The Pythagorean Theorem on the ASVAB: Triples and Traps

The Pythagorean theorem is one of the most useful tools for ASVAB Mathematics Knowledge, which the official program describes as "knowledge of high school mathematics principles." It's one formula, but it shows up as ladders, diagonals, ramps and distances. Here's the formula, the shortcuts that save time without a calculator, and the two traps that cost the most points.

The formula

In a right triangle — one with a 90° angle — the two sides that form the right angle are the legs (a and b), and the longest side, opposite the right angle, is the hypotenuse (c):

a² + b² = c²

Example 1: finding the hypotenuse

A rectangle is 9 inches by 12 inches. How long is its diagonal?

The diagonal splits the rectangle into two right triangles with legs 9 and 12.

1. 9² + 12² = 81 + 144 = 225
2. c = √225 = 15 inches

Example 2: finding a leg (the first trap)

A 13-foot ladder leans against a wall. Its base is 5 feet from the wall. How high up the wall does it reach?

The ladder is the hypotenuse (13). The ground distance is a leg (5). You need the other leg.

1. 13² − 5² = 169 − 25 = 144
2. Height = √144 = 12 feet

The trap: adding the squares out of habit. 13² + 5² = 194, and √194 is about 13.9 — longer than the ladder itself, which is impossible. When you know the hypotenuse, you subtract.

The triples worth memorizing

Some right triangles have whole-number sides. Recognize them and you skip the arithmetic entirely:

Multiples work too. 6-8-10 is 3-4-5 doubled; 9-12-15 is 3-4-5 tripled — which is why Example 1 came out to 15 so cleanly. When you see legs of 9 and 12, you can write 15 without squaring anything.

The second trap: a triangle that isn't right

The formula only works for right triangles. If a problem gives three sides and asks whether the triangle is a right triangle, test the formula:

Is a triangle with sides 4, 5 and 6 a right triangle?

4² + 5² = 16 + 25 = 41, and 6² = 36. Since 41 ≠ 36, it isn't.

Always use the longest side as c.

Squares and roots without a calculator

Since there's no calculator on the ASVAB, it pays to know the perfect squares by heart, at least up to 15² = 225:

1² = 1 · 2² = 4 · 3² = 9 · 4² = 16 · 5² = 25 · 6² = 36 · 7² = 49 · 8² = 64 · 9² = 81 · 10² = 100 · 11² = 121 · 12² = 144 · 13² = 169 · 14² = 196 · 15² = 225

If a root isn't a perfect square, estimate it between two you know: √70 is between √64 = 8 and √81 = 9, a little closer to 8. On a multiple-choice question, that's often enough.

Practice it

Our free practice test includes Mathematics Knowledge with the computer version's time limits (15 questions in 31 minutes in the full version), plus the other three AFQT sections, and ends with an AFQT estimate — a practice estimate, not an official score.

For the full list of MK topics, see Mathematics Knowledge Practice Test.

Quick answers

What is the Pythagorean theorem?

In a right triangle, a² + b² = c², where c is the hypotenuse — the side opposite the right angle.

How do you find a missing leg?

Subtract: leg² = c² − (other leg)². With a hypotenuse of 13 and a leg of 5, the other leg is √(169 − 25) = 12.

What are the most common Pythagorean triples?

3-4-5, 5-12-13, 8-15-17 and 7-24-25 — and their multiples, like 6-8-10 and 9-12-15.

Does the Pythagorean theorem work for every triangle?

No, only right triangles. Check by testing whether the two shorter sides' squares add up to the longest side's square.

Sources

• What to Expect When You Take the ASVAB — officialasvab.com
• FAQs (calculators) — officialasvab.com