Pythagorean Theorem Calculator
Calculate the missing side of a right triangle using the Pythagorean theorem (a² + b² = c²). Find hypotenuse or leg length.
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Pythagorean Theorem Calculator: Find Any Missing Side
The Pythagorean Theorem Calculator is a fast, accurate tool for solving one of the most famous equations in mathematics: a² + b² = c². Whether you need to find the length of the hypotenuse or one of the legs in a right-angled triangle, this calculator does the squaring, adding, subtracting, and square rooting for you instantly.
Discovered independently by multiple ancient civilizations but attributed to the Greek mathematician Pythagoras, this theorem is the foundation of Euclidean geometry. It links geometry to algebra and forms the basis for the distance formula, trigonometry, and countless applications in physics, engineering, and construction.
💡 The Golden Rule: The theorem ONLY applies to right triangles (triangles with exactly one 90° angle). The hypotenuse (c) is always the longest side, and it is always the side exactly opposite the 90° angle.
When to Use This Pythagorean Calculator
🔨 Carpentry & Construction
Ensure corners are perfectly square (90 degrees) using the 3-4-5 rule. Calculate the length of a roof rafter from the rise and run.
📺 Screen Sizes
TVs and monitors are measured diagonally. If you know the width and height of a screen, this theorem gives you the advertised diagonal size.
🧭 Navigation & Distance
Calculate the shortest straight-line distance ("as the crow flies") between two points if you know the distance traveled North/South and East/West.
🎓 Math Homework
Verify your manual calculations for geometry and trigonometry assignments. Understand how radicals and square roots apply to physical shapes.
The Formula Explained
The Pythagorean theorem states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides.
a² + b² = c²
- a and b are the lengths of the legs (the sides that meet at the 90° right angle).
- c is the length of the hypotenuse (the longest side, opposite the 90° angle).
To find the Hypotenuse (c):
c = √(a² + b²)
Square both legs, add them together, then take the square root.
To find a Leg (a or b):
a = √(c² − b²)
Square the hypotenuse and the known leg. Subtract the leg's square from the hypotenuse's square, then take the square root.
Step-by-Step Calculation Guide
Scenario: Finding the Hypotenuse
Given Leg a = 5, Leg b = 12. Find c.
Square Leg a: 5² = 5 × 5 = 25
Square Leg b: 12² = 12 × 12 = 144
Add them (a² + b²): 25 + 144 = 169
Take the square root: c = √169 = 13
Scenario: Finding a Leg
Given Hypotenuse c = 25, Leg b = 24. Find Leg a.
Square Hypotenuse c: 25² = 625
Square Leg b: 24² = 576
Subtract (c² − b²): 625 − 576 = 49
Take the square root: a = √49 = 7
5 Worked Examples
Example 1: The Classic 3-4-5
Given: Legs a = 3, b = 4.
c = √(3² + 4²) = √(9 + 16) = √25 = 5. (This is the most famous Pythagorean triple).
Example 2: Ladder Against a Wall
Given: A 15 ft ladder (hypotenuse c = 15) is placed 9 ft away from a wall (leg b = 9).
Height reached (leg a) = √(15² - 9²) = √(225 - 81) = √144 = 12 ft.
Example 3: TV Screen Size
Given: A TV is 48 inches wide (a) and 27 inches tall (b).
Diagonal (c) = √(48² + 27²) = √(2,304 + 729) = √3,033 ≈ 55 inches. (This is a standard 55-inch TV).
Example 4: Decimal/Fractional Sides
Given: Legs a = 2.5, b = 6.
c = √(2.5² + 6²) = √(6.25 + 36) = √42.25 = 6.5.
Example 5: Isosceles Right Triangle
Given: Legs a = 10, b = 10 (A 45-45-90 triangle).
c = √(10² + 10²) = √(100 + 100) = √200 ≈ 14.142. (Exact answer is 10√2).
Common Mistakes to Avoid
❌ Adding Instead of Subtracting to Find a Leg
Consequence: If you use a = √(c² + b²), your calculated leg will be longer than the hypotenuse. The hypotenuse MUST be the longest side.
✓ Fix: Only ADD when finding the hypotenuse. SUBTRACT when finding a leg.
❌ Forgetting to Take the Square Root
Consequence: Stopping at c² = 25 and saying the hypotenuse is 25, instead of √25 = 5.
✓ Fix: The final step of the Pythagorean theorem is always applying the square root.
❌ Applying to Non-Right Triangles
Consequence: If the triangle doesn't have a 90-degree angle, a² + b² will NOT equal c².
✓ Fix: Verify the triangle has a right angle. If it doesn't, you must use the Law of Cosines.
Tips and Best Practices
- Know the common Pythagorean Triples: 3-4-5, 5-12-13, 8-15-17, and 7-24-25. Multiples of these also work (e.g., 6-8-10 is just 3-4-5 doubled). Recognizing these saves time.
- Sanity Check: Is your calculated hypotenuse longer than both legs? If not, you made a mistake. Is your calculated hypotenuse shorter than the sum of the two legs? (It must be, due to the triangle inequality theorem).
- Use the Converse to check for square: If you measure a triangle and a² + b² exactly equals c², you have proven that the corner is a perfect 90° right angle. Carpenters use this daily.
Frequently Asked Questions
▶What is the Pythagorean Theorem?
▶Does the Pythagorean theorem work for all triangles?
▶How do I find a missing leg instead of the hypotenuse?
▶What is a Pythagorean triple?
▶How do I know which side is the hypotenuse?
▶Can the lengths of a right triangle be negative?
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Conclusion
The Pythagorean Theorem Calculator takes the manual work out of one of the most essential formulas in mathematics. By instantly calculating a² + b² = c², it provides immediate, exact lengths for any missing side of a right triangle.
Whether you are squaring up a room in carpentry, finding the distance between two points, or finishing geometry homework, this tool guarantees accuracy. Remember that this theorem only applies to right triangles, and always verify that your hypotenuse is the longest side of the trio.
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