These 15 Pythagorean theorem examples are solved step by step and ordered from easiest to hardest. Each uses a2 + b2 = c2, where a and b are the legs and c is the hypotenuse: add the squares of the legs to find c, or subtract to find a missing leg. Every example has a diagram and its full working.
Try each example problem yourself before reading the solution, then compare your working line by line. If an idea is new, the link under the example takes you to the lesson that explains it. For problems with hidden answers, go to the practice problems.
The method behind every example
Every example on this page follows the same four moves. Once they are automatic, most right-triangle problems take under a minute.
- Find the right angle and the hypotenuse. The hypotenuse is the side opposite the right angle and is always the longest side. Call it c. The other two sides are the legs a and b. See how to identify the hypotenuse if you're unsure.
- Write the theorem with the numbers in it. Put the known lengths into a2 + b2 = c2.
- Solve. If c is missing, add the squares. If a leg is missing, subtract: a2 = c2 − b2.
- Take the square root and check. The hypotenuse must come out longer than each leg but shorter than the two legs added together.
Finding the hypotenuse or a leg
These first three examples use Pythagorean triples, sets of whole numbers that fit the theorem exactly, so the answers come out as whole numbers.
Example 1: find the hypotenuse
A right triangle has legs of length 9 and 40. Find the hypotenuse.
Answer: c = 41.
Check: 41 is longer than 40 but shorter than 9 + 40 = 49.
Example 2: find a missing leg
The hypotenuse of a right triangle is 37 and one leg is 12. Find the other leg.
Answer: b = 35.
Because the hypotenuse was given, you subtract. Adding would give √1513 ≈ 38.9, which is longer than the hypotenuse and so impossible. More practice: finding a missing side.
Example 3: spot the hypotenuse in a turned triangle
Triangle PQR has a right angle at Q. PR = 65 and QR = 16. Find PQ.
The side opposite the right angle at Q is PR, so PR = 65 is the hypotenuse, even though it is drawn along the bottom.
PQ2= 652 − 162= 4225 − 256= 3969PQ= 63Answer: PQ = 63.
Letters and orientation don't matter. Find the right angle; the side across from it is c.
Exact radical answers, decimals and units
Most right triangles don't have whole-number sides. When c2 is not a perfect square, give the exact answer in simplest radical form and then a rounded decimal.
Example 4: a hypotenuse that is a radical
Find the hypotenuse of a right triangle with legs 4 and 8. Give the exact answer and a decimal to the nearest hundredth.
Answer: c = 4√5 ≈ 8.94.
To simplify, look for the largest perfect square that divides 80. It is 16, so √80 = √16√5 = 4√5.
Example 5: a missing leg that is a radical
A right triangle has hypotenuse 10 and one leg 4. Find the other leg exactly and to the nearest hundredth.
Answer: b = 2√21 ≈ 9.17.
The square root simplifier checks this kind of step.
Example 6: decimal side lengths
The legs of a right triangle are 2.8 m and 4.5 m. How long is the hypotenuse?
Answer: c = 5.3 m exactly.
Decimals follow the same steps. Square carefully (2.82 = 7.84, not 5.6) and round only at the end.
Example 7: mixed units
One leg of a right triangle is 0.65 m and the other is 72 cm. Find the hypotenuse in meters.
Convert first: 0.65 m = 65 cm.
c2= 652 + 722= 4225 + 5184 = 9409c= √9409 = 97 cmAnswer: c = 97 cm = 0.97 m.
Mixing 0.65 and 72 in the formula would give about 72.003, which is nonsense. Always put both legs in the same unit first.
Squares, isosceles and special triangles
Some shapes hide a right triangle: a square's diagonal, the height of an isosceles triangle, a rhombus's diagonals. Draw the right triangle in and the theorem does the rest.
Example 8: the diagonal of a square
A square has sides of 11 cm. How long is its diagonal?
The diagonal splits the square into two right triangles with legs 11 and 11.
d2= 112 + 112 = 242d= √121 · 2= 11√2 ≈ 15.56Answer: d = 11√2 ≈ 15.56 cm.
For any square, diagonal = side ×√2, the 45-45-90 triangle rule.
Example 9: legs of an isosceles right triangle
An isosceles right triangle has a hypotenuse of 12. How long is each leg?
Both legs are equal, so call each one x.
x2 + x2= 1222x2= 144x2= 72x= √72 = 6√2 ≈ 8.49Answer: each leg is 6√2 ≈ 8.49.
This is the reverse of Example 8: to go from hypotenuse to leg, divide by √2, since 12 ÷ √2 = 6√2.
Example 10: the height of an isosceles triangle
An isosceles triangle has two sides of 25 and a base of 14. Find its height and its area.
The height meets the base at a right angle at its midpoint, making a right triangle with hypotenuse 25 and leg 14 ÷ 2 = 7.
h2= 252 − 72= 625 − 49 = 576h= 24Area= 1 over 2 · 14 · 24 = 168Answer: the height is 24 and the area is 168 square units.
Remember to halve the base. Using 14 as the leg is the most common slip. See height of a triangle.
Coordinates and the converse
Example 11: distance between two points
Find the distance between A(1, −3) and B(8, 21).
Draw a right triangle with a horizontal leg and a vertical leg.
Δx= 8 − 1 = 7Δy= 21 − (−3) = 24d2= 72 + 242 = 625d= 25Answer: AB = 25.
This is exactly the distance formula, d = √(x2 − x1)2 + (y2 − y1)2. Watch the double negative: 21 − (−3) = 24.
Example 12: is it a right triangle?
A triangle has sides 33, 56 and 65. Is it a right triangle?
Use the converse: square the two shorter sides and compare with the square of the longest.
332 + 562= 1089 + 3136= 4225652= 4225Answer: yes.
The squares match, so the angle opposite the 65 side is exactly 90°. Compare a near miss: for 10, 15, 18 you get 100 + 225 = 325 against 182 = 324. Because 325 is bigger, that triangle is acute, not right.
3D, multi-step and algebra examples
Harder problems chain two right triangles together or put a variable in the sides. The method doesn't change. You just apply it twice, or solve an equation at the end.
Example 13: the diagonal of a room (3D)
A room is 12 ft long, 9 ft wide and 8 ft high. How long is the longest straight pole that fits, from one floor corner to the opposite ceiling corner?
First the floor diagonal, then use it as a leg with the height.
dfloor= √122 + 92 = √225 = 15d= √152 + 82 = √289= 17Answer: 17 ft.
In one step: d = √122 + 92 + 82 = √289 = 17. That is the 3D Pythagorean theorem.
Example 14: two triangles sharing a height
In triangle ABC, AD is perpendicular to BC. AB = 39, BD = 15 and DC = 77. Find AC.
Step 1, in triangle ABD:
h2= 392 − 152= 1521 − 225 = 1296h= 36Step 2, in triangle ADC:
x2= 362 + 772= 1296 + 5929 = 7225x= 85Answer: AC = 85.
The shared side AD carries the answer from one triangle into the other. As a bonus, the area of ABC is 1 over 2 · 92 · 36 = 1656.
Example 15: sides written with a variable
The legs of a right triangle are x and 2x, and the hypotenuse is 15. Find x.
Answer: x = 3√5 ≈ 6.71, so the legs are about 6.71 and 13.42.
Square the whole term: (2x)2 = 4x2, not 2x2. We reject x = −3√5 because a length can't be negative.
Summary of all 15 Pythagorean theorem examples
| # | What you find | Given | Answer |
|---|---|---|---|
| 1 | Hypotenuse | legs 9, 40 | 41 |
| 2 | Leg | hypotenuse 37, leg 12 | 35 |
| 3 | Leg (turned triangle) | PR = 65, QR = 16 | 63 |
| 4 | Hypotenuse (radical) | legs 4, 8 | 4√5 ≈ 8.94 |
| 5 | Leg (radical) | hypotenuse 10, leg 4 | 2√21 ≈ 9.17 |
| 6 | Hypotenuse (decimals) | legs 2.8 m, 4.5 m | 5.3 m |
| 7 | Hypotenuse (mixed units) | 0.65 m, 72 cm | 0.97 m |
| 8 | Square diagonal | side 11 cm | 11√2 ≈ 15.56 cm |
| 9 | Legs of isosceles right triangle | hypotenuse 12 | 6√2 ≈ 8.49 |
| 10 | Height and area | sides 25, 25, 14 | height 24, area 168 |
| 11 | Distance between points | (1, −3), (8, 21) | 25 |
| 12 | Right triangle or not | 33, 56, 65 | yes |
| 13 | Space diagonal | room 12 × 9 × 8 ft | 17 ft |
| 14 | Side via a shared height | AB = 39, BD = 15, DC = 77 | 85 |
| 15 | Variable sides | x, 2x, hypotenuse 15 | 3√5 ≈ 6.71 |
To check any of these yourself, use the Pythagorean theorem calculator. For problems set in real situations (ladders, ramps, screens, ships), continue with the word problems. To avoid the usual traps, read common mistakes.
Patterns worth remembering
- Add for c, subtract for a leg. If your "leg" comes out longer than the hypotenuse, you added when you should have subtracted.
- Triples save time. 9-40-41, 12-35-37, 16-63-65, 33-56-65 and 36-77-85 all appeared above. Recognizing them, and their multiples, lets you skip the square root. See the list of triples.
- Exact first, decimal last. Keep √80 or 4√5 through every step and round once at the end.
- Hidden right triangles are everywhere. Diagonals, heights, distances on a grid and room diagonals all come down to the same equation.
Frequently asked questions
What is a simple example of the Pythagorean theorem?
A right triangle with legs 12 and 16 has hypotenuse 20, because 144 + 256 = 400 and √400 = 20. Example 1 above (legs 9 and 40, hypotenuse 41) works exactly the same way.
What is an example of the Pythagorean theorem in real life?
Finding how high a ladder reaches up a wall, the diagonal size of a TV screen, the length of a wheelchair ramp, or a straight-line distance on a map. The word problems page has fifteen of these, and real-life uses covers the applications in more depth.
How do I know whether to add or subtract?
Look at what is missing. If the side opposite the right angle (the hypotenuse) is missing, add the squares of the other two sides. If a leg is missing, subtract the square of the known leg from the square of the hypotenuse.
What if the answer is not a whole number?
That's normal. Most right triangles have at least one irrational side. Give the exact value in simplest radical form, such as 2√21, and round the decimal only if the question asks for it.
Does the Pythagorean theorem work for every triangle?
No, only for right triangles. For any other triangle, a2 + b2 is either larger than c2 (acute) or smaller (obtuse), and you need the law of cosines instead.
Keep exploring
- PracticePythagorean Theorem Word Problems15 Pythagorean theorem word problems from real life (ladders, TVs, ramps, ships, kites, roofs) with a 4-step strategy, a sketch for each and full solutions.
- PracticePythagorean Theorem Practice Problems30 Pythagorean theorem practice problems in three levels: missing sides, radicals, the converse, coordinates and algebra, each with a worked solution.
- LearnHow to Do the Pythagorean Theorem, Step by StepHow to do the Pythagorean theorem in 8 steps: label the sides, add or subtract the squares, take the square root and check. Flowchart and worked examples.
- CalculatorsPythagorean Theorem CalculatorFree Pythagorean theorem calculator. Enter any two sides of a right triangle to find the third, with step-by-step working, exact radical answers and a drawing.
- PracticePythagorean Theorem WorksheetsFree Pythagorean theorem worksheets with answers: a printable worksheet generator plus ready-made sheets on the hypotenuse, missing sides and word problems.
- LearnCommon Pythagorean Theorem MistakesTwelve common Pythagorean theorem mistakes, from a + b = c to misusing the converse, each with a wrong-vs-right worked example and a final self-check list.