To solve a Pythagorean theorem word problem, draw the situation, label the right triangle hiding in it (two legs and the hypotenuse opposite the right angle), solve a2 + b2 = c2 for the missing side, and sense-check the answer against reality. The fifteen real-world problems below each come with a sketch and a full solution.
The problems get harder as you go. Problems 1 to 6 need one use of the theorem, 7 to 11 need an extra step, and 12 to 15 need a clever sketch or some algebra. Try each one before you open the solution. If the basic formula is still new, start with the worked examples.
A four-step strategy for any word problem
Here is the strategy at work on a ladder problem: a 20 ft ladder leans against a wall with its foot 5 ft from the wall. How high up the wall does it reach?
- Draw. Sketch the scene, however roughly, and find the right angle. It's usually where something vertical meets something horizontal: a wall and the floor, a pole and the ground, the two edges of a screen.
- Label. Write the known lengths on the sketch, with units, and a letter for the unknown. The hypotenuse is the side opposite the right angle: the ladder, the wire, the diagonal, the straight-line path.
- Solve. Substitute into a2 + b2 = c2. Add the squares if you want the hypotenuse; subtract if you want a leg.
- Sense-check. The hypotenuse must be the longest side, the units must match, and the number must be believable. A 20 ft ladder can't reach 25 ft.
One-step word problems
1. Ladder against a wall
Ladder safety guides recommend the "4-to-1 rule": put the foot 1 ft from the wall for every 4 ft of ladder up to where it rests. A 20 ft ladder is placed with its foot 5 ft from a wall. How high up the wall does it reach, to the nearest tenth of a foot?
Show solution
The wall and the ground meet at a right angle, so the ladder is the hypotenuse (20) and the distance from the wall is a leg (5).
h2 + 52= 202h2= 400 − 25 = 375h= √375 = 5√15 ≈ 19.4Answer: about 19.4 ft. Sense-check: a little less than the ladder's length, as it should be for a steep ladder. The ladder makes an angle of about 75.5° with the ground; the ladder calculator solves problems like this one.
2. What size is this TV?
TVs are sold by the length of the screen's diagonal. A screen measures 48 in wide and 27 in tall. What size would it be advertised as?
Show solution
The width and height are the legs; the diagonal is the hypotenuse.
d2= 482 + 272= 2304 + 729 = 3033d= √3033 ≈ 55.07Answer: about 55.1 in, so it is sold as a 55-inch TV. Sense-check: the diagonal is longer than the 48 in width but much less than 48 + 27 = 75. Try other screens with the screen size calculator.
3. Home plate to second base
A baseball diamond is a square with bases 90 ft apart. The catcher throws from home plate straight to second base. How far is the throw, to the nearest foot?
Show solution
Home, first and second base form a right triangle with the right angle at first base. Both legs are 90 ft.
d2= 902 + 902 = 16,200d= 90√2 ≈ 127.3Answer: about 127 ft. Sense-check: a square's diagonal is always about 1.414 times its side. That's the 45-45-90 triangle shortcut.
4. How far from port?
A ship leaves port and sails 88 km due east, then turns and sails 105 km due north. How far is it from port in a straight line?
Show solution
East and north are perpendicular, so the two legs of the journey are the legs of a right triangle.
d2= 882 + 1052= 7744 + 11,025 = 18,769d= √18,769 = 137Answer: 137 km. Sense-check: the ship sailed 88 + 105 = 193 km, so going straight would save 56 km. Navigation uses the same idea; see navigation and GPS.
5. Swimming across a river
A swimmer aims straight across a river 72 m wide, but the current carries her 21 m downstream by the time she reaches the far bank. How far did she actually swim?
Show solution
The width of the river (straight across) and the drift (along the bank) are at right angles. Her real path is the hypotenuse.
d2= 722 + 212= 5184 + 441 = 5625d= √5625 = 75Answer: 75 m. Sense-check: only 3 m more than the width, which makes sense because the drift is small compared with the width. (21-72-75 is the 7-24-25 triple multiplied by 3.)
6. Corner to corner on a soccer field
A soccer field measures 105 m by 68 m. A player jogs diagonally from one corner flag to the opposite one. How far does she jog, to the nearest tenth of a meter?
Show solution
d2= 1052 + 682= 11,025 + 4624 = 15,649d= √15,649 ≈ 125.1Answer: about 125.1 m. Sense-check: longer than the 105 m length but far shorter than the 173 m it takes to go round two sides. More sports examples: the theorem in sports.
Two-step word problems
In these problems the theorem gives you a number that still needs one more operation: a subtraction, an addition, or a multiplication. The sense-check step matters most here, because it is easy to stop one step early.
7. The park shortcut
A rectangular park is 91 m long and 60 m wide. Instead of walking along two sides from corner A to corner B, you cut straight across the grass. How many meters do you save?
Show solution
Step 1, the diagonal:
d2= 912 + 602= 8281 + 3600 = 11,881d= √11,881 = 109Step 2, compare: along the sides is 91 + 60 = 151 m, and 151 − 109 = 42.
Answer: you save 42 m. Sense-check: the question asked for the saving, not the diagonal. Stopping at 109 is the classic two-step mistake.
8. How high is the kite?
Maya flies a kite on 85 m of string, which we'll treat as a straight line. The kite is directly above a point 36 m away from her, measured horizontally. Her hand is 1.5 m above the ground. How high is the kite above the ground?
Show solution
Step 1, the right triangle has its corner level with her hand, not at the ground. The string is the hypotenuse.
h2= 852 − 362= 7225 − 1296 = 5929h= 77Step 2, add the height of her hand: 77 + 1.5 = 78.5.
Answer: 78.5 m above the ground. Sense-check: a real string sags, so the true height is a little less. The straight-string answer is an upper limit.
9. A wheelchair ramp
The ADA Standards for Accessible Design limit a ramp's slope to 1:12, meaning at most 1 inch of rise for every 12 inches of horizontal run. A porch is 2 ft above the ground. What is the shortest legal run, and how long is the sloping ramp surface?
Show solution
Step 1, the run: 2 × 12 = 24 ft.
Step 2, the ramp surface is the hypotenuse:
L2= 22 + 242 = 4 + 576 = 580L= √580 ≈ 24.08Answer: a 24 ft run and a ramp surface of about 24.08 ft. Sense-check: gentle slopes make the hypotenuse barely longer than the run. Here the difference is about 0.08 ft, roughly 1 inch.
10. Fabric for a tent
The front of an A-frame tent is an isosceles triangle 2.2 m wide and 1.2 m tall at the center pole. The tent is 2.5 m long. How long is each sloping side, and how much fabric covers the two sloping roof panels?
Show solution
Step 1, the pole splits the front into two right triangles with legs 1.2 m and 2.2 ÷ 2 = 1.1 m:
s2= 1.12 + 1.22= 1.21 + 1.44 = 2.65s= √2.65 ≈ 1.628Step 2, each roof panel is a rectangle s by 2.5 m: 2 × 1.628 × 2.5 ≈ 8.14.
Answer: each side is about 1.63 m, and the two panels need about 8.14 m² of fabric (before seams). Sense-check: halving the width is the step people forget. Using 2.2 m as a leg gives about 2.51 m, which is longer than the whole tent is wide.
11. Guy wires for a pole
A pole is held steady by three identical guy wires. Each is attached 18 m up the pole and anchored to the ground 8 m from its base. How much wire is needed in total, to the nearest tenth of a meter?
Show solution
Step 1, one wire is the hypotenuse:
w2= 182 + 82 = 324 + 64 = 388w= √388 = 2√97 ≈ 19.70Step 2, three wires: 3 × √388 = 6√97 ≈ 59.09.
Answer: about 59.1 m (before any extra for fastening). Sense-check: round after multiplying. Rounding one wire to 19.7 first gives exactly the same here, but early rounding often shifts the last digit.
Harder real-world problems
These need a 3D sketch, a little algebra, or the insight that the problem is really asking about a diagonal.
12. Will the umbrella fit in the box?
A shipping box measures 60 cm by 40 cm by 30 cm. Will a straight 76 cm umbrella fit inside: (a) lying flat on the bottom, (b) at any angle?
Show solution
(a) The longest line on the bottom is its diagonal:
√602 + 402 = √5200 ≈ 72.1(b) The longest line in the box is the space diagonal, from a bottom corner to the opposite top corner:
d = √602 + 402 + 302 = √6100 ≈ 78.1Answer: it won't fit flat (72.1 cm < 76 cm), but it fits on the space diagonal (78.1 cm > 76 cm). Sense-check: there's only about 2 cm to spare, so a thick handle may not fit. The 3D Pythagorean theorem explains the formula.
13. Cutting a roof rafter
A house is 28 ft wide and has a "6/12 pitch" roof, meaning it rises 6 in for every 12 in of horizontal run. The rafters meet at the center. How long is a rafter from the ridge to the top of the wall? If it also overhangs 1 ft horizontally past the wall, what is its total length?
Show solution
The run is half the width: 14 ft. A 6/12 pitch means the rise is half the run: 7 ft.
r2= 142 + 72 = 196 + 49 = 245r= √245 = 7√5 ≈ 15.65With the overhang the run is 15 ft and the rise is 7.5 ft (still half):
√152 + 7.52 = √281.25 ≈ 16.77Answer: about 15.65 ft to the wall and about 16.77 ft in total. Sense-check: carpenters do exactly this. The rafter calculator and the page on construction show more.
14. The bookcase that won't stand up
You assemble a bookcase 90 in tall and 20 in deep while it lies on its back. The basement ceiling is 92 in high. When you tip the bookcase upright, will it hit the ceiling?
Show solution
As it tips about its bottom edge, the far top corner swings in a circle. Its radius is the diagonal of the bookcase's side: legs 90 and 20.
d2= 902 + 202 = 8100 + 400 = 8500d= √8500 ≈ 92.2Answer: yes. The corner needs about 92.2 in of clearance, and the ceiling is only 92 in high, so it scrapes by about 0.2 in. Sense-check: the bookcase itself is shorter than the ceiling, which is why this surprises people. The diagonal, not the height, is what matters while it tips.
15. Width and height of a 65-inch TV
A 65-inch TV has a 16:9 screen: its width and height are in the ratio 16 to 9. Find the width and height, to the nearest tenth of an inch.
Show solution
Write the width as 16x and the height as 9x:
(16x)2 + (9x)2= 652256x2 + 81x2= 4225337x2= 4225x= 65 over √337 ≈ 3.541Width = 16x ≈ 56.7 in; height = 9x ≈ 31.9 in.
Answer: about 56.7 in wide and 31.9 in tall. Sense-check: √56.652 + 31.872 ≈ 65, and 56.7 ÷ 31.9 ≈ 1.78 ≈ 16 ÷ 9.
Traps to watch for
- Mixed units. Convert everything to one unit before you square. 6 ft and 30 in is 72 in and 30 in, not 6 and 30.
- The wrong side as hypotenuse. The ladder, the string, the wire and the diagonal are hypotenuses. Heights and ground distances are legs.
- Stopping one step early. Read the question again before you write the answer: savings, totals, heights above the ground.
- Forgetting to halve. Isosceles shapes like tents, roofs and gables split into two right triangles, each with half the base.
- No right angle. The theorem only works if the angle really is 90°. For a sloping ground or a leaning wall, use the law of cosines.
More slips like these are collected on the common mistakes page.
Answers at a glance
| # | Problem | Answer |
|---|---|---|
| 1 | Ladder height | 5√15 ≈ 19.4 ft |
| 2 | TV size | ≈ 55.1 in (a 55-inch TV) |
| 3 | Home to second base | 90√2 ≈ 127 ft |
| 4 | Ship from port | 137 km |
| 5 | River swim | 75 m |
| 6 | Soccer field diagonal | ≈ 125.1 m |
| 7 | Park shortcut saving | 42 m |
| 8 | Kite height | 78.5 m |
| 9 | Ramp | run 24 ft, surface ≈ 24.08 ft |
| 10 | Tent | sides ≈ 1.63 m, fabric ≈ 8.14 m² |
| 11 | Guy wires | 6√97 ≈ 59.1 m |
| 12 | Umbrella in a box | not flat (72.1 cm); yes diagonally (78.1 cm) |
| 13 | Rafter | ≈ 15.65 ft; ≈ 16.77 ft with overhang |
| 14 | Bookcase | hits the ceiling (needs ≈ 92.2 in) |
| 15 | 65-inch TV | ≈ 56.7 in by 31.9 in |
Want a printable set? The word problems worksheet has twelve more with an answer key. For harder puzzles, try the challenge problems.
Frequently asked questions
How do you know when to use the Pythagorean theorem in a word problem?
Look for a right angle and two known lengths. Clues include something vertical meeting something horizontal (walls, poles, towers), perpendicular directions (north and east), rectangles and their diagonals, and the words "straight-line distance" or "diagonal".
What are some real-world examples of the Pythagorean theorem?
Ladder heights, TV and phone screen sizes, ramp and roof lengths, distances on maps, the diagonal of a field or a room, and checking that a corner is square in construction. See what the theorem is used for for many more.
Should I round my answer in a word problem?
Usually yes, because real measurements aren't exact. Round to a sensible precision, often the precision of the measurements you were given, and only at the last step. If the question asks for an exact answer, leave it as a radical such as 5√15.
What if the problem has no picture?
Draw one yourself. That's the first step of the strategy and the most important one. Even a rough sketch shows you where the right angle is and which side is the hypotenuse.
Why is my answer longer than the hypotenuse?
You probably added the squares when you should have subtracted. When the hypotenuse is given (a ladder, a wire, a string) and you want a leg, use a2 = c2 − b2.
Keep exploring
- PracticePythagorean Theorem ExamplesFifteen Pythagorean theorem examples solved step by step, from finding the hypotenuse to 3D diagonals, coordinates and algebra, each with a diagram and answer.
- PracticePythagorean Theorem Word Problems WorksheetPrintable Pythagorean theorem word problems worksheet: 12 real-world problems with room to work, plus an answer key with labeled sketches and full solutions.
- Real-life usesWhat Is the Pythagorean Theorem Used For?What is the Pythagorean theorem used for? 17 real-life uses in building, screens, navigation, GPS, games, physics and sports, each with a worked calculation.
- CalculatorsLadder Calculator: Length, Height and AngleLadder calculator. Enter any two of ladder length, height on the wall and base distance to get the third and the angle, checked against the 4-to-1 safety rule.
- CalculatorsScreen Size Calculator for TVs and MonitorsTV and monitor screen size calculator. Turn a diagonal and aspect ratio into width, height and area in inches and cm, get PPI, and compare two screens.
- PracticePythagorean Theorem Practice Problems30 Pythagorean theorem practice problems in three levels: missing sides, radicals, the converse, coordinates and algebra, each with a worked solution.