Pythagorean theorem calculator
Two values of a right triangle in, the rest worked out.
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Example
Example: how high a 12 ft ladder reaches with its foot 3 ft from the wall
The ladder is the hypotenuse, c = 12; the distance from its foot to the wall is one leg, a = 3; the height it reaches on the wall is the other leg, b. Type 12 in Side c and 3 in Side a.
- b = √(c² − a²) = √(12² − 3²) = √135 = 3√15 ≈ 11.619 ft.
- The angle at the foot of the ladder is β, opposite the wall: tan α = 3 / 11.619 gives α ≈ 14.48°, so β = 90° − α ≈ 75.52°.
- Also shown: area ≈ 17.428 ft², perimeter ≈ 26.619 ft, height on the hypotenuse ≈ 2.905 ft.
| a | b | c | Missing side | Angles |
|---|---|---|---|---|
| 3 | 12 | b = 3√15 ≈ 11.619 | α ≈ 14.48°, β ≈ 75.52° | |
| 4 | 12 | b = 8√2 ≈ 11.314 | α ≈ 19.47°, β ≈ 70.53° | |
| 2 | 12 | b = 2√35 ≈ 11.832 | α ≈ 9.59°, β ≈ 80.41° | |
| 6 | 8 | c = 10 | α ≈ 36.87°, β ≈ 53.13° | |
| 9 | 12 | c = 15 | α ≈ 36.87°, β ≈ 53.13° | |
| 5 | 12 | c = 13 | α ≈ 22.62°, β ≈ 67.38° | |
| 2 | 3 | c = √13 ≈ 3.606 | α ≈ 33.69°, β ≈ 56.31° |
With its foot 3 ft out, the 12 ft ladder reaches 11.619 ft, about 11 ft 7½ in, up the wall and stands at about 75.5°, close to the 75° ladder makers usually recommend.
How it works
- Type any two values of the right triangle: two sides, or a side and an angle. c is the hypotenuse, opposite the right angle.
- The missing sides and angles appear first, exact where they can be (√13) and as decimals.
- Below: the formula used, the area, the perimeter, the height on the hypotenuse and the two segments it cuts.
Frequently asked questions
- What is the Pythagorean theorem?
- In a right triangle the square on the hypotenuse equals the sum of the squares on the other two sides: c² = a² + b². With legs 3 and 4, c = √(9 + 16) = √25 = 5.
- How do I find a leg instead of the hypotenuse?
- Subtract instead of add: b = √(c² − a²). With c = 13 and a = 5, b = √(169 − 25) = √144 = 12.
- How are the angles found?
- With the tangent, sine or cosine: tan α = a / b, sin α = a / c, cos α = b / c. The two acute angles add up to 90°.
- Can it solve a triangle without a right angle?
- Yes — choose any triangle, or use the triangle calculator page, and give three values, at least one a side: it uses the laws of sines and cosines.
- What is the 3-4-5 rule?
- A triangle with sides 3, 4 and 5 always has a right angle, because 3² + 4² = 9 + 16 = 25 = 5². Builders use it to square a corner: mark 3 ft along one wall and 4 ft along the other, and the corner is square when the marks are exactly 5 ft apart. Multiples work too: 6, 8, 10 or 9, 12, 15.
- Why does it show √135 = 3√15?
- That is the exact answer, simplified: 135 = 9 × 15, and √9 = 3. The decimal after it, ≈ 11.619, is the same length rounded to three places: the exact form for schoolwork, the decimal for measuring.
- Is anything uploaded?
- No. It is worked out in your browser, and nothing you type leaves your computer.
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