Pythagorean Theorem

The One Relationship Every Right Triangle Obeys

The Pythagorean theorem connects the three sides of any right triangle. If the two shorter sides — the legs — have lengths $a$ and $b$, and the longest side — the hypotenuse — has length $c$, then

$a^2 + b^2 = c^2.$

The hypotenuse is always the side opposite the right angle, and it is always the longest side. That single fact prevents the most common setup error: putting a leg where the hypotenuse belongs. Before you plug anything in, find the right angle, look straight across from it, and label that side $c$.

The theorem is a two-way street. If you know both legs, you square them, add, and take a square root to get the hypotenuse. If you know the hypotenuse and one leg, you subtract: $b^2 = c^2 - a^2$. Mixing up when to add and when to subtract is the difference between a right answer and a distractor the test writers planted for exactly that mistake.

Pythagorean Triples Save Time

Certain whole-number side lengths appear again and again. Memorizing them lets you skip the arithmetic:

Every multiple of a triple is also a triple. A triangle with legs $9$ and $12$ is just $3,4,5$ scaled by $3$, so its hypotenuse is $5 \times 3 = 15$ — no squaring required. Spotting a scaled triple is one of the fastest moves in SAT geometry.

The Distance Formula Is the Theorem in Disguise

To find the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane, draw the horizontal and vertical legs of a right triangle between them. The horizontal leg has length $|x_2 - x_1|$ and the vertical leg has length $|y_2 - y_1|$. The distance is the hypotenuse:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.$

You do not have to memorize this as a separate rule — it is $a^2 + b^2 = c^2$ applied to the "run" and the "rise." Because both differences get squared, the order you subtract in never matters; a negative difference squares to the same positive value.

Reaching into Three Dimensions

The theorem also finds diagonals inside boxes. The space diagonal of a rectangular box with edge lengths $\ell$, $w$, and $h$ is

$d = \sqrt{\ell^2 + w^2 + h^2}.$

This comes from applying the theorem twice: first to the base to get the face diagonal, then to the right triangle formed by that face diagonal and the vertical edge. Recognizing that a "3D" question is really two ordinary right triangles stacked together turns an intimidating problem into two familiar ones.

When the Legs Are Not Handed to You

Harder SAT questions hide one of the sides behind algebra. You might be told the hypotenuse and that the legs differ by a fixed amount, or that a leg is expressed in terms of a variable. In those cases you still start from $a^2 + b^2 = c^2$, substitute the expressions, and solve the resulting equation — which is often a quadratic. The geometry supplies the equation; your algebra finishes the job. And because a length can never be negative, you discard any negative solution the algebra produces.

The Converse: Is It Even a Right Triangle?

The relationship runs backward too. If three side lengths satisfy $a^2 + b^2 = c^2$ (with $c$ the largest), the triangle must contain a right angle. If $a^2 + b^2 > c^2$, the largest angle is acute; if $a^2 + b^2 < c^2$, it is obtuse. The SAT occasionally asks whether a triangle is right — test the sides rather than trusting how the figure looks, especially since figures are often "not drawn to scale."

Putting It Together

Nearly every Pythagorean problem is one of four moves: find a hypotenuse (add, then root), find a leg (subtract, then root), find a distance (differences, then the same add-and-root), or set up an equation when a side is algebraic. Identify which move the question wants, keep the hypotenuse straight, and check whether your answer is reasonable — the hypotenuse should come out larger than either leg, and a leg should come out smaller than the hypotenuse.