How to Find the Missing Side of a Right Triangle

Pythagoras is usually taught as a single formula, and that is where the trouble starts. It runs in two directions, and treating it as one rule means adding when you should have subtracted. The answer that comes out still looks like a length, which is exactly why the mistake survives.

Identify the hypotenuse first

Before any arithmetic, find the hypotenuse. It is the side opposite the right angle, and it is always the longest of the three. Everything else follows from getting this right, because the hypotenuse is the only side that sits alone on one side of the equation.

Once you know which side is the hypotenuse, the decision makes itself. If the hypotenuse is what you are missing, you are building it out of the other two, so you add. If you already have the hypotenuse and want a leg, you are taking a piece back out of it, so you subtract.

Two worked examples

First, a missing hypotenuse. The legs measure 6 and 8. Square them to get 36 and 64, add to get 100, and the square root is 10. The answer is longer than both legs, which is right, because the hypotenuse always is.

Now a missing leg. The hypotenuse is 13 and one leg is 5. Square them to get 169 and 25, subtract to get 144, and the square root is 12. The answer is shorter than 13, which is right, because a leg always is. Had you added here instead you would have got 13.93, a leg longer than the hypotenuse it sits inside, which is impossible.

A check that costs nothing

Whatever you calculate, compare it against the sides you already had. The hypotenuse must be the largest of the three, and each leg must be smaller than it. This one glance catches the add and subtract mix up every time, and it takes a second.

There is a second useful bound. Any side must be shorter than the other two combined, so on a triangle with legs of 6 and 8 the hypotenuse has to land somewhere below 14. Getting 10 fits comfortably. Getting 15 would mean something went in the wrong slot.

When the answer is not a whole number

Sets like 3, 4, 5 and 5, 12, 13 come up constantly in exercises because they stay whole the whole way through, and that can quietly train the expectation that a decimal means a mistake. It does not. Legs of 4 and 7 give a hypotenuse of the square root of 65, which is 8.06 to two places, and that is simply the answer.

Keep the root exact for as long as you can if the result feeds into further work, and round only at the end. Rounding early and then squaring the rounded value is a reliable way to lose accuracy without noticing.

Mistakes that catch people out

• Adding when a leg is missing. The single most common error, and it gives a leg longer than the hypotenuse.
• Misidentifying the hypotenuse. It faces the right angle and is the longest side. Never one of the sides forming the right angle.
• Forgetting the square root. The sum or difference is the square of the side, not the side itself.
• Squaring after adding rather than before. Square each side first, then combine. 6 plus 8 squared is not 10.
• Using it on a triangle with no right angle. Pythagoras only holds for right triangles. Otherwise reach for the cosine rule.

Practice questions

Common questions

How do I know whether to add or subtract?

Look at which side is missing. If it is the hypotenuse, add the squares of the two legs. If it is a leg, subtract its known partner from the hypotenuse squared.

Which side is the hypotenuse?

The one opposite the right angle, and always the longest of the three. Identify it before doing any arithmetic.

What if my leg comes out longer than the hypotenuse?

You added when you should have subtracted. A leg is always shorter than the hypotenuse, so that result is impossible.

Does Pythagoras work on any triangle?

No, only on right triangles. Without a right angle you need the cosine rule instead, which reduces to Pythagoras when the angle is 90 degrees.

What if I get a negative number under the square root?

The side you called the hypotenuse is shorter than the leg you subtracted, so the two have been swapped. Check which side faces the right angle.

Do the answers have to be whole numbers?

No. Whole number sets like 3, 4, 5 and 5, 12, 13 are common in textbooks, but most real measurements give a decimal answer.

Related lessons and calculators

• How to Find the Height of a Right Triangle to the Hypotenuse for the altitude onto the side you just calculated.
• Area of a Triangle: Every Formula and When to Use Each for the overview of every area method in one place.
• How to Find the Area of a Triangle From Three Sides for the area once all three sides are known.
• Distance and Pythagoras Calculator which runs this both ways and extends it into three dimensions.
• Triangle Solver for the angles once the third side is found.
• National Council of Teachers of Mathematics for curriculum standards and how this topic is sequenced in school.
• Area formulas for every common shape, a calculator covering fifteen shapes and an index of every geometry lesson here.