Example 1
- a: 3
- b: 4
- Find c
c = 5
The most famous Pythagorean triple.
Pythagorean theorem calculator
Pick which side you need and type the two you know. Pythagorean theorem: a² + b² = c². With a = 3 and b = 4 you get c = 5. With a = 5 and b = 12 you get c = 13.
The angle between a and b must be right. Hypotenuse c is always the longest side.
Result
Choose which side to find and enter the two known lengths.
How it works
The Pythagorean theorem: in a right triangle a² + b² = c², where c is the side opposite the right angle. Solving for c: c = √(a² + b²). Solving for a: a = √(c² - b²). Same idea for b. With 3 and 4: 9 + 16 = 25, square root 5.
People check a right angle with a 3-4-5 stick, or they find a rectangle diagonal. 3 and 4 are the legs, 5 is longest. 5-12-13 is another clean triple: 25 + 144 = 169. The angle between a and b must be right, or the formula lies.
What are you solving for? has three modes: hypotenuse c when you know a and b, leg a when you know b and c, or b when you know a and c. Type 3 and 4, leave mode c, click Calculate. A comma and a period both parse. c must be longer than the known leg: b = 4 and c = 3 have no result.
Vector length (3, 4) is the same 5, only in coordinates. Sine 3/5 = 0.6 from this triangle lives on the sine page. Area of 3-4-5 is (3 × 4)/2 = 6, another calculator.
Both typed sides share one unit: centimeters with centimeters. The header switch does not turn a 3 into inches. Without two known sides the calculator waits.
a = 6 and b = 8 give c = 10, a doubled 3-4-5. a = 8 and b = 15 give c = 17. With a = 9 and c = 15 you want b: √(225 - 81) = 12.
Formula
a² + b² = c²
c = √(a² + b²) ··· a = √(c² − b²) ··· b = √(c² − a²)
How to use
- Pick What are you solving for?: c when you know a and b, a when you know b and c, or b when you know a and c.
- Type the two known sides. For c, leave 3 in leg a and 4 in leg b.
- Click Calculate. 3 and 4 give c = 5. 5 and 12 give c = 13.
- c must be longer than the known leg. b = 4 and c = 3 is not a right triangle.
- Vector (3, 4) of length 5 lives on vector length. This page stays with three triangle sides.
a, b, and c on 3-4-5
The Pythagorean theorem here is a² + b² = c². 3 and 4 give c = 5. 5 and 12 give c = 13. The angle between a and b is right.
- a
- A leg. 3 in the 3-4-5 triangle. In mode a you compute √(c² − b²).
- b
- The other leg. 4. Doubling 3-4-5 gives 6-8-10.
- c
- The hypotenuse, the longest side. 5, 13 or 17. A c shorter than a leg will not run.
Examples
Example 2
- a: 5
- b: 12
c = 13
Another clean triple.
Example 3
- b: 8
- c: 10
a = 6
√(100 − 64) = √36 = 6.
Example 4
- a: 6
- c: 10
b = 8
The 6-8-10 triangle (double 3-4-5).
Related calculators
Frequently asked questions
What is c when a = 3 and b = 4?
The hypotenuse is 5. 9 + 16 = 25, square root 5. That is the 3-4-5 triangle, the first clean example.
What is c at 5 and 12?
You get 13. 25 + 144 = 169. Another clean Pythagorean triple, same formula a² + b² = c².
How do I find a leg when I know c?
Pick mode a or b. a = √(c² − b²). With c = 15 and a = 9 you get b = 12. c must be the longest side.
Must the angle between a and b be right?
Yes. a² + b² = c² is for 90°. Any other angle needs cosine, another calculator.
Do 6 and 8 give 10?
Yes. That is a doubled 3-4-5. 36 + 64 = 100, square root 10. Scale does not break the right angle.
Where is the length of vector (3, 4)?
On the vector-length page. There √(9+16) = 5 as well, from coordinates x and y, not from triangle sides.
What is the area of triangle 3-4-5?
The area is 6, because (3 × 4)/2. This calculator finds the missing side, not the area. Area lives on another page.
Does b = 4 and c = 3 work?
No. The hypotenuse must be longest. 3 cannot carry a leg of 4, and the calculator refuses that layout.
What is c at 8 and 15?
You get 17. 64 + 225 = 289, square root 17. Another integer triple.
Does sine 3/5 belong here?
No. 3/5 = 0.6 is on sine, sides mode. Here you hunt the missing side, not a ratio.
Knowledge sources
The calculator computes the same formula as the definition below.
Page updated in 2026.