Geometry desk
Right Triangle Calculator
Calculate the hypotenuse and area of a right triangle from its two legs.
Right Triangle Calculator: Calculate the side opposite a right angle and the area enclosed by the triangle from its two perpendicular legs. This helps check a diagonal across a rectangular opening or a simple right-angle layout without measuring the diagonal directly. Runs 100% locally in your browser with zero server file uploads.
- Category
- Calculators
- Runs
- In your browser
- Cost
- Free · no sign-up
- Availability
- Ready to use
Hypotenuse
5 cm
- Area
- 6 cm²
Hypotenuse = √(leg a² + leg b²); Area = (leg a × leg b) ÷ 2- The two supplied legs meet at a 90° angle.
- Dimensions are measured in the same unit.
- Values must be finite and greater than zero.
Formula and worked example
Hypotenuse = √(leg a² + leg b²); Area = (leg a × leg b) ÷ 2
A worked example with sample values:
| Inputs | |
|---|---|
| Leg A | 3 cm |
| Leg B | 4 cm |
| Results | |
|---|---|
| Hypotenuse | 5 cm |
| Area | 6 cm² |
Areas of squares give the diagonal
Let a and b denote the legs and c the hypotenuse. Pythagoras gives c² = a² + b²: the area of the square on the hypotenuse equals the combined areas of squares on the legs. Taking the positive square root gives c = √(a² + b²). Triangle area is A = a × b ÷ 2 because the perpendicular legs serve as base and altitude.
With a = 3 cm and b = 4 cm, the squared lengths are 9 and 16 cm². Their sum is 25 cm², so c = √25 = 5 cm. The area is 3 × 4 ÷ 2 = 6 cm². The result sheet shows Hypotenuse = 5 cm and Area = 6 cm².
OpenStax explains the square-area relationship (https://openstax.org/books/prealgebra-2e/pages/9-3-use-properties-of-angles-triangles-and-the-pythagorean-theorem) for triangles with a right angle. Equal-looking legs alone do not prove that angle is 90°.
Scaling a layout
Scaling the example by ten gives legs 30 and 40 cm, hypotenuse 50 cm and area 600 cm². All lengths grow tenfold, whereas area grows a hundredfold. For checking a measured layout, compare its actual diagonal with the calculated one; the calculator assumes perpendicular legs and cannot verify the corner from two lengths alone. Convert both legs to a common unit before entering them. A positive output with three displayed decimals is a rounded length, not evidence that your measurements were accurate to that precision.
How to use it
- Enter positive dimensions in the same unit.
- Choose the unit used for every dimension.
- Calculate, then review the result and formula.
Privacy & limitations
Every value and calculation stays in your browser. No dimensions are uploaded or sent to an external service.
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Frequently asked questions
Which sides belong in the two right-triangle fields?
Enter the sides meeting at the 90° corner. The hypotenuse is opposite that corner and is the longest side. Entering a hypotenuse as a leg asks the tool to calculate a different triangle.
What are Pythagorean triples, and can I scale them?
They are positive whole-number side sets satisfying a² + b² = c². Examples include 3–4–5, 5–12–13 and 8–15–17. Multiplying every side by the same factor preserves the right angle; 6–8–10 is twice 3–4–5.
Can I recover a leg when I know the hypotenuse?
By hand, use b = √(c² − a²), with c the hypotenuse and a the known leg. For c = 13 and a = 5, b = √(169 − 25) = 12. This widget accepts two legs, so it does not perform that inverse entry directly.
Embed this tool
Free to use on any website. Paste the code where the tool should appear; the link under it lets readers open the full page.
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