Geometry

Applying the Pythagorean Theorem

Using the Pythagorean Theorem to find unknown side lengths and distances.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Proving the Pythagorean Theorem

Measuring Distances You Can't Measure Directly

A ladder leans against a wall, its base 6 feet from the wall and its top reaching 8 feet up. Before reading on, predict the length of the ladder itself — a distance that would be awkward to measure directly while the ladder is in use, but that the Pythagorean Theorem can find without touching the ladder at all.

The ladder, the wall, and the ground form a right triangle — the wall and ground as the two legs, and the ladder itself as the hypotenuse, exactly the setup the Pythagorean Theorem was built for.

Worked Example — Finding the Ladder's Length

Using legs of 6 feet (base distance) and 8 feet (height), find the ladder's length. 6² + 8² = 36 + 64 = 100. Taking the square root: √100 = 10. The ladder is 10 feet long.

Worked Example — Finding a Missing Leg

A right triangle has a hypotenuse of 13 and one leg of 5. Find the other leg. Rearranging the theorem: b² = c² − a² = 13² − 5² = 169 − 25 = 144. Taking the square root: b = √144 = 12.

The theorem also finds the straight-line distance between two points on the coordinate plane — the horizontal and vertical differences between the points form the two legs of a right triangle, and the direct distance between them is the hypotenuse.

Worked Example — Finding the Distance Between Two Coordinate Points

Find the distance between (1, 2) and (4, 6). The horizontal leg is 4 − 1 = 3, and the vertical leg is 6 − 2 = 4. Apply the theorem: 3² + 4² = 9 + 16 = 25, so the distance is √25 = 5.

Tip

For a coordinate-distance problem, find the horizontal and vertical differences between the two points first — those two differences become the legs a and b, and the straight-line distance between the points is simply the resulting hypotenuse.

Common Mistakes

Solving for a leg using the same setup as solving for the hypotenuse, adding both known values instead of subtracting.
When the hypotenuse is already known and a leg is missing, rearrange the theorem to subtract: (missing leg)² = c² − (known leg)², not add the two known values.
Using the raw coordinate values instead of their differences when finding a distance between two points.
The legs of the right triangle formed between two coordinate points are the differences in x and the differences in y — not the coordinates themselves.

Key Takeaways

The Pythagorean Theorem finds an unmeasurable distance, like a ladder's length, from two measurable legs.
Rearranging the theorem solves for a missing leg when the hypotenuse and one leg are already known.
The distance between two coordinate points is the hypotenuse of a right triangle formed by their horizontal and vertical differences.

Summary

Applying the Pythagorean Theorem to real distances, including coordinate distances, completes this grade's geometry. The final unit shifts to statistics, investigating relationships between two related sets of data.

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