This Pythagorean theorem calculator solves one missing side of a right triangle. Enter legs a and b and hypotenuse c, with exactly one value set to 0 as the unknown. The default uses a=3, b=4, and c=0, which solves to hypotenuse 5.
When you always have both legs and only need the hypotenuse, the right triangle calculator is a shorter path. For area from three sides, use the triangle calculator.
How the formula works
If c is zero, return √(a² + b²). If a is zero, return √(c² – b²). If b is zero, return √(c² – a²). The hypotenuse must be longer than each known leg when a leg is missing.
Worked example
a = 3, b = 4, c = 0. Missing hypotenuse = √(9 + 16) = √25 = 5.
| Input | Value |
|---|---|
| Leg a | 3 |
| Leg b | 4 |
| Hypotenuse c | 0 (solve) |
| Missing side | 5 |
How to use the fields
- Leg a: enter the length, or 0 to solve for a.
- Leg b: enter the length, or 0 to solve for b.
- Hypotenuse c: enter the length, or 0 to solve for c.
Solving a missing leg
Example pattern: a=0, b=4, c=5 yields a=3. Always confirm c > b before trusting the square root of c² – b².
Common mistakes
- Leaving two zeros instead of one
- Putting the longest side in a leg field when it is the hypotenuse
- Using the theorem on non-right triangles
- Mixing units across sides
Construction layout
Carpenters still use 3-4-5 checks to square a corner. Measure 3 and 4 along the edges and confirm the diagonal is 5 in matching units. This calculator is the digital version of that check when dimensions grow beyond easy mental math.
Classroom practice
Give three variants: missing c, missing a, and missing b, always with a true Pythagorean triple. Students should mark the zero field clearly before calculating.
Then scale to 6-8-10 to show similarity. The zero still marks the unknown on the scaled set.
Distance on a grid
Horizontal and vertical runs on a map form legs. The straight line distance is the hypotenuse. Enter the runs as a and b with c=0 to get that travel length in grid units.
If you already know the straight distance and one run, solve for the other run with a zero in that leg field.
Marking the unknown clearly
Circle the zero field on paper before you type. Most user errors come from leaving the wrong side at zero or zeroing two sides. One unknown keeps the theorem determined.
After you solve, plug all three sides back into a² + b² and compare to c² as a quick check.
Survey style offsets
When a path goes east a meters and north b meters, the straight flight distance is the hypotenuse with c set to 0. That grid reading is the same theorem in map clothing.
If GPS gives the straight distance and one offset, solve for the other offset with a zero in that leg field.
Non integer sides
Real cuts are rarely perfect integers. Enter decimals in matching units and expect a decimal hypotenuse. Rounding too early on the legs can shift the final cut by more than your saw tolerance.
Keep full precision through the square root, then round once for the cut list.
Classroom triple drill
Practice 3-4-5, 5-12-13, and 8-15-17 with the zero in different fields. Students should predict which side is longest before calculating. That prediction habit reduces hypotenuse and leg mix ups.
After each solve, verify a² + b² equals c² within ordinary rounding.
When a result looks impossible, check that the hypotenuse field holds the longest side and that only one field is zero. Those two checks fix most stuck attempts before you change any numbers.
Limitations
Only right triangles with exactly one unknown side are supported. Angles, altitude to the hypotenuse, and oblique triangles are out of scope.