Formula
Hypotenuse c = √(a² + b²). Square check: c² = a² + b². Optional difference = measured hypotenuse − calculated hypotenuse.
Math & Geometry
Pythagorean Theorem Calculator
Find the hypotenuse of a right triangle from two legs, with square-sum proof, optional check length and a printable classroom or layout record.
Math & Geometry
Pythagorean Theorem Calculator
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Live resultReadyCalculator queued
Formula used
Hypotenuse c = √(a² + b²). Square check: c² = a² + b². Optional difference = measured hypotenuse − calculated hypotenuse.
This is the method behind the answer, so the result can be checked rather than simply trusted.Visual grid
This number is one point on a larger pattern
Pythagorean Theorem is not just a final answer. It is a step on a line: before and after, input and output, assumption and result.
Micro-timehours, minutes, shiftsHuman scaledays, weeks, projectsMacro-timemonths, years, calendars
InputFormulaResult
ReadyCalculationTime keeps the path visible: the input, the method and the final number belong together.
CalculationTime
Pythagorean Theorem Calculation Report
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Inputs
- Leg a
- 3 units
- Leg b
- 4 units
- Optional measured hypotenuse
- 5 units
- Round display to
- 3 decimal places
Method
Hypotenuse c = √(a² + b²). Square check: c² = a² + b². Optional difference = measured hypotenuse − calculated hypotenuse.
- For legs 3 and 4: a² + b² = 3² + 4² = 9 + 16 = 25. The square root of 25 is 5, so the hypotenuse is 5 units.
Assumptions
- The triangle is a right triangle: leg a and leg b meet at a 90° angle.
- Both legs use the same unit. Do not mix inches and centimetres without converting first.
- The optional measured hypotenuse is a comparison line, not an input to the theorem result.
- Rounding affects the displayed answer only; keep more precision for layout, machining, engineering or assessed work.
Notes
Use this space on the printed report for client, supplier, classroom, job-location, measurement, quote or approval notes.
Worked example
For legs 3 and 4: a² + b² = 3² + 4² = 9 + 16 = 25. The square root of 25 is 5, so the hypotenuse is 5 units.
Professional note
Master’s Tip: for set-out work, mark the two legs and the diagonal on the same record. A 3-4-5 triangle is useful because the diagonal confirms the corner is square, not merely that two lengths were measured.
Regional and unit assumptions
Standard or basis: Euclidean right-triangle geometry using consistent units. The page does not replace surveying, engineering, building-code or site-tolerance requirements.
Assumptions and limitations
- The triangle is a right triangle: leg a and leg b meet at a 90° angle.
- Both legs use the same unit. Do not mix inches and centimetres without converting first.
- The optional measured hypotenuse is a comparison line, not an input to the theorem result.
- Rounding affects the displayed answer only; keep more precision for layout, machining, engineering or assessed work.
Methodology & Accuracy
How this calculator is checked
CalculationTime pages are built around visible arithmetic: the formula, assumptions, worked example and practical limitations are shown so the result can be checked rather than simply trusted.
Formula used
Hypotenuse c = √(a² + b²). Square check: c² = a² + b². Optional difference = measured hypotenuse − calculated hypotenuse.
Standard or basis
Standard or basis: Euclidean right-triangle geometry using consistent units. The page does not replace surveying, engineering, building-code or site-tolerance requirements.
Where a calculator follows a named legal, trade or industry standard, that standard is cited visibly. Otherwise the page uses transparent general arithmetic and states its limits.Master's Tip
Master’s Tip: for set-out work, mark the two legs and the diagonal on the same record. A 3-4-5 triangle is useful because the diagonal confirms the corner is square, not merely that two lengths were measured.
Related calculators
Questions
How do you use the Pythagorean theorem?
For a right triangle, square both legs, add those squares, then take the square root to find the hypotenuse.
What is the hypotenuse for 3 and 4?
The hypotenuse is 5 because 3² + 4² = 9 + 16 = 25, and √25 = 5.
Can I use different units for the two legs?
No. Convert both legs into the same unit first, then apply the formula.
Does this prove my corner is square?
It can help. If the measured diagonal matches the calculated hypotenuse for two perpendicular legs, the layout is consistent with a right angle within your measurement tolerance.
What if I know the hypotenuse and one leg?
Use the rearranged form: missing leg = √(hypotenuse² − known leg²). This page focuses on finding the hypotenuse from two legs.
Calculation note
The Pythagorean theorem is one of the most useful bridges between classroom geometry and practical layout. It turns two perpendicular sides into a diagonal, making it useful for construction set-out, ramps, screens, maps, ladders, drawing checks and distance problems.
The theorem is a square-area statement
The familiar formula a² + b² = c² says that the square built on the hypotenuse has the same area as the two squares built on the legs combined. The calculator keeps that square-sum step visible before taking the square root.
The 3-4-5 triangle is a practical squaring tool
A triangle with legs in the ratio 3 and 4 has a diagonal of 5. Builders and students often use that relationship because it gives a quick check that two measured legs form a right angle.
Print the diagonal with the assumptions
A printable report is useful for homework, timber or tile layout, site notes and drawing checks because it preserves the two legs, calculated diagonal, optional measured diagonal and rounding basis on one page.
Sources and further readingEncyclopaedia Britannica: Pythagorean theoremKhan Academy: Pythagorean theorem