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Grades 8 to 10 (Years 8 to 11 in England)

The Pythagorean theorem: a² + b² = c², with a worksheet

Here is how to use the Pythagorean theorem to find a missing side of a right triangle, whether it is the hypotenuse or one of the legs, and how to test whether a triangle has a right angle. Try the free Pythagorean theorem practice below, then print a fresh worksheet with an answer key.

The method, step by step

  1. Check the triangle has a right angle, and find the hypotenuse: the longest side, opposite the right angle.
  2. Write the formula a² + b² = c², where c is the hypotenuse and a and b are the legs.
  3. To find the hypotenuse, square both legs, add them, then take the square root.
  4. To find a leg, square the hypotenuse and the known leg, subtract, then take the square root.
  5. Check that the hypotenuse is the longest side, and round only at the end.

CCSS 8.G.B.6CCSS 8.G.B.7England: GCSE Foundation GeometryEngland: GCSE Higher GeometryEngland: KS3 Geometry and measuresEngland: GCSE Foundation Geometry and measures

Worked example

A right triangle has legs of 9 cm and 12 cm. How long is the hypotenuse?

  1. The legs are a = 9 and b = 12, and the hypotenuse is c.
  2. Square the legs: 9² = 81 and 12² = 144.
  3. Add them: 81 + 144 = 225, so c² = 225.
  4. Take the square root: c = √225 = 15.
  5. Check: 15 is longer than both legs, as the hypotenuse must be.
  6. The hypotenuse is 15 cm.

Practice now

Question 1 of 100 correct

A ladder leans against a wall. Its foot is 10 m from the wall and it reaches 2 m up the wall. How long is the ladder? Round to 1 decimal place.

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What the Pythagorean theorem says

In any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. As a formula, a² + b² = c². The hypotenuse, c, is the side opposite the right angle, and it is always the longest side. The two sides that make the right angle are called the legs.

The squares in the formula are real squares. Draw a square on each side of a triangle with sides 3, 4 and 5. The squares on the legs have areas 9 and 16, and the square on the hypotenuse has area 25. Since 9 + 16 = 25, the two smaller squares exactly fill the big one.

Finding the hypotenuse

When you know both legs, add their squares, then take the square root. Many answers are not whole numbers, so round at the very end.

  • Legs 6 and 8: 36 + 64 = 100, and √100 = 10.
  • Legs 5 and 12: 25 + 144 = 169, and √169 = 13.
  • Legs 7 and 10: 49 + 100 = 149, and √149 ≈ 12.2, to 1 decimal place.

Finding a missing leg

When you know the hypotenuse and one leg, subtract instead of adding. Rearranged, the formula reads a² = c² − b², so you always take the smaller square away from the square of the hypotenuse.

  • Hypotenuse 17, leg 8: 289 − 64 = 225, and √225 = 15.
  • Hypotenuse 25, leg 7: 625 − 49 = 576, and √576 = 24.
  • Hypotenuse 11, leg 6: 121 − 36 = 85, and √85 ≈ 9.2, to 1 decimal place.

A quick rule tells you which to do: finding the longest side, add; finding a shorter side, subtract.

Pythagorean triples

Some right triangles have whole-number sides. These sets are called Pythagorean triples: 3, 4, 5 and 5, 12, 13 and 8, 15, 17 and 7, 24, 25. Multiply a triple by any whole number and you get another, so 6, 8, 10 and 9, 12, 15 work too. Spotting a triple can save a calculation.

Pythagorean theorem word problems

Real problems hide the right triangle. Look for a square corner: a wall meeting the ground, two sides of a rectangle, or north meeting east.

  • A 13 ft ladder stands 5 ft from a wall. How high does it reach? The ladder is the hypotenuse: 169 − 25 = 144, and √144 = 12 ft.
  • A TV screen is 48 inches wide and 27 inches tall. Its diagonal is √(2304 + 729) = √3033 ≈ 55.1 inches.
  • A park is 80 m long and 60 m wide. Walking along two sides is 140 m, but the diagonal is √(6400 + 3600) = √10,000 = 100 m, a shortcut of 40 m.
  • A box is 3 cm by 4 cm by 12 cm. The diagonal of the base is 5 cm, so the longest rod that fits from corner to corner is √(25 + 144) = 13 cm.

The converse: is it a right triangle?

The theorem also works backward. If the squares of the two shorter sides add up to the square of the longest side, the triangle has a right angle. For sides 20, 21 and 29: 400 + 441 = 841 = 29², so it is a right triangle. For sides 6, 7 and 9: 36 + 49 = 85, but 9² = 81, so it is not.

How it is taught in school

In the US, Common Core places the Pythagorean theorem in grade 8. Students explain a proof of the theorem and its converse (8.G.B.6), use it to find unknown side lengths in right triangles in two and three dimensions (8.G.B.7), and use it to find the distance between two points on a coordinate grid (8.G.B.8). It comes back in high school geometry and trigonometry.

In England, Pythagoras' theorem is part of the Key Stage 3 programme of study, where pupils use it to solve problems involving right-angled triangles. At GCSE, Foundation tier applies it in two dimensions, and Higher tier extends it to three-dimensional shapes.

How parents can help

  • Check squares and square roots first. If 12² = 144 and √81 = 9 come quickly, the theorem is easy.
  • Ask your child to point to the hypotenuse before every question. Most mistakes come from mixing it up with a leg.
  • Find right triangles at home: a ladder against a wall, a TV screen, the corner of a table. Measure two sides and predict the third.
  • Learn the 3, 4, 5 triangle. Builders use it to check that a corner is square.
  • Print a fresh worksheet, let your child mark it with the answer key, and read the worked steps together for any they missed.

Common mistakes

  • Adding the legs instead of their squares. With legs 6 and 8, the hypotenuse is 10, not 14.
  • Adding when you should subtract. With hypotenuse 13 and leg 5, the other leg is √(169 − 25) = 12. Adding gives √194 ≈ 13.9, longer than the hypotenuse, which is impossible.
  • Forgetting the square root. c² = 100 means c = 10, not 100.
  • Putting a leg in place of c. The hypotenuse is always opposite the right angle and always the longest side.
  • Using the theorem on a triangle with no right angle. It only works for right triangles.
  • Rounding too early. Keep the full value until the last step, then round once.

Questions people ask

What is the Pythagorean theorem formula?
a² + b² = c², where a and b are the legs of a right triangle and c is the hypotenuse, the side opposite the right angle.
How do you find the hypotenuse?
Square both legs, add them, and take the square root. With legs 9 and 40: 81 + 1600 = 1681, and √1681 = 41.
How do you find a missing leg?
Square the hypotenuse and the known leg, subtract, then take the square root. With hypotenuse 10 and leg 6: 100 − 36 = 64, so the other leg is 8.
Does the Pythagorean theorem work for every triangle?
No, only for right triangles. For other triangles, high school trigonometry uses the law of cosines instead.
What is a Pythagorean triple?
Three whole numbers that fit a² + b² = c², such as 3, 4, 5 or 5, 12, 13. Multiples of a triple, like 6, 8, 10, are triples too.
What grade learns the Pythagorean theorem?
In the US, grade 8 (8.G.B.6 to 8.G.B.8). In England, it is taught in Key Stage 3 and used again throughout GCSE.
Is there a free Pythagorean theorem worksheet with answers?
Yes. Use the Print a worksheet button on this page. Each worksheet is freshly made, and the answer key can show answers only or answers with worked steps.

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