Calculator
Enter the two known sides for a right triangle. The result appears above this form after submission.
Formula Used
Main theorem: a² + b² = c²
Find the hypotenuse: c = √(a² + b²)
Find leg a: a = √(c² - b²)
Find leg b: b = √(c² - a²)
When the square root is not a perfect square, the calculator simplifies the radical by taking perfect-square factors out of the root.
How to Use This Calculator
- Select which side you want to solve for.
- Enter the two known sides of the right triangle.
- Optionally add a unit label such as cm or m.
- Choose how many decimal places you want displayed.
- Press Calculate to see the exact radical result.
- Review the decimal form and the worked steps.
- Use the export buttons to save the answer as CSV or PDF.
Example Data Table
| Case | Known Values | Formula | Exact Result | Decimal Result |
|---|---|---|---|---|
| Find c | a = 3, b = 4 | c = √(a² + b²) | 5 | 5.000000 |
| Find c | a = 1, b = 1 | c = √(a² + b²) | √2 | 1.414214 |
| Find c | a = 3, b = 6 | c = √(a² + b²) | 3√5 | 6.708204 |
| Find a | b = 8, c = 10 | a = √(c² - b²) | 6 | 6.000000 |
| Find b | a = 7, c = 25 | b = √(c² - a²) | 24 | 24.000000 |
Article
Understanding Exact Radical Answers
The Pythagorean theorem links the three sides of a right triangle. It states that a² + b² = c². The longest side is the hypotenuse. The other two sides are legs. This calculator finds a missing side and keeps the answer in exact radical form whenever possible.
Why Radical Form Matters
Many triangle problems do not end with whole numbers. They produce square roots. A decimal is useful for measurement. An exact radical is better for algebra. It keeps precision. It also shows structure. Students can simplify roots and compare patterns more clearly.
What This Calculator Does
This page solves for any one missing side. You choose whether to find a leg or the hypotenuse. Then you enter the known sides. The tool checks that the values are positive. It also checks that the hypotenuse is longer than a leg when needed. If the result is irrational, the calculator simplifies the radical. It also shows a decimal approximation for quick reference.
How Exact Simplification Works
Suppose the missing hypotenuse is found from legs 5 and 12. The expression becomes √(25 + 144). That equals √169. The final answer is 13. Now try legs 1 and 1. The expression becomes √2. That cannot simplify further. With legs 3 and 6, the result is √45. The calculator rewrites that as 3√5. This step removes perfect-square factors from inside the root.
Helpful for Study and Checking
This tool is useful in homework, quizzes, and revision. It helps verify hand calculations. It also supports teachers who need quick examples. The example table shows common right-triangle cases. The export buttons make it easier to save results. That helps when preparing notes or practice sheets. Use the exact form first. Then compare the decimal to confirm scale and reasonableness.
Common Mistakes to Avoid
Do not place the hypotenuse in a leg field. Do not subtract in the wrong order. A negative value under the root means the inputs do not form a valid right triangle. Always check units before solving. Keep units consistent. Simplify the radical fully. Then round only at the end if you need a decimal answer for clear reporting or measurement tasks.
FAQs
1. What is a radical answer?
A radical answer keeps the square root symbol instead of converting everything into decimals. For example, √2 is exact, while 1.414214 is an approximation.
2. When should I use this calculator?
Use it when you know two sides of a right triangle and need the third. It works for finding either leg or the hypotenuse.
3. Why does the tool show both exact and decimal results?
The exact result is better for algebra. The decimal result is better for measurements, checking size, and comparing practical values quickly.
4. Can the calculator simplify roots like √45?
Yes. It looks for perfect-square factors inside the root. That is why √45 becomes 3√5 instead of staying unsimplified.
5. What happens if my inputs do not make a right triangle?
The page shows a validation message. This usually happens when the supposed hypotenuse is not longer than the leg you entered.
6. Can I enter decimals?
Yes. Decimal entries are converted into fractions behind the scenes. That lets the calculator keep an exact radical form whenever possible.
7. Are units required?
No. Units are optional. Add them when you want the result label to show cm, m, inches, or another measurement unit.
8. What file name should I save this page as?
Save it as pythagorean_theorem_radical_answers.php. The same name is also placed in the opening comment for convenience.