A free printable Pythagorean theorem worksheet for Grade 8 math.
This free pythagorean theorem worksheet gives Grade 8 students lots of practice with a² + b² = c². It starts with spotting the hypotenuse on triangles turned at different angles, then moves on to finding the hypotenuse and missing legs on triangles drawn to scale.
Students then use the theorem in reverse to test whether triangles are right triangles, and finish with five real-world word problems: a ladder, a TV screen, a shortcut across a park, a kite and a wheelchair ramp. Every answer in the key shows the working and was checked by computer.
| Part | Skill practised | Marks |
|---|---|---|
| A | Identify the hypotenuse | 3 |
| B | Find the hypotenuse (4 drawn triangles) | 4 |
| C | Find a missing leg (4 drawn triangles) | 4 |
| D | Is it a right triangle? | 2 |
| E | Five real-world word problems | 10 |
| F | Spot the mistake | 2 |
It supports the Ontario Grade 8 math curriculum expectations in the Spatial Sense strand. Each triangle has a work box for showing steps.
In a right triangle, the two sides that form the right angle are the legs (a and b), and the longest side, across from the right angle, is the hypotenuse (c). The Pythagorean theorem says:
a² + b² = c²
The first skill on this Pythagorean theorem worksheet is finding the hypotenuse even when the triangle is rotated: it's always the side opposite the little square that marks the right angle. For background and proofs, see the Pythagorean theorem on Wikipedia.
Square both legs, add them, then take the square root. For legs of 6 m and 8 m: c² = 6² + 8² = 36 + 64 = 100, so c = 10 m. When the answer isn't a perfect square, round: legs of 7 m and 9 m give c² = 130, so c ≈ 11.4 m.
When you know the hypotenuse, subtract instead of adding: b² = c² − a². With a hypotenuse of 10 cm and a leg of 6 cm, b² = 100 − 36 = 64, so b = 8 cm. A quick check: a leg must always be shorter than the hypotenuse. Part C of the Pythagorean theorem worksheet practises this on four triangles.
The theorem also works backwards. If the squares of the two shorter sides add up to the square of the longest side, the triangle has a right angle.
| Sides | a² + b² | c² | Right triangle? |
|---|---|---|---|
| 9, 12, 15 | 9² + 12² = 225 | 15² = 225 | Yes |
| 5, 6, 8 | 5² + 6² = 61 | 8² = 64 | No |
| 8, 15, 17 | 8² + 15² = 289 | 17² = 289 | Yes |
| 7, 10, 12 | 7² + 10² = 149 | 12² = 144 | No |
Builders use the 3-4-5 version of this test to check that corners are square. To revise solving equations like these, try the algebra basics quiz.
These questions match the Pythagorean theorem worksheet PDF, which has every triangle drawn to scale and work boxes for each answer.
Questions 1–3: write the letter of the hypotenuse in three rotated right triangles.
Open each part of the Pythagorean theorem worksheet answers to check your work. The printable key is at the end of the PDF.
For more practice, try the geometry shapes and properties quiz and the multi-step word problems quiz.
Yes. The worksheet and its answer key are free to download and print as a PDF, with no account needed.
In any right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a² + b² = c². The hypotenuse is the longest side, opposite the right angle.
Rearrange the formula and subtract: b² = c² − a². Then take the square root. For a hypotenuse of 10 and a leg of 6, b² = 100 − 36 = 64, so b = 8.
Square the two shorter sides and add them. If the total equals the square of the longest side, the triangle is a right triangle. For 9, 12 and 15: 81 + 144 = 225 = 15², so yes.
It is written for Grade 8 math and also works for Grade 9 review.
Ready to start? Download the free Pythagorean theorem worksheet PDF (use the Download button above), print it double-sided, and keep the answer key pages for marking.