Pythagorean Theorem Calculator — a² + b² = c²
Solve for any missing side of a right triangle using the Pythagorean theorem. Includes a Pythagorean triple checker and common triples reference. See also Right Triangle Calculator and Trigonometry Calculator.
Solve for a Missing Side
Pythagorean Triple Checker
Enter three side lengths to check if they form a right triangle (satisfy a² + b² = c²).
How to Use the Pythagorean Theorem
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: a² + b² = c². To find a missing side, select what you want to solve for, enter the two known values, and click Calculate. The calculator shows the missing side, all three sides, a step-by-step verification, plus the area and perimeter.
Pythagorean Theorem Formula
a² + b² = c²
Solve for c: c = √(a² + b²)
Solve for a: a = √(c² − b²)
Solve for b: b = √(c² − a²)
Area = ½ × a × b
Perimeter = a + b + c
Example Calculation
Find the hypotenuse when a = 3 and b = 4:
c² = a² + b²
c² = 3² + 4² = 9 + 16 = 25
c = √25 = 5
Find side a when b = 12 and c = 13:
a² = c² − b²
a² = 13² − 12² = 169 − 144 = 25
a = √25 = 5
Common Pythagorean Triples
| a | b | c | a² + b² | c² | Type |
|---|---|---|---|---|---|
| 3 | 4 | 5 | 25 | 25 | Primitive |
| 5 | 12 | 13 | 169 | 169 | Primitive |
| 8 | 15 | 17 | 289 | 289 | Primitive |
| 7 | 24 | 25 | 625 | 625 | Primitive |
| 9 | 40 | 41 | 1681 | 1681 | Primitive |
| 11 | 60 | 61 | 3721 | 3721 | Primitive |
| 12 | 35 | 37 | 1369 | 1369 | Primitive |
| 13 | 84 | 85 | 7225 | 7225 | Primitive |
| 20 | 21 | 29 | 841 | 841 | Primitive |
| 28 | 45 | 53 | 2809 | 2809 | Primitive |
| 6 | 8 | 10 | 100 | 100 | 3, 4, 5 × 2 |
| 9 | 12 | 15 | 225 | 225 | 3, 4, 5 × 3 |
| 10 | 24 | 26 | 676 | 676 | 5, 12, 13 × 2 |
| 15 | 20 | 25 | 625 | 625 | 3, 4, 5 × 5 |
| 20 | 48 | 52 | 2704 | 2704 | 5, 12, 13 × 4 |
Solved Examples
Example 1: Classic 3-4-5 Triangle Verification
Verify that a triangle with sides 3, 4, and 5 is a right triangle.
Check if a² + b² = c² (where c is the longest side)
3² + 4² = 9 + 16 = 25
5² = 25
25 = 25. Yes, it is a right triangle.
Example 2: Finding the Diagonal of a Rectangle (TV Screen Size)
A TV screen measures 40 inches wide and 30 inches tall. What is the diagonal (screen size)?
The diagonal forms the hypotenuse of a right triangle with the width and height as legs.
d² = 40² + 30²
d² = 1600 + 900 = 2500
d = √2500
d = 50 inches (diagonal screen size)
Example 3: Distance Between Two Points on a Map
Two towns are 12 km apart east-west and 9 km apart north-south. What is the straight-line distance between them?
The east-west and north-south distances form the legs of a right triangle.
d² = 12² + 9²
d² = 144 + 81 = 225
d = √225
d = 15 km
Example 4: Ladder Reaching a Window
A ladder must reach a window 12 feet above the ground. The base of the ladder is 5 feet from the wall. How long must the ladder be?
The ladder is the hypotenuse; the wall height and ground distance are the legs.
L² = 12² + 5²
L² = 144 + 25 = 169
L = √169
L = 13 feet
Practice Questions
1. A right triangle has legs of 8 and 15. Find the hypotenuse.
Answer: c = √(8² + 15²) = √(64 + 225) = √289 = 17
2. Is (7, 24, 25) a Pythagorean triple?
Answer: 7² + 24² = 49 + 576 = 625. 25² = 625. Yes, 625 = 625, so it is a Pythagorean triple.
3. A rectangular garden is 24 meters long and 10 meters wide. How long is the diagonal path across it?
Answer: d = √(24² + 10²) = √(576 + 100) = √676 = 26 meters
4. The hypotenuse of a right triangle is 26 and one leg is 10. Find the other leg.
Answer: b = √(26² - 10²) = √(676 - 100) = √576 = 24
5. A ship sails 6 km north and then 8 km east. How far is it from its starting point?
Answer: Distance = √(6² + 8²) = √(36 + 64) = √100 = 10 km
6. Find the length of a wire stretched from the top of a 20-meter pole to a point on the ground 15 meters from the base.
Answer: Wire = √(20² + 15²) = √(400 + 225) = √625 = 25 meters
Common Mistakes to Avoid
Applying the theorem to non-right triangles
The Pythagorean theorem only works for right triangles. If the triangle does not have a 90° angle, you must use the law of cosines instead: c² = a² + b² - 2ab cos(C).
Squaring errors
A common arithmetic error is computing a + b instead of a² + b². Remember to square each side individually before adding. For example, 3 + 4 = 7 is wrong; you need 3² + 4² = 9 + 16 = 25.
Forgetting to take the square root
After calculating c² = a² + b², you must take the square root to get c. A frequent mistake is reporting c² as the answer. For instance, if c² = 25, the answer is c = 5, not c = 25.
Key Takeaways
- The Pythagorean theorem (a² + b² = c²) applies exclusively to right triangles, where c is the hypotenuse.
- To find a missing leg, rearrange to a = √(c² - b²). The hypotenuse must always be larger than either leg.
- Pythagorean triples are integer solutions like (3, 4, 5), (5, 12, 13), and (8, 15, 17) that satisfy the theorem exactly.
- The theorem extends to practical applications including finding diagonals, distances between points, and required lengths in construction.
- Any multiple of a Pythagorean triple is also a triple: multiplying (3, 4, 5) by 2 gives (6, 8, 10), which also satisfies the theorem.
- To check whether three sides form a right triangle, verify that the sum of squares of the two shorter sides equals the square of the longest side.
Frequently Asked Questions
What is the Pythagorean theorem?
The Pythagorean theorem states that in any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c². It is one of the most fundamental theorems in mathematics, attributed to the ancient Greek mathematician Pythagoras.
What is a Pythagorean triple?
A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a² + b² = c². The most famous is (3, 4, 5). A primitive triple has no common factor greater than 1. Any multiple of a Pythagorean triple is also a triple — for example, (6, 8, 10) = 2 × (3, 4, 5).
Does the Pythagorean theorem work for all triangles?
No — it only applies to right triangles (triangles with one 90° angle). For non-right triangles, use the law of cosines: c² = a² + b² − 2ab·cos(C), which generalizes the Pythagorean theorem. When C = 90°, cos(C) = 0 and it reduces to a² + b² = c².
How can I tell if a triangle is right, acute, or obtuse?
Given three sides where c is the longest: if a² + b² = c², it is a right triangle. If a² + b² > c², it is acute (all angles less than 90°). If a² + b² < c², it is obtuse (one angle greater than 90°). The Pythagorean triple checker above performs this test.
What are real-world applications of the Pythagorean theorem?
The Pythagorean theorem is used in construction (ensuring walls are square), navigation (calculating straight-line distances), computer graphics (distance between points), surveying, architecture, and physics. Any time you need to find a distance in 2D or 3D space, the Pythagorean theorem is involved.