Pythagorean Theorem

In a right triangle, the area of the square on the hypotenuse equals the combined areas of the squares on the legs.

Relative to θ, sine, cosine, and tangent compare opposite, adjacent, and hypotenuse sides. A right triangle has one acute angle labelled theta. The vertical side is opposite theta, the horizontal leg is adjacent, and the sloping longest side is the hypotenuse; a square marks the ninety-degree angle.

In every right triangle, the square of the hypotenuse equals the sum of the squares of the legs.

The relationship

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

$a$ and $b$ label the legs, which form the right angle. $c$ labels the hypotenuse, opposite the right angle and always the longest side.

Why the sides are squared

The theorem compares areas. Build a square on each side of a right triangle: the large square on the hypotenuse has the same area as the other two combined.

Three grid squares with areas 9, 16, and 25, showing that 3 squared plus 4 squared equals 5 squared.

For a $3$–$4$–$5$ triangle,

$$ 3^2+4^2=9+16=25=5^2. $$

Finding a hypotenuse

Finding a missing leg

When the hypotenuse is known, subtract the known leg-square from $c^2$.

Common mistakes

Self-check

Is the triangle right?
Which side is opposite the right angle?
Is my hypotenuse the longest value?
Did I take the square root at the end?
Did I include units?