Solve Real-World Problems with the Pythagorean Theorem

Students apply the Pythagorean Theorem to multi-step real-world and mathematical problems, such as finding the length of a diagonal, the height of a building using shadow and distance, or whether a given triangle is right-angled. They draw and label a diagram before setting up the equation.

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Grade 8 · Geometry: Pythagorean Theorem · question 1 of 3

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20 rungs. A star at the top.

√(6^2 + 8^2) =

How it is taught

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A right triangle's long side squared is the sum of the two legs squared.

A rectangle's diagonal is that long side.

Read it: Right triangles in stories

A path runs 45 m straight across a rectangular park from one corner to the opposite corner, and one side is 36 m. Two sides of the park meet at a square corner, so the path is the longest side. 45 × 45 = 2025 and 36 × 36 = 1296, so 2025 − 1296 = 729, and 27 × 27 = 729: the other side is 27 m.

Find the square corner in the story: where a wall meets the ground, or where two sides of a rectangle meet.
The side across from it is the longest side. Decide whether the story asks for it or for a leg.
Work it out from the squares, then answer in the story's unit.

Two sides of a rectangle meet at a square corner, so those two sides and the path make a right triangle.

The path is across from the square corner, so its square, 2025, holds both leg squares.

Taking the known side's square away leaves the other side's: 2025 − 1296 = 729, and 27 × 27 = 729.

Name the longest side first, the one across from the square corner; then use the squares.

Treating the longest side as a leg: 45 × 45 + 36 × 36 = 3321.

The path is across from the square corner, so it is the longest side: 2025 − 1296 = 729, and 27 × 27 = 729.

A rectangular field is 28 m by 45 m. A path runs straight across it from one corner to the opposite corner. The path is ___ m long. — 53. Two sides of the field meet at a square corner, so the path is the long side: 784 + 2025 = 2809, and 53 × 53 = 2809.

13 ft ladder, 5 ft from wall — how high?

A 13-foot ladder leans against a wall. Its bottom sits 5 feet out from the wall. How high up the wall does the ladder reach?

Here's the picture. The wall, the ground, and the ladder form a right triangle.

Grade 8 · Drawing a diagram, then solving with a² + b² = c²

Label what we know. The ground is 5 feet, the ladder is 13 feet, and the height h up the wall is what we want to find.

The Pythagorean Theorem says: a-squared plus b-squared equals c-squared. Here, c is the ladder — the hypotenuse, the side across from the right angle.

Substitute the numbers we know. Five squared plus h squared equals thirteen squared.

Square the numbers. Five squared is twenty-five, and thirteen squared is one hundred sixty-nine.

To get h squared by itself, subtract twenty-five from both sides.

One sixty-nine minus twenty-five is one hundred forty-four. So h squared equals one hundred forty-four.

Take the square root of both sides. The square root of one hundred forty-four is twelve.

The ladder reaches twelve feet up the wall. Always draw the triangle first, then use a-squared plus b-squared equals c-squared. Great work!

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics

Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal: 1 star — one skill mastered, every rung of its ladder climbed
Climbing now: Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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