Pythagorean theorem
Pythagorean theorem reference and worked examples
A six-page printable guide to right-triangle parts, square areas, missing sides, irrational lengths and the coordinate distance formula.
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Pythagorean theorem worksheet: missing side lengths
Print a four-question Pythagorean theorem worksheet for missing hypotenuses or legs. Choose whole-number triples or irrational results from whole or decimal givens.
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Find the missing length. Show the squared-length equation and your calculation. Round noninteger answers to four decimal places.
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Choose the number type and the missing side
Whole-number practice uses exact Pythagorean triples. Decimal-answer practice keeps the given sides whole but requires a square root that is not an integer. Decimal-side practice introduces tenths. Choose a missing hypotenuse, a missing leg, or a mix of both.
Keep the squared calculations exact until the final step. The worked key distinguishes an exact answer from an approximation and rounds noninteger roots to four decimal places. Length and distance are always nonnegative.
Learning objective
What this activity builds
Choose addition or subtraction of squared lengths from the position of the missing side, then take the positive square root.
Worked example
A right triangle has hypotenuse 13 and one leg 5. Find the other leg.
12 units.
The missing side is a leg, so subtract the known leg square from the hypotenuse square: 13² − 5² = 169 − 25 = 144. The positive square root of 144 is 12. Adding the squares here would solve the wrong problem.
Teaching notes
Make the practice count
- Mark the hypotenuse before substituting any numbers.
- Start with whole-number triples so learners can check an answer by squaring it.
- Move to decimal answers only after the choice between addition and subtraction is secure.
- Retain the exact squared result until the final square-root calculation; round once at the end.
Choose the right Pythagorean theorem worksheet settings
Start with Whole-number triples for exact integer answers. Move to Whole sides, decimal answer when learners are ready to calculate irrational lengths from whole-number givens, then use Decimal sides for givens in tenths. Every setting generates four right triangles.
Choose Hypotenuse for four addition problems or Leg for four subtraction problems. Mixed, the default, includes two missing hypotenuses and two missing legs so learners must identify the unknown side before choosing an operation.
When should I add the squared side lengths, and when should I subtract?
Add the two leg squares when the hypotenuse is missing. When a leg is missing, subtract the known leg square from the hypotenuse square. Identify the hypotenuse opposite the right angle before writing the equation.
Which answers are exact, and which should be rounded?
Whole-number triples produce exact integer answers. The Whole sides, decimal answer and Decimal sides settings both produce an irrational length: keep its square-root form exact during the work, then round the final length to four decimal places. The worked answer key shows both forms.
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Customize the practice
Choose your options, update the preview, then download the exact same pages.
Each sheet has four original right triangles. Whole-number mode uses generated Pythagorean triples. Decimal modes keep exact squared quantities and round only an irrational final length to four decimal places.
Enlarged pages scroll. PDF size stays the same.
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