Data Source and Methodology
Authoritative source: Euclid’s Elements, Book I, Proposition 47 (ca. 300 BCE). See: Clark University – Euclid’s Elements I.47.
Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.
The Formula Explained
Pythagorean theorem (right triangles):
LaTeX: c = \sqrt{a^2 + b^2}
LaTeX: a = \sqrt{c^2 - b^2}, \quad b = \sqrt{c^2 - a^2}
LaTeX: \text{Area} = \frac{a \cdot b}{2}, \quad \text{Perimeter} = a + b + c
LaTeX: \alpha = \arctan\!\left(\frac{a}{b}\right), \quad \beta = \arctan\!\left(\frac{b}{a}\right)
Glossary of Variables
| Symbol | Name | Description |
|---|---|---|
a | Leg a | One side forming the right angle. |
b | Leg b | The other side forming the right angle. |
c | Hypotenuse | Longest side, opposite the right angle. |
\alpha | Angle alpha | Angle opposite leg a; α = arctan(a/b). |
\beta | Angle beta | Angle opposite leg b; β = arctan(b/a). |
| Area | Right triangle area | Computed as (a·b)/2. |
| Perimeter | Triangle perimeter | Sum of all sides, a + b + c. |
How It Works: A Step-by-Step Example
Example Suppose a = 3 and b = 4. To find c:
- Apply the theorem: c = √(a² + b²).
- Substitute: c = √(3² + 4²) = √(9 + 16) = √25.
- Compute: c = 5.
- Extras: Area = (3·4)/2 = 6; Perimeter = 3 + 4 + 5 = 12. Angles: α = arctan(3/4) ≈ 36.87°, β ≈ 53.13°.
Frequently Asked Questions (FAQ)
Which sides are a, b, and c?
a and b are the legs meeting at 90°. c is the hypotenuse, opposite the right angle and always the largest side.
Can I find a leg if I know the hypotenuse and the other leg?
Yes. Use a = √(c² − b²) or b = √(c² − a²). Make sure c is greater than the known leg; otherwise the square root would be invalid.
What units should I use?
Any consistent length units (m, cm, in, ft, etc.). The results keep the same unit.
How many decimals does the tool support?
You can choose 0–10 decimal places for display. Internally, the computation uses full precision.
Does the tool verify if three given sides make a right triangle?
Yes. If you enter all three, it checks whether c² ≈ a² + b² within a small tolerance and reports the classification.
Why do I see an error about the hypotenuse?
The hypotenuse must be the largest side. If c ≤ a or c ≤ b, adjust your values or solve for another side.
Can I share the calculation?
Use “Share results” to copy a URL with your inputs and rounding. Anyone visiting the link will see the same setup pre-filled.