Pythagorean Calculator
Calculate triangle sides with a² + b² = c²
Find hypotenuse or leg
Common triples
High precision
You're laying out a deck and need it perfectly square. The walls meet at what should be a 90° angle—but is it? The screen diagonal is 55 inches, but what are the width and height for a 16:9 ratio?
The Pythagorean theorem (a² + b² = c²) solves right triangle problems everywhere—construction, navigation, screen sizing, distance calculation. This calculator handles all the variations.
What is the Pythagorean Theorem?
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides. It's fundamental to geometry, construction, and countless practical applications.
The formula:
a² + b² = c²
Where:
a and b = the two shorter sides (legs)
c = the longest side (hypotenuse)
Solving for each:
c = √(a² + b²)
a = √(c² - b²)
b = √(c² - a²)
The simplest Pythagorean triple: 3² + 4² = 5². Use 3-4-5 (or multiples like 6-8-10) to verify right angles in construction.
Why People Actually Need This Tool
Buildings, furniture, screens, layouts—most human-made objects have right angles. Verifying them requires the Pythagorean theorem.
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Construction layout — Verify walls are square using 3-4-5 method.
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Screen sizing — Calculate TV/monitor dimensions from diagonal.
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Distance calculation — Find straight-line distance from coordinates.
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Stair building — Calculate stringer length from rise and run.
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Navigation — Determine direct distance from offset coordinates.
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Landscaping — Lay out square corners for patios and beds.
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Woodworking — Verify cuts and assemblies are square.
How to Use the Pythagorean Calculator
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Enter known sides — Any two of the three sides.
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Select what to solve — Hypotenuse or missing leg.
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View result — Calculated side length.
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Optional: Verify triple — Check if sides form integer triple.
| Common Pythagorean Triples | Ratio | Common Use |
|---|---|---|
| 3, 4, 5 | 3:4:5 | Basic square verification |
| 5, 12, 13 | 5:12:13 | Longer layouts |
| 8, 15, 17 | 8:15:17 | Large construction |
| 7, 24, 25 | 7:24:25 | Extended measurements |
| 6, 8, 10 | 3:4:5 ×2 | Scaled version of basic |
| 9, 12, 15 | 3:4:5 ×3 | Larger scaled version |
The Pythagorean theorem requires a 90° angle. For other triangles, use the law of cosines: c² = a² + b² - 2ab·cos(C).
Real-World Use Cases
1. The Square Corner Check
Context: Laying out 12×16 foot deck, need corners at 90°.
Problem: How to verify corners are square?
Solution: 12² + 16² = 400. √400 = 20. Measure diagonal—should be exactly 20 feet.
Outcome: Deck layout verified square before building.
2. The TV Size
Context: 65-inch TV (diagonal), 16:9 aspect ratio.
Problem: What are the actual width and height?
Solution: For 16:9, if width = 16x and height = 9x, then (16x)² + (9x)² = 65². x = 3.54. Width = 56.7", Height = 31.9".
Outcome: Know actual TV dimensions for furniture planning.
3. The Ladder Safety
Context: 20-foot ladder against wall, base 5 feet from wall.
Problem: How high up the wall does it reach?
Solution: h² = 20² - 5² = 400 - 25 = 375. h = 19.36 feet.
Outcome: Ladder reach calculated for safe placement.
4. The Stair Stringer
Context: Deck stairs: 36" total rise, 48" total run.
Problem: How long is the stringer board?
Solution: c = √(36² + 48²) = √(1296 + 2304) = √3600 = 60 inches = 5 feet.
Outcome: Correct lumber length for stair construction.
5. The Walking Distance
Context: Walk 3 blocks east, then 4 blocks north.
Problem: How far is direct distance back?
Solution: c = √(3² + 4²) = √25 = 5 blocks.
Outcome: Shortcut distance calculated (classic 3-4-5).
6. The Picture Frame
Context: Frame inner dimensions 8" × 10". Cutting diagonal brace.
Problem: Diagonal brace length?
Solution: c = √(64 + 100) = √164 = 12.8 inches.
Outcome: Correct cut for frame reinforcement.
7. The Survey Distance
Context: Two points: one 300m east, 400m north of reference.
Problem: Direct distance from reference?
Solution: c = √(300² + 400²) = √250000 = 500 meters.
Outcome: Survey distance calculated from coordinates.
Common Mistakes and How to Avoid Them
If your calculated "hypotenuse" is shorter than a leg, you made an error. C must be greater than both a and b.
Privacy and Data Handling
This Pythagorean Calculator operates entirely in your browser.
- No calculations are sent to any server.
- No data is stored.
- No account required.
- Works completely offline.
Your measurements stay private.
Conclusion
The Pythagorean theorem is 2,500 years old and still used daily. Construction, navigation, screen sizing, and countless applications depend on this simple relationship.
This calculator handles all variations—finding hypotenuse, finding legs, verifying triples. Enter any two sides, get the third.
Ancient math. Modern applications. Timeless utility.