Pythagorean Theorem Calculator
2-in-1 Right Triangle Tools
Tool A: Find hypotenuse c from legs a and b
Tool B: Find leg a from hypotenuse c and leg b
How to Use This Pythagorean Theorem Calculator
Right-triangle homework usually asks for one missing side—and the square-root step is where scratchpad errors pile up. This page splits the job into two blocks: find the hypotenuse from both legs, or find a missing leg when you know the hypotenuse and one leg.
- Tool A — hypotenuse. Enter positive leg lengths a and b, then press Calculate. The result is c = √(a² + b²).
- Tool B — missing leg. Enter hypotenuse c and known leg b. The tool computes a = √(c² − b²) and rejects inputs where c ≤ b.
- Copy the result. Use the copy button beside each answer for worksheets or CAD notes.
- Cross-check related tools. For coordinate distance (same math in disguise), try the Distance Calculator. For full triangle solving, use the Triangle Calculator.
Pythagorean Formulas and Practical Applications
Picture a carpenter squaring a corner with a 3-4-5 layout: the longest board sits across the right angle. That longest side is always the hypotenuse c; the two legs meet at 90°.
Core relationship
a² + b² = c²
Legs 3 and 4 give c = √(9 + 16) = 5. Scale to 6-8-10 for a room diagonal check—the ratio stays valid.
Finding the hypotenuse
c = √(a² + b²)
Use Tool A when both legs are known—roof run and rise, ladder height and base offset, or any right triangle where c is the unknown longest side.
Finding a missing leg
a = √(c² − b²)
Tool B rearranges the same identity. If c is not strictly longer than the known leg, the square root would need an imaginary number—so the calculator stops with an error instead of a nonsense length.
Where this shows up on site
- Checking a rectangular pad is square before a pour.
- Converting horizontal and vertical offsets into a diagonal brace length.
- Verifying ladder reach against a wall height and setback.
Standard Units and Conversion Tables
All three sides must share one length unit. Results use high-precision decimal arithmetic with up to 14 decimal places and trailing zeros trimmed.
| Input rule | Notes |
|---|---|
| Same unit on a, b, c | Do not mix inches and feet in one run |
| Positive lengths only | Zero or negative sides are rejected |
| c > known leg | Required for Tool B (missing leg) |
For slope between two coordinate points, see the Slope Calculator. Browse the full math calculators hub for related geometry tools.
Frequently Asked Questions
What is the Pythagorean theorem formula?
a² + b² = c² links the two legs and hypotenuse of a right triangle. Solve for c with √(a² + b²) or for a leg with √(c² − b²).
Why must the hypotenuse be the longest side?
It sits opposite the 90° corner, so it spans the widest opening in the triangle. Label the longest known side as c in Tool B.
Do all three sides need the same unit?
Yes—square the numbers first, then take the root. Mixed units would produce a length that does not match any real measurement.
What is a 3-4-5 right triangle?
The smallest integer triple satisfying the theorem. Double it to 6-8-10 for larger layouts; the right angle check still closes.
Can I use this for non-right triangles?
No. Use the Triangle Calculator with SSS, SAS, ASA, AAS, or SSA when the triangle is not right-angled.
Disclaimer. RapidRatio is informational only—not structural, surveying, or professional engineering advice. Verify results with qualified professionals before field use.