How do you solve the Pythagorean theorem?

Q: How do you solve the Pythagorean theorem?

Square the two sides you know, add them for the hypotenuse or subtract for a leg, then take the square root. For legs 3 and 4: 9 + 16 = 25, and √25 = 5.

Pythagorean theorem calculator

Solve for

Leg a

Leg b

Quick triples

Tap a side to solve for it - the dashed side is the one being calculated.

Hypotenuse c: 5
Perimeter: 12
Area: 6
Angle opposite leg a: 36.87°
Angle opposite leg b: 53.13°

c = √(3² + 4²) = √25 = 5

The full answer

The three steps

Write the formula $a^2+b^2=c^2$ and substitute the two sides you know.
Square them, then add if you are finding the hypotenuse or subtract if you are finding a leg.
Take the square root of the result.

Finding the hypotenuse from legs 3 and 4:

$c=\sqrt{3^2+4^2}=\sqrt{9+16}=\sqrt{25}=5$

Finding a leg from hypotenuse 13 and leg 5:

$a=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12$

Which version to use

You know	You want	Do this
Both legs	Hypotenuse $c$	Add the squares, then root
One leg and the hypotenuse	The other leg	Subtract the leg square from the hypotenuse square, then root

When subtracting, the hypotenuse square always comes first - it is the longest side, so its square is the biggest number. If your subtraction goes negative, the side you called the hypotenuse is not actually the hypotenuse.

The mistakes that cost marks

Mistake	Why it fails	Fix
Adding before squaring	$(3+4)^2=49$ , not $25$	Square each side separately, then add
Adding when solving for a leg	Gives a side longer than the hypotenuse	Subtract from the hypotenuse square
Mixing units	Feet on one side, inches on the other	Convert everything to one unit first
Forgetting the square root	Leaves you with $c^2$ , not $c$	The last step is always the root

When it does not apply

The theorem only works when the triangle has a 90° angle. For any other triangle you need the law of cosines, which adds a correction term for the actual angle.