Pythagorean Theorem Word Problems Calculator

a² + b² = c²

Translate the story into a right triangle, identify the two known values, then let the calculator finish the missing side.

Most Pythagorean theorem word problems ask for a straight-line distance, a diagonal, or a missing leg. The workflow is always the same: draw the right triangle, label a, b, and c, then apply a² + b² = c².

Worked word-problem cards

Each card shows the real-world situation, known values, formula setup, final answer, and a calculator shortcut for checking the result.

Ladder problem

How long is the ladder?

A ladder reaches a window 12 ft high and its base is 5 ft from the wall.

Known values: a = 5 ft, b = 12 ft

Formula: c = √(5² + 12²) = √169

Answer: The ladder is 13 ft long.

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Rectangle diagonal

Find the diagonal across a room

A room is 9 m wide and 12 m long. What is the floor diagonal?

Known values: a = 9 m, b = 12 m

Formula: c = √(9² + 12²) = √225

Answer: The diagonal is 15 m.

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Screen size

Convert width and height to diagonal

A screen is 16 in wide and 9 in tall. What is the diagonal?

Known values: a = 16 in, b = 9 in

Formula: c = √(16² + 9²) = √337

Answer: The diagonal is about 18.36 in.

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Construction layout

Check a square corner

Two layout marks are 6 ft and 8 ft from a corner. How long should the diagonal be?

Known values: a = 6 ft, b = 8 ft

Formula: c = √(6² + 8²) = √100

Answer: The diagonal should be 10 ft.

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Coordinate distance

Find distance between two points

Point A is (2, 3) and point B is (10, 9). What is AB?

Known values: horizontal = 8, vertical = 6

Formula: distance = √(8² + 6²) = √100

Answer: The distance is 10 units.

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Calculator workflow for word problems

Use this four-step pattern before entering values. It prevents the common mistake of treating the hypotenuse as a leg.

Step 1

Sketch the right triangle

Mark the 90° angle and decide which side is the longest straight-line distance.

longest side = c

Step 2

List the known numbers

Put the two values from the word problem beside a, b, or c.

known: two sides

Step 3

Choose the equation

Use c = √(a² + b²) for a diagonal or √(c² - a²) for a missing leg.

a² + b² = c²

Step 4

Enter the calculator inputs

Type the two known side lengths, calculate, then round to the unit requested.

answer ≈ rounded result

Related guides

Pythagorean theorem formula

Review the core formula and calculator phrases for right triangle problems.

Pythagorean theorem examples

Walk through classic substitutions, squaring, and square-root steps.

Converse theorem calculator

Check whether three side lengths make a right, acute, or obtuse triangle.

Word Problems FAQ

Q1.How do I know which side is c in a word problem?

c is the hypotenuse: the side opposite the right angle and the longest side. Diagonals, ladders leaning against walls, and straight-line distances are often c.

Q2.What if the word problem gives the hypotenuse and one leg?

Rearrange the formula. Use a = √(c² - b²) or b = √(c² - a²) to find the missing leg.

Q3.Can coordinate-distance problems use this calculator?

Yes. Find the horizontal change and vertical change first, then use them as the two legs in a² + b² = c².

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