Pythagorean Theorem Word Problems

Word problems show you how the Pythagorean theorem is used in real life. The following examples are typical and often appear in tests and exams.

> 💡 Approach for every word problem:

> 1. Read the problem carefully.

> 2. Sketch a picture – a right triangle with the sides labeled.

> 3. Identify which side is the hypotenuse and which are the legs.

> 4. Plug into the formula and compute.

> 5. Don't forget to write the answer in words with units.

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Table of contents

• A ladder leaning against a wall
• The diagonal of a rectangle
• The diagonal of a square
• Distance between two points
• Height of an isosceles triangle
• The shortest route

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A ladder leaning against a wall

Problem: A ladder $5$ m long is leaning against a wall so that its foot is $3$ m from the wall. To what height $h$ does the ladder reach on the wall?

The ladder forms the hypotenuse of a right triangle; the distance from the foot to the wall and the height on the wall form the legs.

Answer: The ladder reaches a height of $4$ m.

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The diagonal of a rectangle

Problem: A rectangle has dimensions $a=6$ cm and $b=8$ cm. Compute the length of its diagonal $u$.

The diagonal of a rectangle divides the rectangle into two right triangles. The sides of the rectangle are the legs, and the diagonal is the hypotenuse.

Answer: The diagonal of the rectangle has length $10$ cm.

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The diagonal of a square

Problem: A square has side $a=5$ cm. What is the length of its diagonal?

In a square, both legs of the right triangle are equal:

Answer: The diagonal of the square has length approximately $7.07$ cm.

> 💡 Formula for the diagonal of a square: $u=a\sqrt{2}$

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Distance between two points

Problem: City $A$ is $30$ km north and $40$ km east of city $B$. What is the straight-line distance between the cities?

Imagine a right triangle: one leg is $30$ km (north), the other is $40$ km (east), and the hypotenuse is the straight-line distance.

Answer: The straight-line distance is $50$ km.

> 👉 More detailed explanation: Distance between two points in a plane

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Height of an isosceles triangle

Problem: An isosceles triangle has base $z=12$ cm and legs $r=10$ cm. Compute its height $v$.

The height of an isosceles triangle bisects the base. This creates a right triangle where the leg of the isosceles triangle is the hypotenuse, half the base is one leg, and the height is the other leg.

Answer: The height of the triangle is $8$ cm.

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The shortest route

Problem: A park has the shape of a rectangle with dimensions $80$ m and $60$ m. A path runs around the perimeter of the park, but you can also walk diagonally across the park. By how many meters is the diagonal path shorter than the path along two adjacent sides?

Step 1: The diagonal of the park.

Step 2: The path along two adjacent sides.

Step 3: The difference.

Answer: The diagonal path is $40$ m shorter.

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Related articles

• The Pythagorean theorem – complete guide
• The Pythagorean theorem formula
• Pythagorean theorem examples
• Distance between two points
• The Pythagorean theorem in 3D

Practice

• 📊 Word problems
• 🔢 Hypotenuse calculation
• 🧮 Leg calculation
• 🎯 Mixed problems