Slope between two points
Calculate slope, rise, run, intercept, line equation and distance from two points. Handle vertical lines and identical points explicitly.
Pythagorean theorem calculator: find a right-triangle side
Use a² + b² = c² for a right triangle, where c is the hypotenuse opposite the right angle. Choose whether the two supplied lengths are both legs or one leg and the hypotenuse.
Free to use, with no sign-in. Inputs and results stay in this page and are not sent with the form. Download a result before leaving; this tool does not save your work.
Worked result
Hypotenuse 5; area 6; perimeter 12.
- Square the two leg lengths.
- Add to obtain the squared hypotenuse.
- Use the positive square root for a physical length.
Find the hypotenuse of a 3-by-4 right triangle
The two legs are perpendicular and use the same length unit.
c² = 3² + 4² = 9 + 16 = 25
c = 5
Area = 3×4/2 = 6 square units
Perimeter = 3 + 4 + 5 = 12 unitsThe hypotenuse is longer than either leg. If your result is shorter than a supplied leg, check which field represents the hypotenuse and whether the geometry is actually a right triangle.
The right angle is an assumption, not a result
For perpendicular legs, the Pythagorean theorem relates the sum of their squared lengths to the squared hypotenuse. If the hypotenuse and one leg are known, subtract the squared leg from the squared hypotenuse before taking a square root.
Once all three lengths are known, area is half the product of the legs and perimeter is their sum. These calculations use Euclidean geometry; they do not establish that real measurements are exactly perpendicular.
c = √(a² + b²)
b = √(c² − a²), c > a > 0
area = a×b/2
perimeter = a + b + cInput limits and common mistakes
Lengths must be strictly positive, finite and in the same unit. In missing-leg mode, the hypotenuse must be greater than the known leg. Equal values would make a degenerate triangle, which is rejected.
Values are bounded to 1e100 and calculated with floating-point arithmetic. A hypotenuse extremely close to a leg can make the missing leg sensitive to rounding. This tool is for right triangles, not arbitrary triangles or geodesic distances.
Questions about pythagorean theorem
Can I use this for any triangle?
No. The squared-side equality applies to a right triangle. Other triangles require other relationships and sufficient geometric information.
Which side is the hypotenuse?
It is opposite the right angle and is the longest side of a nondegenerate right triangle.
Do units matter?
Yes. Convert all lengths to the same unit first. Area then uses that unit squared.
Sources and verification
These references explain the formulas or formats. The worked examples above were created for this tool and checked against independent calculations or format round trips. AI-assisted development does not imply a named human expert review.
- OpenStax: the Pythagorean theorem — Right-triangle assumptions and the squared-side relationship.
Published 2026-10-23. Examples are illustrations, not measurements of a service or claims about search rankings.
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Further reading
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