How to solve for a missing side: finding a, b, or c step by step

Every Pythagorean theorem problem is one of exactly three jobs: find the hypotenuse, or find one of the two legs. The formula never changes — a² + b² = c² — but the algebra does, and mixing up the versions is where most wrong answers are born. I have marked enough of these to know the pattern: students rarely fail on the square roots; they fail on adding when they should subtract. So this guide treats the three jobs separately, each with its own rearranged formula, fully worked examples with every line of arithmetic shown, and the sanity checks that catch the classic error before it costs you.

First, name the sides correctly

Before any formula, label the triangle. The side called c is always the hypotenuse: the side opposite the right angle, and always the longest of the three. The other two sides, a and b, are the legs — and here is the freedom that confuses people: it does not matter which leg you call a and which you call b. The formula is symmetric in the legs, so swapping their names changes nothing. What you may never do is let c be anything other than the hypotenuse. If a problem gives you a triangle with sides 5 and 13 where 13 is opposite the right angle, then c = 13 and one of your legs is 5 — even if the diagram labels the 13 side “a”. Trust the right angle, not the lettering. Every worked example below starts from that one habit.

Job 1 — finding c: the hypotenuse

When both legs are known, use the formula as written and solve for c:

Worked example 1. Legs of 8 and 15. Square them: 8² = 64 and 15² = 225. Add: 64 + 225 = 289. Root: √289 = 17. The hypotenuse is 17 — you may recognise 8-15-17 from the triples list; textbook problems are full of it.

Worked example 2. Legs of 9 and 12. Squares: 81 and 144. Sum: 225. Root: √225 = 15. (That is 3-4-5 scaled by 3, and recognising the family gets you the answer before the arithmetic does.)

Worked example 3 — the honest kind. Legs of 4 and 7. Squares: 16 and 49. Sum: 65. Root: √65 ≈ 8.06. Most real triangles end like this, in a decimal, and that is fine: report the root exactly (√65) for exact work, or round sensibly — 8.06 to two decimals, 8.1 to one. What you should never do is round mid-problem and then feed the rounded value into a later step; keep the full value until the final line.

It is no accident that “pythagorean theorem calculator find c” is the most-typed version of this search: the hypotenuse is the side people need in the real world — diagonals, distances, ladder lengths — and it is also the friendliest job, because the formula needs no rearranging. The other two jobs are where the theorem earns its reputation.

Job 2 — finding a leg: subtract, never add

When the hypotenuse and one leg are known, the formula turns inside out. You are peeling one square away from the hypotenuse’s square:

Worked example 1. Hypotenuse 25, known leg 24; find the other leg. Squares: 25² = 625 and 24² = 576. Subtract: 625 − 576 = 49. Root: √49 = 7. The triangle is 7-24-25.

Worked example 2. Hypotenuse 13, known leg 5. Squares: 169 and 25. Subtract: 144. Root: √144 = 12 — the 5-12-13 triple again.

Worked example 3 — decimal answer. Hypotenuse 17, known leg 9. Squares: 289 and 81. Subtract: 208. Root: √208 ≈ 14.42. Notice that the missing leg (14.42) is shorter than the hypotenuse (17) — hold that thought, because it is your built-in error detector: a leg can never be longer than the hypotenuse. If your subtraction version ever produces a leg longer than c, the arithmetic is wrong, full stop. Searches like “pythagorean theorem calculator find a” spike every exam season precisely because this job punishes one specific slip, which the next section isolates.

Job 3 — how to find b in the Pythagorean theorem

Finding b is the same job as finding a — the legs are interchangeable — but it deserves its own walk-through, because textbook diagrams so often label the horizontal leg b and the hypotenuse c, leaving b as the unknown. The rearranged formula mirrors the a version exactly:

Worked example 1. Hypotenuse 26, leg a = 10; find b. Squares: 26² = 676 and 10² = 100. Subtract: 676 − 100 = 576. Root: √576 = 24. The triangle is 10-24-26 — the 5-12-13 family doubled.

Worked example 2. Hypotenuse 10, leg a = 6; find b. Squares: 100 and 36. Subtract: 64. Root: √64 = 8 — the 6-8-10 triangle, 3-4-5 doubled.

So the routine for how to find b in pythagorean theorem questions is identical to the routine for a, and the search phrase “pythagorean theorem find b” leads to the same three lines every time: square c, subtract the known square, root the result. If you remember nothing else, remember the order — the hypotenuse is squared first and the known leg is taken away from it. Reversing the subtraction (leg squared minus hypotenuse squared) gives a negative number under the root, which is the theorem’s way of telling you the steps are backwards.

The mistake box: the error I see more than all others combined

Three more checks I run on every missing-side answer, and I recommend them until they are reflexes:

• The longest-side check. c must be the largest number in the finished triangle. Any result that breaks this is wrong, however confident the arithmetic felt.
• The plug-back check. Square your answer and the two given sides and confirm a² + b² = c² holds exactly (or to rounding, for decimals). Fifteen seconds, total certainty.
• The size sense check. A leg close in length to the hypotenuse means the other leg is short (24 with hypotenuse 25 leaves just 7); a leg near the hypotenuse’s half means the other leg is long. If the picture in your head disagrees with the number on the page, redo the number.

Before solving: check the triangle is right-angled at all

The theorem — and this whole page — applies only to right triangles, and the converse of the theorem gives you a test for that. Take the three sides, square the two shorter ones, and compare their sum with the square of the longest:

The converse test on four triangles. Equal means the angle opposite the longest side is exactly 90°.

This is more than an exam trick. It is how builders verify a corner is square (the 3-4-5 check is the converse in work boots), and it is the first thing to run when a problem hands you all three sides and asks about the angle: the numbers tell you which kind of triangle you are holding before you solve anything.

Two real problems, start to finish

The throw to second base. A baseball diamond is a square with 90 ft sides, so the catcher’s throw from home plate to second base is a hypotenuse with both legs equal to 90. c = √(90² + 90²) = √(8,100 + 8,100) = √16,200 ≈ 127.28 ft — 127 ft and about 3.35 in. Notice this is the missing-side machinery running Job 1 with a = b, and notice the answer is not a whole number: the diamond is a square, and a square’s diagonal is side × √2, which never lands even.

The map distance. Two points on a grid, (1, 2) and (4, 6): the horizontal gap is 4 − 1 = 3 and the vertical gap is 6 − 2 = 4, so the straight-line distance is √(9 + 16) = √25 = 5. That is the distance formula from coordinate geometry, and it is nothing but Job 1 wearing a grid — the theorem again, which is why I treat “distance between two points” and “hypotenuse” as the same skill.

Sources: formulas and the converse test follow the standard statements in Euclid’s Elements, Book I, Propositions 47 and 48; every worked example in this article was recomputed and plug-back verified in Python before publication.