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Pythagorean Calculator

Calculate triangle sides with a² + b² = c²

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Last Updated: March 2, 2026
avatarBy Viblaa Team

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 3-4-5 Rule

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

Right Angles Are Everywhere

Buildings, furniture, screens, layouts—most human-made objects have right angles. Verifying them requires the Pythagorean theorem.

  1. Construction layout — Verify walls are square using 3-4-5 method.

  2. Screen sizing — Calculate TV/monitor dimensions from diagonal.

  3. Distance calculation — Find straight-line distance from coordinates.

  4. Stair building — Calculate stringer length from rise and run.

  5. Navigation — Determine direct distance from offset coordinates.

  6. Landscaping — Lay out square corners for patios and beds.

  7. Woodworking — Verify cuts and assemblies are square.

How to Use the Pythagorean Calculator

  1. Enter known sides — Any two of the three sides.

  2. Select what to solve — Hypotenuse or missing leg.

  3. View result — Calculated side length.

  4. Optional: Verify triple — Check if sides form integer triple.

Common Pythagorean TriplesRatioCommon Use
3, 4, 53:4:5Basic square verification
5, 12, 135:12:13Longer layouts
8, 15, 178:15:17Large construction
7, 24, 257:24:25Extended measurements
6, 8, 103:4:5 ×2Scaled version of basic
9, 12, 153:4:5 ×3Larger scaled version
Only Works for Right Triangles

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

Hypotenuse Is Always Longest

If your calculated "hypotenuse" is shorter than a leg, you made an error. C must be greater than both a and b.

Adding Instead of Squaring
❌ The Mistake
Calculating c = a + b instead of c = √(a² + b²).
✅ The Fix
Each side must be squared first. 3 + 4 = 7, but √(9 + 16) = 5.
Forgetting Square Root
❌ The Mistake
Getting c² = 25 and reporting c = 25 instead of c = 5.
✅ The Fix
The formula gives c². Take the square root to get c.
Using on Non-Right Triangle
❌ The Mistake
Applying a² + b² = c² to any triangle.
✅ The Fix
Theorem only works with exactly 90° angle. Other triangles need law of cosines.
Measuring Inaccurately
❌ The Mistake
Using the 3-4-5 method with sloppy measurements—claiming it's square.
✅ The Fix
The method is precise. Measure carefully. Even 1% error compounds.
Wrong Side as Hypotenuse
❌ The Mistake
Treating a leg as hypotenuse in the formula.
✅ The Fix
Hypotenuse is ALWAYS opposite the right angle. It's always the longest side.

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.

Frequently Asked Questions