In every right triangle, a² + b² = c²: the squares of the two legs add up to the square of the hypotenuse. Give the calculator two sides and it finds the third, with the steps, the two acute angles, the area, the perimeter and the height to the hypotenuse.
Example: a ladder must reach a window 4 m up, and its foot stands 1.5 m from the wall. The ladder is the hypotenuse: c = √(4² + 1.5²) = √(16 + 2.25) = √18.25 = 4.27 m. The angle with the ground is 69.4° – a little under the 75° that ladder makers recommend, so move the foot slightly closer.
Finding the hypotenuse or a leg
| You know | Formula | Example |
|---|---|---|
| both legs a and b | c = √(a² + b²) | a = 6, b = 8 → c = 10 |
| hypotenuse c and leg a | b = √(c² − a²) | c = 13, a = 5 → b = 12 |
| two legs, for the angles | α = arctan(a ÷ b), β = 90° − α | a = 3, b = 4 → α = 36.87°, β = 53.13° |
| two legs, for the area | area = a × b ÷ 2 | a = 3, b = 4 → area = 6 |
Pythagorean triples
| a | b | c | Angles |
|---|---|---|---|
| 3 | 4 | 5 | 36.87°, 53.13° |
| 5 | 12 | 13 | 22.62°, 67.38° |
| 8 | 15 | 17 | 28.07°, 61.93° |
| 7 | 24 | 25 | 16.26°, 73.74° |
| 20 | 21 | 29 | 43.60°, 46.40° |
| 9 | 40 | 41 | 12.68°, 77.32° |
Everyday uses
- Squaring a corner: the 3-4-5 rule above, for decks, slabs, tiles and garden beds.
- TV and screen sizes: a screen is measured on the diagonal. A 16:9 screen 48 in wide and 27 in high has a diagonal of √(48² + 27²) = 55.1 in.
- Distances on a map: 3 km east and 4 km north is 5 km in a straight line.
- Ramps and roofs: a ramp rising 0.5 m over 6 m of run is √(36 + 0.25) = 6.02 m long.
- Diagonals of rooms: a 4 × 3 m room is 5 m corner to corner – the longest straight thing you can lay on the floor.
Why it works
Draw a square on each side of a right triangle. The area of the square on the hypotenuse equals the two other squares together. One of the oldest proofs puts four copies of the triangle inside a square of side a + b: the space left over is either one square c² or the two squares a² and b², so they must be equal. The relationship was known in Babylon more than a thousand years before Pythagoras; the Greek tradition gave it his name.
For circles, see the circle calculator. Once zet4u is saved on your device, the calculator works offline, so it is there for homework or a workshop without Wi-Fi.
Frequently asked questions
What is the Pythagorean theorem?
In a right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides: a² + b² = c². With legs of 3 and 4, c² = 9 + 16 = 25, so the hypotenuse is 5.
How do I find a missing leg?
Rearrange: b = √(c² − a²). With a hypotenuse of 13 and a leg of 5, b = √(169 − 25) = √144 = 12. The hypotenuse is always the longest side; if your "hypotenuse" is shorter than the leg, the numbers cannot form a right triangle.
How do builders check a right angle with 3-4-5?
Measure 3 units along one side from the corner and 4 along the other, then measure between the two marks: if it is exactly 5, the corner is square. Bigger multiples are more accurate – 6-8-10 ft or 0.9-1.2-1.5 m for a deck or a slab.
What is a Pythagorean triple?
Three whole numbers that fit a² + b² = c², such as 3-4-5, 5-12-13, 8-15-17 and 7-24-25. Any multiple of one is another (6-8-10, 9-12-15). The calculator tells you when your three sides form one.
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