Pythagorean Theorem Word Problems Worksheet

Twelve real-world problems (ladders, screens, ships, a baseball diamond, a roof) with room to sketch and work. The answer key gives a labeled sketch and a full solution for each.

By the PythagoreanTheorem.net teamUpdated 5 min read

This Pythagorean theorem word problems worksheet has 12 real-world questions. In each one students find the hidden right triangle, sketch and label it, decide whether the missing length is a leg or the hypotenuse, and solve with a2 + b2 = c2. Most take one step; problems 3 and 9–12 need two. The answer key with sketches prints on separate pages.

Jump to the answer key

What the worksheet practices

Word problems test the step that pure triangle drills skip: turning a description into a picture. A ladder against a wall, a screen's diagonal and a ship's course all hide the same shape. Students who draw the sketch first get most of these right; students who grab the numbers and guess usually add when they should subtract.

  • Find the hypotenuse: the TV screen, the park shortcut, the baseball diamond, the ship, the support wire and the roof rafter.
  • Find a leg: the ladder's height, the kite's height and the phone screen's height. The missing side worksheet drills this step without the words.
  • Two steps or a decision: problem 3 compares two routes, 9 asks whether a panel fits through a door, 10 uses the converse to judge a corner, 11 halves the width of a roof, and 12 solves two triangles to see how far a slipping ladder drops.

Answers are whole numbers except problems 2, 4 and 9, which need rounding. Units are feet, inches, meters and kilometers; no unit conversion is needed except in 10(b), where 3 inches becomes 0.25 ft.

A four-step plan for every problem

  1. Sketch. Draw the situation and mark the right angle (wall meets ground, north meets east, a rectangle's corner).
  2. Label. Write the known lengths on the sketch and put x on the unknown.
  3. Choose. If x is across from the right angle, add the squares; if it is a leg, subtract.
  4. Answer the question. Give units, round as asked, and check the result makes sense: a ladder cannot reach higher than its own length.

The Pythagorean theorem word problems lesson works through more examples step by step.

The worksheet: word problems

NameDateScore

Directions: Sketch a right triangle for each problem and label the lengths you know. Solve, show your work and include units. Round to the nearest tenth unless the problem says otherwise.

  • A 13-foot ladder leans against a wall. Its foot is 5 feet from the bottom of the wall. How high up the wall does the ladder reach?
  • A TV screen is 48 inches wide and 27 inches tall. TV sizes are measured along the diagonal. What size is this TV, to the nearest inch?
  • A rectangular park is 400 m long and 300 m wide. Maria walks along two edges from one corner to the opposite corner. Jake takes the straight diagonal path. How much shorter is Jake’s walk?
  • A baseball diamond is a square with 90-foot sides. How far is it from home plate straight across to second base? Round to the nearest tenth of a foot.
  • Lena holds a kite string at ground level. The string is pulled straight and is 50 m long, and the kite is directly above a point 30 m away from her. How high is the kite?
  • A ship sails 8 km due north and then 15 km due east. How far is it from its starting point?
  • A support wire is attached to a pole 24 ft above the ground and staked to the ground 10 ft from the base of the pole. How long is the wire?
  • A phone screen has a 6.5-inch diagonal and is 2.5 inches wide. How tall is the screen?
  • A doorway is 3 ft wide and 7 ft tall. Can a thin, flat panel 7.5 ft wide fit through it if you tilt it along the diagonal? Show why.
  • To check that a corner is square, a builder marks 9 ft along one wall and 12 ft along the other. (a) If the corner is exactly 90°, how far apart are the marks? (b) She measures 15 ft 3 in. Is the corner angle more or less than 90°?
  • A roof is 24 ft wide from wall to wall. Its peak is in the middle, 5 ft above the top of the walls. How long is each sloping rafter from wall to peak? (Ignore any overhang.)
  • A 25-ft ladder leans against a wall with its foot 7 ft from the wall. The foot slips 8 ft farther away from the wall. How far down the wall does the top of the ladder slide?

Answer key

Each answer starts with the sketch a student should draw; the unknown is shown in color. Boxed values are the answers.

  1. x5 ft13 ftx2 = 132 − 52 = 169 − 25 = 144
    x = √144 = 12 ft
    The ladder reaches 12 ft up the wall.
  2. 48 in27 inxx2 = 482 + 272 = 2,304 + 729 = 3,033
    x = √3,033 ≈ 55.07
    To the nearest inch it is a 55-inch TV.
  3. 400 m300 mdd2 = 4002 + 3002 = 160,000 + 90,000 = 250,000
    d = √250,000 = 500 m
    Maria: 400 + 300 = 700 m
    700 − 500 = 200 m shorter
  4. home2nd1st3rd90 ft90 ftxx2 = 902 + 902 = 8,100 + 8,100 = 16,200
    x = √16,200 = 90√2 ≈ 127.3 ft
  5. h30 m50 mh2 = 502 − 302 = 2,500 − 900 = 1,600
    h = √1,600 = 40 m
  6. 8 km15 kmdd2 = 82 + 152 = 64 + 225 = 289
    d = √289 = 17 km
  7. 24 ft10 ftww2 = 242 + 102 = 576 + 100 = 676
    w = √676 = 26 ft
  8. h2.5 in6.5 inh2 = 6.52 − 2.52 = 42.25 − 6.25 = 36
    h = √36 = 6 in
  9. 7 ft3 ftdd2 = 32 + 72 = 9 + 49 = 58
    d = √58 ≈ 7.6 ft
    The diagonal is longer than 7.5 ft, so yes, it fits (with about 1.4 inches to spare).
  10. 9 ft12 ft?(a) d2 = 92 + 122 = 81 + 144 = 225
    d = √225 = 15 ft
    (b) 15 ft 3 in = 15.25 ft
    15.252 = 232.5625 > 225, so the angle is more than 90°.
  11. 12 ft5 ftxHalf the width: 24 ÷ 2 = 12 ft
    x2 = 122 + 52 = 144 + 25 = 169
    x = √169 = 13 ft
  12. 2420715both 25 ftBefore: h = √252 − 72 = √625 − 49 = √576 = 24 ft
    After: foot at 7 + 8 = 15 ft
    h = √252 − 152 = √625 − 225 = √400 = 20 ft
    The top slides 24 − 20 = 4 ft.

Teacher notes

  • Grade the sketch. A correct, labeled sketch shows the student understood the situation even if the arithmetic slips. Many teachers give a point for it.
  • Problem 2 explains why a "55-inch" TV is not 55 inches wide: screen sizes are diagonals. Students can check their own screens with the screen size calculator.
  • Problem 4 is an isosceles right triangle, so the answer is 90√2, the 45-45-90 triangle rule. The Pythagorean theorem in sports page has more examples from the field.
  • Problem 10 is the 3-4-5 rule builders use, scaled up by 3. A longer diagonal means the corner is opened wider than 90°.
  • Problem 12 surprises students: the foot moves 8 ft but the top drops only 4 ft. The ladder calculator lets them test other positions.
  • Standard: Common Core 8.G.B.7, applying the theorem in real-world problems.

Next steps: the real-life uses section shows where these triangles appear in jobs and sports, and the worksheet generator makes new word-problem sheets with fresh numbers.

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