Pythagorean Theorem Calculator — Find Hypotenuse or Leg Instantly
Calculate the missing side of a right triangle using the formula a² + b² = c². Enter any two sides to find the third with step-by-step explanation. Learn the proof, applications, and common mistakes. For related geometry tools, see our geometry calculators hub.
Table of Contents
- 1What Is the Pythagorean Theorem?
- 2The Formula a² + b² = c²
- 3Hypotenuse vs. Legs
- 4How to Recognize a Right Triangle
- 5Step-by-Step Examples
- 6More Worked Examples
- 7Visual Proof
- 8Pythagorean Triples
- 9Common Mistakes
- 10Real-World Applications
- 11Practice Problems
- 12Tips for Mastery
- 13Final Thoughts
- 14FAQ: Common Questions
Calculate Missing Side of a Right Triangle
Enter two known sides of a right triangle. Leave the side you want to find blank. For example, enter 3 and 4 for legs to find the hypotenuse 5, or enter 5 (hypotenuse) and 3 (leg) to find the other leg.
Pythagorean Theorem – Complete Explanation
The Pythagorean theorem is one of the most fundamental results in geometry. It relates the three sides of a right triangle — a triangle with one 90° angle. The formula a² + b² = c² allows you to find any missing side when the other two are known.
Pythagorean Theorem Formula
a² + b² = c²
Where a and b are the legs (the two shorter sides), and c is the hypotenuse (the longest side, opposite the right angle).
1. What Is the Pythagorean Theorem?
The theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (the legs). This relationship holds true for every right triangle, regardless of size.
Key characteristics:
- Right triangle only: The theorem applies only to triangles with one angle exactly 90°.
- Hypotenuse is always opposite the right angle: It is always the longest side.
- Legs are perpendicular: The two legs meet at the 90° angle.
- Works in both directions: If you know the hypotenuse and one leg, you can find the other leg.
2. The Formula a² + b² = c²
The formula can be rearranged to solve for any side:
To find the hypotenuse (c):
c = √(a² + b²)
To find leg a:
a = √(c² − b²)
To find leg b:
b = √(c² − a²)
Notice that when finding a leg, you subtract the square of the other leg from the square of the hypotenuse, then take the square root.
3. Hypotenuse vs. Legs
Understanding the roles of each side is crucial:
Hypotenuse (c): Always the longest side, opposite the right angle. Never touches the right angle.
Legs (a and b): The two shorter sides that form the right angle. They are perpendicular to each other. Their order doesn't matter (a and b are interchangeable).
4. How to Recognize a Right Triangle
A right triangle is any triangle with one angle measuring exactly 90°. You can identify it by:
1. The right angle symbol: A small square drawn in the corner where the legs meet.
2. The word "right triangle": Problems will often state it explicitly.
3. Pythagorean triples: Sides like 3-4-5, 5-12-13, or 8-15-17 indicate a right triangle.
4. Perpendicular lines: If two sides are perpendicular (form a 90° angle), it's a right triangle.
5. Step-by-Step Examples
Example 1: Find the Hypotenuse (3-4-5 Triangle)
Find c when a = 3 and b = 4.
Step 1: Apply the formula c² = a² + b²
c² = 3² + 4²
c² = 9 + 16
c² = 25
Step 2: Take the square root
c = √25
c = 5
Answer: The hypotenuse is 5.
Example 2: Find a Leg (5-12-13 Triangle)
Find a when c = 13 and b = 12.
Step 1: Rearrange the formula to solve for a
a² = c² − b²
a² = 13² − 12²
a² = 169 − 144
a² = 25
Step 2: Take the square root
a = √25
a = 5
Answer: The missing leg is 5.
6. More Worked Examples
Example 3: a = 6, b = 8 → c = √(36+64) = √100 = 10
Example 4: c = 10, a = 6 → b = √(100−36) = √64 = 8
Example 5: a = 5, c = 13 → b = √(169−25) = √144 = 12
Example 6: b = 1, c = √2 → a = √(2−1) = √1 = 1 (45-45-90 triangle)
Example 7: a = 7, b = 24 → c = √(49+576) = √625 = 25
7. Visual Proof
The Pythagorean theorem has hundreds of proofs — one of the most for any mathematical theorem. A classic visual proof uses squares constructed on each side of the right triangle.
Visualizing a² + b² = c²:
Draw a square with side length (a + b). Inside it, place four identical right triangles with legs a and b and hypotenuse c. The remaining area in the center forms a smaller square.
The area of the large square is (a+b)². The area of the four triangles is 4 × (½ab) = 2ab. The central square has area c².
Therefore: (a+b)² = 2ab + c² → a² + 2ab + b² = 2ab + c² → a² + b² = c²
8. Pythagorean Triples
Pythagorean triples are sets of three positive integers that satisfy a² + b² = c². They form right triangles with whole-number sides.
Common Pythagorean triples:
3-4-5 (and multiples: 6-8-10, 9-12-15, 12-16-20)
5-12-13 (and multiples: 10-24-26, 15-36-39)
8-15-17 (and multiples: 16-30-34, 24-45-51)
7-24-25 (and multiples: 14-48-50, 21-72-75)
20-21-29 (and multiples: 40-42-58)
Recognizing these triples can save you time in calculations. Any multiple of a Pythagorean triple is also a Pythagorean triple.
9. Common Mistakes
- Using the theorem for non-right triangles: The Pythagorean theorem only works for right triangles. For other triangles, use the law of cosines.
- Misidentifying the hypotenuse: The hypotenuse is always opposite the right angle and is always the longest side. If you substitute the hypotenuse as a leg, you'll get an incorrect result.
- Forgetting to take the square root: After computing a² + b², you must take the square root to get c, not just stop at c².
- Subtracting incorrectly for legs: When finding a leg, you subtract the square of the other leg from the square of the hypotenuse, not add them.
- Using negative numbers: Side lengths are always positive. If you get a negative under the square root, you've probably mislabeled the sides.
- Not simplifying square roots: √20 = 2√5, not just 20. The calculator above gives decimal approximations, but exact form is often preferred.
10. Real-World Applications
- Construction: Ensuring walls are perpendicular, calculating roof lengths, diagonal bracing.
- Navigation: Finding the shortest distance between two points (straight-line distance).
- Computer graphics: Calculating distances between pixels or points on a screen.
- Physics: Calculating resultant vectors, displacement, and forces at right angles.
- Surveying: Measuring distances across inaccessible terrain.
- Sports: Calculating the distance from home plate to second base in baseball (a perfect 90° angle).
- Diagonals of rectangles: The diagonal of any rectangle with length l and width w is √(l² + w²).
11. Practice Problems
Problem 1: A right triangle has legs of 9 and 12. Find the hypotenuse.
Solution: c = √(81 + 144) = √225 = 15
Problem 2: A right triangle has a hypotenuse of 17 and one leg of 8. Find the other leg.
Solution: b = √(289 − 64) = √225 = 15
Problem 3: The diagonal of a rectangle measures 13 inches, and one side is 5 inches. Find the other side.
Solution: Other side = √(169 − 25) = √144 = 12 inches
Problem 4: A ladder 20 feet long leans against a wall. The foot of the ladder is 12 feet from the wall. How high up the wall does the ladder reach?
Solution: Height = √(400 − 144) = √256 = 16 feet
12. Tips for Mastery
- Memorize the formula and its rearrangements: c = √(a²+b²), a = √(c²−b²), b = √(c²−a²).
- Recognize common Pythagorean triples to check your work quickly.
- Always identify the hypotenuse first — it's opposite the right angle and is the longest side.
- Draw a diagram whenever possible; it helps prevent mislabeling sides.
- Check that your answer is reasonable: the hypotenuse must be larger than either leg, and each leg must be smaller than the hypotenuse.
- Simplify square roots when possible (e.g., √20 = 2√5).
13. Final Thoughts
The Pythagorean theorem is a cornerstone of geometry that has been known for over 2,500 years. Its beauty lies in its simplicity and remarkable usefulness. The relationship a² + b² = c² connects geometry and algebra, giving us a powerful tool to solve problems involving right triangles in countless real-world situations.
Whether you're calculating the length of a diagonal, verifying a right angle, or solving an abstract geometry problem, the Pythagorean theorem is an essential mathematical tool. Use the calculator above to check your answers, but practice the manual steps until the process becomes second nature.
14. Frequently Asked Questions
a² + b² = c², where a and b are legs of a right triangle, and c is the hypotenuse.
No. It only applies to right triangles. For other triangles, use the law of cosines or law of sines.
The hypotenuse is the longest side, always opposite the right angle (90°).
Sets of three positive integers that satisfy a² + b² = c², like 3-4-5, 5-12-13, and 8-15-17.
Yes. Use the rearranged formula: leg = √(hypotenuse² − other leg²).
Irrational side lengths are common (e.g., √2 ≈ 1.414). Round to the desired precision.
Yes, as long as all sides use the same unit of measurement (meters, feet, inches, etc.).
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Related Calculators
Explore these other geometry tools:
- 🔹 Area of Triangle Calculator — Find the area using base and height or sides.
- 🔹 Distance Between Two Points — Uses the Pythagorean theorem in coordinate geometry.
- 🔹 Rectangle Diagonal Calculator — Find the diagonal of a rectangle.
- 🔹 Trigonometry Calculators — Calculate angles and sides of right triangles.
- 🔹 Law of Cosines Calculator — For non-right triangles.
Conclusion
The Pythagorean theorem is a simple yet profound relationship that connects geometry and algebra. Master it, and you'll have a tool for life.
Readers who wish to continue practicing can explore the companion learning resource for guided exercises.