Pythagorean Theorem Calculator
A Pythagorean theorem calculator finds a missing side of a right triangle when the other two side lengths are known. It is useful in geometry, construction, engineering, coordinate problems, and many calculations involving right-angled triangles.
What Is the Pythagorean Theorem?
The Pythagorean theorem applies specifically to right triangles. It states that the square of the hypotenuse equals the sum of the squares of the two shorter sides.
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In the formula, a and b are the legs of the right triangle, while c is the hypotenuse, which is always opposite the right angle.
Finding the Hypotenuse
If both legs are known, rearrange the formula to find the hypotenuse:
c = β(a2 + b2)
For example, if the two legs are 3 units and 4 units:
c = β(32 + 42)
c = β(9 + 16) = β25 = 5
The hypotenuse is therefore 5 units.
Finding a Missing Leg
If the hypotenuse and one leg are known, the other leg can be found by rearranging the formula.
a = β(c2 β b2)
For example, if the hypotenuse is 13 meters and one leg is 5 meters:
a = β(132 β 52)
a = β(169 β 25) = β144 = 12
The missing leg is 12 meters.
Common Pythagorean Triples
| Leg a | Leg b | Hypotenuse c |
|---|---|---|
| 3 | 4 | 5 |
| 5 | 12 | 13 |
| 6 | 8 | 10 |
| 8 | 15 | 17 |
| 7 | 24 | 25 |
When Can the Pythagorean Theorem Be Used?
The theorem can only be applied directly to a triangle that contains a 90-degree angle. It does not apply to arbitrary triangles unless additional relationships or methods are used.
The hypotenuse must also be the longest side of the triangle. If you know the side lengths, the largest one should be opposite the right angle.
Where Is It Used?
The Pythagorean theorem is used to calculate diagonal distances, determine unknown side lengths, solve geometry problems, measure direct distances, and work with right-angled structures in practical applications.
Common Mistakes to Avoid
Do not treat any side as the hypotenuse. The hypotenuse is always opposite the 90-degree angle. Also make sure the measurements use compatible units before performing the calculation.
When finding a missing leg, subtract the known leg's square from the hypotenuse's square. Using addition in this situation will produce an incorrect result.
Quick Summary
The Pythagorean theorem is a2 + b2 = c2 and applies to right triangles. It can be rearranged to find a missing leg or the hypotenuse when the other two sides are known.