The Pythagorean theorem is one of the most versatile tools on the SAT. It shows up in geometry, coordinate geometry, and even word problems.

The Formula

For any right triangle with legs and and hypotenuse :

The hypotenuse is always the longest side and sits opposite the angle.

Pythagorean Triples You Must Know

These integer combinations appear constantly on the SAT:

The Big Three:

Recognizing these triples is faster than computing every time.

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Example 1: Basic Application

A right triangle has legs of length 6 and 8. What is the hypotenuse?

Step 1: Recognize this is a triple scaled by 2.

Step 2: Hypotenuse .

No calculation needed!

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Example 2: Finding a Leg

A ladder 13 feet long leans against a wall. The base is 5 feet from the wall. How high does the ladder reach?

Step 1: The ladder is the hypotenuse (13), the ground distance is a leg (5).

Step 2: Recognize the triple.

Step 3: The height is feet.

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Example 3: Coordinate Geometry

The distance between two points and is:

This IS the Pythagorean theorem applied to coordinates.

What is the distance from to ?

Another triple!

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Common Mistakes to Avoid

  • Squaring the hypotenuse on the wrong side: , NOT .
  • Using it on non-right triangles: The theorem ONLY works for right triangles. Check for the angle first.
  • Confusing legs and hypotenuse: The hypotenuse is always the longest side. If you are solving for a leg, rearrange: .
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SAT Pro Tip

On the SAT, if you see a right triangle with two known sides, your first instinct should be to check for a Pythagorean triple before reaching for your calculator. Triples appear in roughly 70% of SAT right triangle problems.

Ready to practice? Try our Pythagorean Theorem lessons with interactive exercises.