Quick Answer
Quick Answer: The Pythagorean theorem, a² + b² = c², solves any right triangle: square the two known sides, add or subtract, and take the square root. The classic examples are 6² + 8² = 10² (36 + 64 = 100), the ladder that reaches 8 feet, and the 3 × 4 rectangle whose diagonal is 5. Calcrio’s free pythagorean theorem calculator checks every example in this post.
What Is the Pythagorean Theorem, in One Example?
Take the triangle with legs 3 and 4. Square the legs: 3² = 9 and 4² = 16. Add them: 9 + 16 = 25. The hypotenuse is √25 = 5. That is the entire theorem in one line: a² + b² = c².
Every example in this post follows the same three moves — square, add or subtract, square root. What changes is the story: a ladder, a rectangle, a map, a ramp. The numbers change too, but the method never does. Once you have worked through the six examples below, you will recognize the pattern in any new problem: find the right triangle hiding inside it, name the two sides you know, and solve for the third.
Example 1: The Ladder Problem — 6-8-10
A 10-foot ladder leans against a wall with its foot 6 feet from the wall. How high does it reach?
The ladder is the hypotenuse (c = 10), the distance from the wall is one leg (a = 6), and the height is the other leg (b). We know the hypotenuse and one leg, so we subtract: b = √(c² − a²).
Working: b = √(10² − 6²) = √(100 − 36) = √64 = 8 feet.
Check: 6² + 8² = 36 + 64 = 100 = 10². The ladder reaches 8 feet up the wall. This is the 6-8-10 triple — the 3-4-5 triple doubled — and it is the most common ladder problem in textbooks because the answer comes out whole.
Example 2: Finding the Hypotenuse — 5-12-13
A right triangle has legs 5 and 12. Find the hypotenuse.
We know both legs, so we add: c = √(a² + b²).
Working: c = √(5² + 12²) = √(25 + 144) = √169 = 13.
Check: 13² = 169, and 25 + 144 = 169. The hypotenuse is 13. The 5-12-13 triple is the second triple worth memorizing after 3-4-5 — it appears in roof pitches, ramps, and screen diagonals. Try it yourself in Calcrio’s pythagorean theorem calculator and watch the steps match.
Example 3: Finding a Missing Leg — 8-15-17
A right triangle has hypotenuse 17 and one leg 8. Find the other leg.
We know the hypotenuse and one leg, so we subtract: b = √(c² − a²).
Working: b = √(17² − 8²) = √(289 − 64) = √225 = 15.
Check: 8² + 15² = 64 + 225 = 289 = 17². The missing leg is 15. The trap in this problem is adding instead of subtracting: 289 + 64 = 353 has no whole square root, and the answer would be longer than the hypotenuse — which is impossible.
Example 4: The Rectangle Diagonal — 3 × 4
What is the diagonal of a 3 × 4 rectangle?
A rectangle’s diagonal splits it into two right triangles, and the diagonal is the hypotenuse of a triangle whose legs are the rectangle’s sides.
Working: diagonal = √(3² + 4²) = √(9 + 16) = √25 = 5.
Check: 5² = 25, and 9 + 16 = 25. The diagonal is 5. The same method gives the diagonal of a 5 × 12 rectangle: √(25 + 144) = √169 = 13. This is how screen sizes are measured — a “5-inch screen” is the diagonal, not the width — and how builders check that a frame is square.
Example 5: Distance Between Two Points — 5 Units
How far apart are the points (3, 4) and (7, 1)?
The horizontal distance is dx = 7 − 3 = 4 and the vertical distance is dy = 1 − 4 = −3. The straight-line distance is the hypotenuse of a right triangle with legs 4 and 3.
Working: distance = √(4² + (−3)²) = √(16 + 9) = √25 = 5.
Check: 4² + 3² = 16 + 9 = 25 = 5². The points are 5 units apart. This is the distance formula, and it is the Pythagorean theorem in coordinate form — every map distance, every screen distance, every GPS calculation uses it. The sign of dy does not matter, because (−3)² = 9 either way.
Example 6: Is This Triangle Right-Angled? — Testing 3-4-5 and 2-3-4
Do the sides 3, 4, 5 form a right triangle? Test a² + b² = c² with the longest side as c.
Working: 3² + 4² = 9 + 16 = 25, and 5² = 25. Since 25 = 25, yes — it is a right triangle, and it is the 3-4-5 Pythagorean triple.
Now test 2, 3, 4. Working: 2² + 3² = 4 + 9 = 13, but 4² = 16. Since 13 ≠ 16, no — it is not a right triangle. The test is the theorem run backwards: if the squares add up, the angle is 90°; if they don’t, it isn’t. Builders use exactly this test to square a corner: mark 3 feet one way, 4 feet the other, and the diagonal must measure exactly 5 feet.
Summary Table: All Six Examples
| Example | Known | Formula | Working | Answer |
|---|---|---|---|---|
| Ladder 6-8-10 | c = 10, a = 6 | b = √(c² − a²) | √(100 − 36) = √64 | 8 ft |
| Hypotenuse 5-12-13 | a = 5, b = 12 | c = √(a² + b²) | √(25 + 144) = √169 | 13 |
| Missing leg 8-15-17 | c = 17, a = 8 | b = √(c² − a²) | √(289 − 64) = √225 | 15 |
| Rectangle 3 × 4 | sides 3, 4 | d = √(a² + b²) | √(9 + 16) = √25 | 5 |
| Distance (3,4)-(7,1) | dx = 4, dy = −3 | d = √(dx² + dy²) | √(16 + 9) = √25 | 5 |
| Right-triangle test | sides 3, 4, 5 | a² + b² = c²? | 9 + 16 = 25 = 5² | Yes, right |
How Do You Solve a Pythagorean Theorem Word Problem?
Word problems hide the right triangle inside a story. The method is always the same, in four steps.
Step 1: Draw the Right Triangle
Sketch the situation and mark the right angle. The ladder, the wall, and the ground make a right triangle. The rectangle’s diagonal makes two. The two points and their horizontal and vertical gaps make one. If you cannot see the triangle, you cannot solve the problem.
Step 2: Name the Hypotenuse
The hypotenuse is the longest side, opposite the right angle. In the ladder problem it is the ladder itself. In the rectangle it is the diagonal. In the distance problem it is the straight line between the points. Naming it first prevents the most common error — using a leg as c.
Step 3: Add or Subtract the Squares
Finding the hypotenuse (the longest side) means adding: c = √(a² + b²). Finding a leg (a shorter side) means subtracting: b = √(c² − a²). The operation follows from which side is missing.
Step 4: Check by Squaring Back
Square your answer and confirm a² + b² = c². For the ladder: 6² + 8² = 36 + 64 = 100 = 10². If the check fails, the error is in step 2 or 3 — the wrong side named as the hypotenuse, or addition where subtraction was needed. Our guide on how to find the hypotenuse works through the addition case in full.
Pythagorean Theorem Examples in Real Life
Construction. The 3-4-5 method squares corners, foundations, and frames: measure 3 feet along one edge, 4 feet along the other, and the diagonal must be exactly 5 feet. Any other diagonal means the corner is not square.
Ladders and roofs. A ladder’s reach, a roof’s slope, and a ramp’s length are all hypotenuses. A 10-foot ladder 6 feet from a wall reaches 8 feet — the same 6-8-10 triangle as the textbook problem.
Screens and pages. A screen’s size is its diagonal. A 3 × 4 rectangle has diagonal 5; a 5 × 12 rectangle has diagonal 13. Designers and manufacturers use the theorem to convert between width, height, and diagonal size.
Maps and navigation. The distance between two points (3, 4) and (7, 1) is 5 units by the distance formula — the Pythagorean theorem in coordinate form. GPS, games, and mapping software all compute distances this way.
Sports. The distance from home plate to second base on a baseball diamond, the diagonal pass across a soccer field, and the shortest path across a court are all hypotenuses of right triangles drawn on the field.
Common Mistakes in Pythagorean Theorem Problems
1. Adding When You Should Subtract
To find a leg, subtract the known leg’s square from the hypotenuse’s square. Adding gives an answer longer than the hypotenuse, which is impossible.
2. Forgetting the Square Root
36 + 64 = 100 is c², not c. The answer is √100 = 10, not 100. If your answer is bigger than the sum of the two known sides, you forgot the square root.
3. Using the Theorem Without a Right Angle
The theorem only works for right triangles. If the problem does not give or imply a 90° angle — a ladder against a wall, a rectangle’s corner, two points on a grid — look for the right angle before applying the formula.
4. Mixing Up the Sides in a Word Problem
In the ladder problem, the ladder is the hypotenuse, not a leg. In the rectangle, the diagonal is the hypotenuse, not a side. Name the hypotenuse before writing any numbers down.
5. Testing the Wrong Side as c
When checking whether a triangle is right-angled, c must be the longest side. Testing 2, 3, 4 with c = 3 gives 4 + 16 = 20 versus 9 — a meaningless comparison. Always put the longest side on the right of the equals sign.
Frequently Asked Questions
What is an example of the Pythagorean theorem?
The classic example is 3² + 4² = 5²: 9 + 16 = 25. Another is the ladder problem — a 10-foot ladder with its foot 6 feet from a wall reaches √(100 − 36) = 8 feet up the wall.
How do you solve a Pythagorean theorem word problem?
Draw the right triangle, name the hypotenuse, then add the squares if you need the hypotenuse or subtract if you need a leg, take the square root, and check by squaring back into a² + b² = c².
What is the diagonal of a 3 × 4 rectangle?
5, because the diagonal is the hypotenuse of a right triangle with legs 3 and 4: √(3² + 4²) = √(9 + 16) = √25 = 5.
How far apart are the points (3, 4) and (7, 1)?
5 units. The horizontal gap is 4 and the vertical gap is 3, so the distance is √(4² + 3²) = √(16 + 9) = √25 = 5.
What are 5-12-13 and 8-15-17?
They are Pythagorean triples — whole-number sets that satisfy a² + b² = c². 5² + 12² = 25 + 144 = 169 = 13², and 8² + 15² = 64 + 225 = 289 = 17².
How do you check if a triangle is a right triangle?
Test whether a² + b² = c² with the longest side as c. For 3, 4, 5: 9 + 16 = 25 = 5², so it is right-angled. For 2, 3, 4: 4 + 9 = 13 ≠ 16, so it is not.