The ladder against the wall
A 10 foot ladder leans against a wall with its base 6 feet out from the wall. How high does it reach? You cannot measure the top directly without climbing up, but because the wall meets the ground at a right angle, the ladder, the wall, and the ground form a right triangle, and one famous relationship unlocks the missing side. This is the everyday shape of the Pythagorean theorem.
The theorem in plain form
Formula: a^2 + b^2 = c^2, where a and b are the two legs of a right triangle and c is the hypotenuse, the side opposite the right angle and always the longest side.
Each term is a square: a^2 means the area of a square built on leg a. The theorem says the two smaller squares always add up exactly to the square on the hypotenuse. Rearranged, it also finds a missing leg: a = sqrt(c^2 - b^2).
How to use this calculator
- Enter any two of the three sides.
- Mark which one is the hypotenuse if you know it; the longest side always is.
- Run the calculation to get the third side.
- Check that the hypotenuse is longer than either leg, or something went wrong.
Worked example
Back to the ladder: the hypotenuse c = 10 feet, one leg b = 6 feet, and the height a is unknown. Step 1: square the known sides, 10^2 = 100 and 6^2 = 36. Step 2: subtract, a^2 = 100 - 36 = 64. Step 3: take the square root, a = sqrt(64) = 8. The ladder reaches 8 feet up the wall, the classic 6-8-10 triangle. As a second check, a 3-4-5 triangle gives 3^2 + 4^2 = 9 + 16 = 25 = 5^2, confirming the pattern.
Where this comes in handy
Builders use the theorem to square up foundations with the 3-4-5 trick, navigators and game developers compute straight-line distances from coordinates, TV and monitor sizes are just the hypotenuse of the screen rectangle, and roofers check rafter lengths against spans. Anywhere a right angle appears, the theorem turns two measurements into three.
Mistakes people often make
- Using it on triangles without a right angle. For those you need the law of cosines instead.
- Adding the sides before squaring. The squares come first: a^2 + b^2, never (a + b)^2.
- Putting a leg where the hypotenuse goes. The hypotenuse c is always opposite the right angle and is always the longest side.
What this number does not tell you
The theorem gives exact side lengths for a mathematically perfect right triangle. Real walls are rarely perfectly plumb and tape measures have slack, so on a job site the answer is as good as your angle and your two measurements. For rough carpentry, rounding to the nearest eighth of an inch is plenty.
Exploring related geometry? The slope calculator handles steepness between points, the distance formula calculator measures straight-line gaps, and the curved line slope calculator extends the idea to curves.