for accuracy, usability, and real-world value. Shakeel personally oversees content quality, formula verification workflows, and the platform's commitment to publishing tools that are genuinely useful for students, professionals, and everyday users worldwide.
