What is the formula for the circumradius of a right triangle?
The formula is R = c / 2, where c is the hypotenuse and R is the circumradius.
Radius Calculator
Right Triangle Circumradius Calculator
Use this right triangle circumradius calculator when you know the hypotenuse c and want to find the circumradius R. The circumradius is the radius of the circle that passes through all three vertices of the right triangle.
In a right triangle, the hypotenuse is the diameter of the circumcircle. That makes the circumradius especially simple: it is one half of the hypotenuse.
Enter a positive value for hypotenuse c. This calculator assumes c is the side opposite the right angle.
Calculate Circumradius R
This calculator finds Circumradius R using R = c / 2.
Enter inputs to calculate Circumradius R.
Circumradius R
Result-
Solution Steps
Formula: R = c / 2
Formula
R = c / 2
For every right triangle, the hypotenuse is the diameter of the circumcircle.
Here, c is the hypotenuse and R is the circumradius.
- Start with the hypotenuse c.
- Divide c by 2.
- The result is the circumradius R.
Triangle Diagram
The diagram shows the circumcircle around a right triangle. The hypotenuse c acts as the diameter of the circle.
- The circumcenter is the midpoint of the hypotenuse.
- The segment marked R is the circumradius.
- The circle passes through all three triangle vertices.
How to Use This Calculator
- Enter the length of hypotenuse c.
- Make sure c is greater than 0.
- Select Calculate Circumradius.
- Read the result as the circumradius R.
Step-by-Step Example
Given:
- Hypotenuse c = 10
- R = c / 2
- R = 10 / 2
- R = 5
For a right triangle with hypotenuse 10, the circumradius is 5 units.
What the Result Means
The result R is the radius of the circumcircle. If R = 5, then the circle passing through the three vertices has radius 5 units.
Because the hypotenuse is the diameter of the circumcircle, the full diameter is c and the radius is half of c.
Right Triangle Circumradius Relation R
On the main right triangle radius calculator, this same calculation appears as the Circumradius and Radius Relation R mode.
The relation uses the same formula, R = c / 2, so this page is the focused version when you only need circumradius R.
When to Use This Calculator
- Finding the radius of the circle through all three vertices of a right triangle.
- Using a known hypotenuse to work with circumcircle geometry.
- Checking whether a radius or diameter value matches a right triangle diagram.
- Comparing circumradius R with inradius r.
- Solving problems that mention the midpoint of the hypotenuse.
Common Mistakes
- Entering a leg length instead of the hypotenuse c.
- Forgetting that R is half of c, not the full hypotenuse.
- Confusing circumradius R with inradius r.
- Calling the hypotenuse c a leg.
- Using zero or a negative value for c.
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FAQs
What does the circumradius result mean?
The result is the radius of the circle that passes through all three vertices of the right triangle.
Why is the circumcenter at the midpoint of the hypotenuse?
In a right triangle, the hypotenuse is the diameter of the circumcircle, so its midpoint is the center of that circle.
Do I need leg a or leg b to find the circumradius?
No. If you already know the hypotenuse c, the circumradius is simply R = c / 2.
Is the circumradius the same as the diameter?
No. The hypotenuse c is the diameter of the circumcircle, and the circumradius R is half of that diameter.