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Pythagoras’ theorem: solved examples

Worked examples of pythagoras’ theorem, from easy to hard, with every step written out. Try each one yourself first, then check the working.

✓ Every answer here is computed and checked by the math engine

Easy

  1. Example 1. A right triangle has legs of 7 and 24. How long is the hypotenuse?

    1. Square and add the legs.
      c2=72+242=625c^2 = 7^2 + 24^2 = 625
    2. Take the square root.
      c=625=25c = \sqrt{625} = 25

    Answer: 2525

  2. Example 2. A right triangle has legs of 3 and 4. How long is the hypotenuse?

    1. Square and add the legs.
      c2=32+42=25c^2 = 3^2 + 4^2 = 25
    2. Take the square root.
      c=25=5c = \sqrt{25} = 5

    Answer: 55

  3. Example 3. A right triangle has legs of 60 and 63. How long is the hypotenuse?

    1. Square and add the legs.
      c2=602+632=7569c^2 = 60^2 + 63^2 = 7569
    2. Take the square root.
      c=7569=87c = \sqrt{7569} = 87

    Answer: 8787

Medium

  1. Example 4. A right triangle has a hypotenuse of 25 and one leg of 7. How long is the other leg?

    1. Subtract the squares.
      b2=252−72=576b^2 = 25^2 - 7^2 = 576
    2. Take the square root.
      b=576=24b = \sqrt{576} = 24

    Answer: 2424

  2. Example 5. A right triangle has a hypotenuse of 5 and one leg of 3. How long is the other leg?

    1. Subtract the squares.
      b2=52−32=16b^2 = 5^2 - 3^2 = 16
    2. Take the square root.
      b=16=4b = \sqrt{16} = 4

    Answer: 44

  3. Example 6. A right triangle has a hypotenuse of 87 and one leg of 60. How long is the other leg?

    1. Subtract the squares.
      b2=872−602=3969b^2 = 87^2 - 60^2 = 3969
    2. Take the square root.
      b=3969=63b = \sqrt{3969} = 63

    Answer: 6363

Hard

  1. Example 7. A right triangle has legs of 8 and 5. Find the exact length of the hypotenuse.

    1. Square and add the legs.
      c2=82+52=89c^2 = 8^2 + 5^2 = 89
    2. Simplify the square root.
      c=89=89c = \sqrt{89} = \sqrt{89}

    Answer: 89\sqrt{89}

  2. Example 8. A right triangle has legs of 2 and 4. Find the exact length of the hypotenuse.

    1. Square and add the legs.
      c2=22+42=20c^2 = 2^2 + 4^2 = 20
    2. Simplify the square root.
      c=20=25c = \sqrt{20} = 2\sqrt{5}

    Answer: 252\sqrt{5}

  3. Example 9. A right triangle has legs of 10 and 12. Find the exact length of the hypotenuse.

    1. Square and add the legs.
      c2=102+122=244c^2 = 10^2 + 12^2 = 244
    2. Simplify the square root.
      c=244=261c = \sqrt{244} = 2\sqrt{61}

    Answer: 2612\sqrt{61}

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