To find the missing side of a right triangle when you know the hypotenuse and one leg, square both, subtract, and take the square root: a = √c2 − b2. For c = 13 and b = 5, a = √169 − 25 = √144 = 12. The hypotenuse must be the longest side, or no triangle exists.
Find a, b or c: what to enter
Homework questions often read like a fill-in-the-blank: find a when b = 5 and c = 13. The unknown is the side marked "?" or "n/a", and you only need the other two. Choose Find leg a, type b and c, and the box for a fills itself in.
| You need | You know | Formula | Example |
|---|---|---|---|
| Leg a | b and c | a = √c2 − b2 | b = 5, c = 13 ⇒ a = 12 |
| Leg b | a and c | b = √c2 − a2 | a = 9, c = 41 ⇒ b = 40 |
| Hypotenuse c | a and b | c = √a2 + b2 | a = 6, b = 8 ⇒ c = 10 (use the hypotenuse calculator) |
The two legs are interchangeable: calling a leg a or b doesn't change the answer. The hypotenuse is different. It has to go in box c, so find it first.
Why you subtract: rearranging the theorem
Start from the Pythagorean theorem, subtract b2 from both sides to get a2 on its own, then take the positive square root:
a2 + b2= c2a2= c2 − b2a= √c2 − b2Subtracting makes the number smaller, which is exactly right: a leg is always shorter than the hypotenuse. For the full lesson with practice questions, see how to find the missing side.
Worked examples
Example 1: find a when b = 7 and c = 25
a2= 252 − 72 = 625 − 49 = 576a= √576 = 24a = 24. The sides 7, 24, 25 form a Pythagorean triple.
Example 2: find b when a = 6 and c = 10
b2= 102 − 62 = 100 − 36 = 64b= √64 = 8b = 8: the 3-4-5 triangle doubled.
Example 3: an answer in simplest radical form
The hypotenuse is 12 and one leg is 8 (the default example in the calculator above).
a2= 144 − 64 = 80a= √80 = √16 · 5 = 4√5 ≈ 8.9443a = 4√5 ≈ 8.94, rounded to the nearest hundredth. The square root simplifier shows how to find the square factor 16.
Example 4: decimals
The hypotenuse is 7.5 m and one leg is 4.5 m.
a2= 7.52 − 4.52 = 56.25 − 20.25 = 36a= 6a = 6 m exactly: the triangle is 3-4-5 scaled by 1.5.
Example 5: a ladder against a wall
A 10 ft ladder leans against a wall with its foot 6 ft from the wall. How high up the wall does it reach? The ladder is the hypotenuse; the ground and the wall are the legs.
h = √102 − 62 = √64 = 8The ladder reaches 8 ft up the wall. The ladder calculator also checks the safe angle.
Common mistakes and how the calculator catches them
- Adding instead of subtracting. √132 + 52 = √194 ≈ 13.93 is longer than the hypotenuse, which is impossible for a leg.
- Subtracting before squaring. √13 − 5 = √8 ≈ 2.83 is wrong. Square each length first, then subtract.
- Putting a leg in the hypotenuse box. If c is not the largest number, c2 − b2 is zero or negative and there is no real answer. The calculator stops, explains why, and shows what the answer would be if the two numbers were swapped.
- Mixing units. Convert first. A hypotenuse of 2 m and a leg of 120 cm become c = 200 cm and b = 120 cm, so a = √40,000 − 14,400 = 160 cm.
- Rounding too early. Keep the exact value, or every digit, until the final step. More slips like these are on the common mistakes page.
A shortcut: the difference of two squares
Because c2 − b2 = (c − b)(c + b), you can often skip the big squares. When the hypotenuse and the leg are close together, this is much faster in your head:
a2 = 252 − 242 = (25 − 24)(25 + 24) = 1 · 49 = 49, a = 7In the same way, 412 − 402 = 1 · 81 = 81, so a right triangle with hypotenuse 41 and a leg of 40 has a missing leg of 9.
Table: hypotenuse, known leg and missing leg
| Hypotenuse c | Known leg | Missing leg (exact) | Decimal |
|---|---|---|---|
| 2 | 1 | √3 | 1.7321 |
| 3 | 2 | √5 | 2.2361 |
| 5 | 3 | 4 | 4 |
| 5 | 4 | 3 | 3 |
| 6 | 2 | 4√2 | 5.6569 |
| 10 | 5 | 5√3 | 8.6603 |
| 10 | 6 | 8 | 8 |
| 10 | 7 | √51 | 7.1414 |
| 12 | 8 | 4√5 | 8.9443 |
| 13 | 5 | 12 | 12 |
| 13 | 12 | 5 | 5 |
| 17 | 8 | 15 | 15 |
| 25 | 7 | 24 | 24 |
| 25 | 15 | 20 | 20 |
| 29 | 20 | 21 | 21 |
| 41 | 9 | 40 | 40 |
Decimals are rounded to four places. To practice, try the missing side worksheet, which has a full answer key.
Frequently asked questions
How do you find the missing side of a right triangle?
If the missing side is a leg, use a = √c2 − b2. If it is the hypotenuse, use c = √a2 + b2. If you know a side and an angle rather than two sides, use trigonometry; the right triangle solver does both.
Does it matter which leg I call a and which I call b?
No. The formula treats the legs the same way, so swapping them gives the same missing length. Only the hypotenuse has a fixed place: it is always c.
Why does the calculator say the hypotenuse must be longer than the leg?
Because c2 − b2 would be zero or negative, and no real length has a negative square. In a right triangle the hypotenuse is always the longest side, so check that the biggest number is in box c.
Can I find a missing side without knowing the hypotenuse?
Yes, if you know both legs: then the missing side is the hypotenuse. With one leg and an acute angle, use sine, cosine or tangent. One length on its own is never enough.
Does this work for triangles without a right angle?
No. For other triangles, use the law of cosines, c2 = a2 + b2 − 2abcosC, which reduces to the Pythagorean theorem when C = 90°.
Why is the answer sometimes a square root?
When c2 − b2 is not a perfect square, the exact length is irrational. For example √22 − 12 = √3. The calculator gives it in simplest radical form and as a decimal.
Keep exploring
- CalculatorsHypotenuse CalculatorFree hypotenuse calculator: find the longest side of a right triangle from two legs, a leg and an angle, or the area and a leg, with steps and exact radicals.
- CalculatorsPythagorean Theorem CalculatorFree Pythagorean theorem calculator. Enter any two sides of a right triangle to find the third, with step-by-step working, exact radical answers and a drawing.
- LearnHow to Find the Missing Side of a Right TriangleHow to find a missing leg of a right triangle with a = √(c² − b²): why you subtract, five worked examples, a mental-math shortcut and impossible inputs.
- PracticeFind the Missing Side WorksheetPrintable find the missing side worksheet: 16 Pythagorean theorem problems to find a missing leg or the hypotenuse, with a step-by-step answer key.
- CalculatorsRight Triangle Calculator and SolverRight triangle solver: enter any two values, such as two sides or a side and an angle, to get all three sides, both angles, area, perimeter, altitude and radii.
- LearnCommon Pythagorean Theorem MistakesTwelve common Pythagorean theorem mistakes, from a + b = c to misusing the converse, each with a wrong-vs-right worked example and a final self-check list.