Find the Missing Side Worksheet

Sixteen right triangles with one side missing. Most ask for a leg, a = √c2 − b2, and a few hide the hypotenuse, so read each figure first. Worked answer key on its own page.

By the PythagoreanTheorem.net teamUpdated 4 min read

This find the missing side worksheet gives 16 right triangles with two sides known and one missing. Thirteen problems ask for a missing leg, which means subtracting squares: a2 = c2 − b2. Three ask for the hypotenuse, so students must decide which side is unknown before choosing to add or subtract. The worked answer key prints on a separate page.

Jump to the answer key

What the worksheet practices

Finding a missing leg is the step most students get wrong after they learn a2 + b2 = c2: they add when they should subtract. This sheet drills the decision. Each problem starts the same way: find the right-angle mark, name the side across from it the hypotenuse, and check whether that is the side you know or the one you need.

  • Problems 1–9, whole-number answers from Pythagorean triples. Problems 6 and 9 give both legs, so the hypotenuse is missing. Problem 8 names the sides with letters: in triangle XYZ the right angle is at Y, so XZ is the hypotenuse.
  • Problems 10–13, radical answers in simplest radical form with a decimal, such as √48 = 4√3 ≈ 6.93. Problem 12 is a hypotenuse problem in disguise.
  • Problems 14–15, decimal sides (6.5 ft and 2.5 ft; 4.1 m and 0.9 m) with whole-number answers.
  • Problem 16, a short algebra step: both legs are x, so x2 + x2 = 102.

The triangles appear in many orientations, and the unknown is labeled x, a, b or a question mark, so students cannot rely on a picture always looking the same. Figures are not drawn to scale.

How to use it

  • Warm-up: review the three forms of the theorem on the find the missing side lesson: c = √a2 + b2, a = √c2 − b2 and b = √c2 − a2.
  • Ask for a label first. Before any arithmetic, have students write "leg" or "hypotenuse" next to each x. It takes ten seconds and prevents most errors.
  • Calculators: optional for 1–9 and 14–15; useful for the decimals in 10–13 and 16.
  • Checking: the missing side calculator shows the same steps as the key. Time needed: about 25–35 minutes.

The worksheet: find the missing side

NameDateScore

Directions: Find the missing side of each right triangle. Decide first whether it is a leg or the hypotenuse. Give exact answers; when an answer is not a whole number, also round it to the nearest hundredth. Figures are not drawn to scale.

  • Find x.3x5
  • Find x.12x13
  • Find x.6 cmx10 cm
  • Find b.8b17
  • Find x.24 mx25 m
  • Find x.940x
  • Find x.10 inx26 in
  • Find YZ.20?29YZX
  • Find x.1235x
  • Find x.4x8
  • Find a.3a7
  • Find x.57x
  • Find x.6x12
  • Find x.2.5 ftx6.5 ft
  • Find x.0.9 mx4.1 m
  • Both legs are equal. Find x.xx10

Answer key

Missing leg: subtract the squares. Missing hypotenuse (problems 6, 9 and 12): add them. The boxed value is the answer; ≈ marks a decimal rounded to the nearest hundredth.

  1. x2 = 52 − 32 = 25 − 9 = 16
    x = √16 = 4
  2. x2 = 132 − 122 = 169 − 144 = 25
    x = √25 = 5
  3. x2 = 102 − 62 = 100 − 36 = 64
    x = √64 = 8 cm
  4. b2 = 172 − 82 = 289 − 64 = 225
    b = √225 = 15
  5. x2 = 252 − 242 = 625 − 576 = 49
    x = √49 = 7 m
  6. x2 = 92 + 402 = 81 + 1,600 = 1,681
    x = √1,681 = 41
  7. x2 = 262 − 102 = 676 − 100 = 576
    x = √576 = 24 in
  8. YZ2 = 292 − 202 = 841 − 400 = 441
    YZ = √441 = 21
  9. x2 = 122 + 352 = 144 + 1,225 = 1,369
    x = √1,369 = 37
  10. x2 = 82 − 42 = 64 − 16 = 48
    x = √48 = 4√3 ≈ 6.93
  11. a2 = 72 − 32 = 49 − 9 = 40
    a = √40 = 2√10 ≈ 6.32
  12. x2 = 52 + 72 = 25 + 49 = 74
    x = √74 ≈ 8.60
  13. x2 = 122 − 62 = 144 − 36 = 108
    x = √108 = 6√3 ≈ 10.39
  14. x2 = 6.52 − 2.52 = 42.25 − 6.25 = 36
    x = √36 = 6 ft
  15. x2 = 4.12 − 0.92 = 16.81 − 0.81 = 16
    x = √16 = 4 m
  16. x2 + x2 = 102
    2x2 = 100,  x2 = 50
    x = √50 = 5√2 ≈ 7.07

Teacher notes

  • The classic error: adding when a leg is missing. On problem 1 that gives √52 + 32 = √34 ≈ 5.83, a "leg" longer than the hypotenuse 5. Teach the sense check: a leg is always shorter than the hypotenuse. The common mistakes page lists the others.
  • Subtract squares, not sides. Some students compute √13 − 12 = 1 on problem 2. Ask them to write the squared numbers (169 and 144) before subtracting.
  • Problem 16 gives x = 5√2, the same result as dividing the hypotenuse by √2 in a 45-45-90 triangle.
  • Problem 15 is the 9-40-41 triple divided by 10; problem 14 is 5-12-13 halved.
  • Standard: Common Core 8.G.B.7.

For more practice, the hypotenuse worksheet is a gentler start, the word problems worksheet puts missing legs in real situations, and the worksheet generator makes new missing-leg sheets on demand.

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