What is the formula for the inradius of a right triangle?
The formula is r = (a + b − c) / 2, where a and b are the legs, c is the hypotenuse, and r is the inradius.
Radius Calculator
Right Triangle Inradius Calculator
Use this right triangle inradius calculator when you know leg a, leg b, and hypotenuse c and want to find the inradius r. The inradius is the radius of the incircle, the circle that fits inside the right triangle and touches all three sides.
This focused calculator is helpful when a geometry problem gives all three side lengths and asks for the circle inside the triangle. It uses the standard right triangle inradius formula, so the side labeled c must be the hypotenuse.
Enter positive values for a, b, and c. The hypotenuse c should be the longest side, and the three side lengths should form a valid right triangle.
Calculate Inradius r
This calculator finds Inradius r using r = (a + b − c) / 2.
Enter inputs to calculate Inradius r.
Inradius r
Result-
Solution Steps
Formula: r = (a + b − c) / 2
Formula
r = (a + b − c) / 2
For a right triangle, the inradius can be found directly from the two legs and the hypotenuse.
Here, a and b are the legs of the right triangle, c is the hypotenuse, and r is the inradius.
- Add the two legs: a + b.
- Subtract the hypotenuse: a + b − c.
- Divide the result by 2 to get the inradius r.
Triangle Diagram
The diagram shows the incircle inside a right triangle. The circle touches leg a, leg b, and hypotenuse c.
- The segment marked r is the inradius.
- The small square marks the right angle.
- The hypotenuse c is opposite the right angle and is the longest side.
How to Use This Calculator
- Enter the length of leg a.
- Enter the length of leg b.
- Enter the length of hypotenuse c.
- Check that c is the longest side and that the side lengths describe a right triangle.
- Select Calculate Inradius to get r.
Step-by-Step Example
Given:
- Leg a = 6
- Leg b = 8
- Hypotenuse c = 10
- r = (a + b − c) / 2
- r = (6 + 8 − 10) / 2
- r = 4 / 2
- r = 2
For a 6-8-10 right triangle, the inradius is 2 units.
What the Result Means
The result r is the radius of the incircle. If r = 2, then a circle with radius 2 units can fit inside the right triangle while touching all three sides.
The inradius is useful when you need the size of the inscribed circle, the distance from the incenter to each side, or a quick way to connect side lengths with area and semiperimeter.
Right Triangle Inradius Relation r
On the main right triangle radius calculator, this same calculation appears as the Inradius and Radius Relation r mode.
The relation uses the same formula, r = (a + b − c) / 2, so this page is the focused version when you only need inradius r.
When to Use This Calculator
- Finding the radius of the incircle in a right triangle.
- Checking homework problems that give all three side lengths.
- Using area and semiperimeter relationships with right triangles.
- Comparing the inradius r with the circumradius R.
- Solving geometry problems where an inner circle touches all three sides.
Common Mistakes
- Using c as a leg instead of the hypotenuse.
- Forgetting to divide a + b − c by 2.
- Using the circumradius formula R = c / 2 when the question asks for inradius r.
- Entering side lengths that do not form a valid right triangle.
- Mixing units, such as entering two sides in inches and one side in centimeters.
Related Calculators
right triangle radius calculator
Use the parent calculator for inradius, circumradius, and radius relation modes.
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right triangle circumradius calculator
Find circumradius R from the hypotenuse c.
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right triangle area calculator
Calculate area from the right triangle side lengths.
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right triangle semiperimeter calculator
Find semiperimeter before using formulas that involve s.
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right triangle side calculator
Find a missing side before calculating the inradius.
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FAQs
What does the inradius result mean?
The result is the radius of the circle that fits inside the right triangle and touches all three sides.
Why does c have to be the hypotenuse?
This formula assumes c is the side opposite the right angle. In a right triangle, that side is the hypotenuse and it is the longest side.
Can I find the inradius if I only know the two legs?
Yes, but first find the hypotenuse c using the Pythagorean theorem or a right triangle side calculator, then use r = (a + b − c) / 2.
Is the inradius the same as the circumradius?
No. The inradius r belongs to the circle inside the triangle, while the circumradius R belongs to the circle passing through the three vertices.