Pythagorean Theorem
Calculate hypotenuse and side lengths using the Pythagorean theorem a^2 + b^2 = c^2. Evaluate sqrt expressions for right triangle problems with the math evaluator.
Geometry
Detailed Explanation
Pythagorean Theorem
The Pythagorean theorem states that in a right triangle:
a^2 + b^2 = c^2
Where c is the hypotenuse (longest side) and a, b are the other two sides.
Finding the Hypotenuse
Given sides a=3 and b=4:
sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5
Finding a Missing Side
If the hypotenuse is 13 and one side is 5:
sqrt(13^2 - 5^2) = sqrt(169 - 25) = sqrt(144) = 12
Common Pythagorean Triples
| a | b | c | Verification |
|---|---|---|---|
| 3 | 4 | 5 | sqrt(3^2+4^2) = 5 |
| 5 | 12 | 13 | sqrt(5^2+12^2) = 13 |
| 8 | 15 | 17 | sqrt(8^2+15^2) = 17 |
| 7 | 24 | 25 | sqrt(7^2+24^2) = 25 |
3D Distance (Extended Pythagorean)
For the distance between two points in 3D space:
sqrt(dx^2 + dy^2 + dz^2)
Example: distance from (0,0,0) to (1,2,3):
sqrt(1^2 + 2^2 + 3^2) = sqrt(14) = 3.742
Real-World Applications
- Construction: Calculating ladder reach, roof pitch, or diagonal measurements
- Navigation: Finding straight-line distance between two coordinates
- Screen size: Computing the diagonal of a monitor from width and height
Use Case
A carpenter calculating the diagonal measurement of a rectangular frame to verify it is square, or a student solving geometry homework problems.