Calculator inputs
Choose a method, enter values, and calculate the circumradius instantly.
Plotly graph
This chart compares the main values from the current calculation.
Example data table
| Method | Example inputs | Triangle area | Circumradius |
|---|---|---|---|
| Three sides | a = 13, b = 14, c = 15 | 84 | 8.125 |
| Side and angle | Side = 10, Opposite angle = 40° | Not required | 7.779 |
| Coordinates | (0,0), (6,0), (2,5) | 15 | 3.448 |
Formula used
Using three sides
Heron’s area: Δ = √(s(s-a)(s-b)(s-c))
Semi-perimeter: s = (a + b + c) / 2
Circumradius: R = abc / 4Δ
Using one side and its opposite angle
Circumradius: R = a / 2sin(A)
Using coordinates
First find side lengths from the distance formula. Then use Heron’s formula and compute R = abc / 4Δ.
How to use this calculator
- Select the method that matches your known values.
- Enter valid side lengths, angle values, or point coordinates.
- Click the calculate button.
- Read the result summary above the form.
- Review the chart and geometry view for extra insight.
- Use the CSV or PDF buttons to export the result.
About triangle circumradius
What the circumradius means
The circumradius is the radius of the circle passing through all three vertices of a triangle. That circle is called the circumcircle. It is a core geometry concept. It also appears in trigonometry and coordinate geometry. A larger circumradius means a larger surrounding circle.
Why this value matters
The circumradius helps compare triangle size and shape. It is useful in proofs. It supports drawing, surveying, and modeling tasks. Students often use it to connect side lengths, angles, and area. Engineers may use it during layout checks. Designers use it in circular fitting work.
Common ways to find it
There is more than one valid route. If all three sides are known, Heron’s formula works well. If one side and its opposite angle are known, the sine rule form is faster. If coordinates are known, side lengths can be derived first. Then the same circumradius formula can be applied.
Accuracy tips
Always check that the triangle is valid. Side lengths must satisfy triangle inequality. Angles must stay between zero and one hundred eighty degrees. Coordinate points must not be collinear. Small input errors can change the final radius. Use consistent units throughout the calculation.
What this page gives you
This calculator does more than show one number. It returns the circumradius, diameter, circle circumference, and circle area. It also shows triangle area when possible. The included graph gives a quick visual comparison. The geometry plot helps confirm the shape. Export tools make record keeping easier.
When to use each method
Use the three side method when complete edge data is available. Use the side and angle method for direct trigonometry problems. Use coordinates when working on graph paper or analytic geometry. Each option fits a different classroom or practical need. That flexibility makes the tool useful for many users.
FAQs
1. What is the circumradius of a triangle?
It is the radius of the unique circle that passes through all three triangle vertices. That circle is called the circumcircle.
2. Can I calculate circumradius from sides only?
Yes. Use the three side method. The calculator first finds triangle area with Heron’s formula and then applies R = abc / 4Δ.
3. What happens for a right triangle?
For a right triangle, the circumradius equals half of the hypotenuse. This is a helpful shortcut and a good way to check your answer.
4. Why does the calculator reject some side values?
The sides must form a real triangle. If triangle inequality fails, no valid area exists, so the circumradius cannot be computed.
5. Can coordinates be used directly?
Yes. Enter three non collinear points. The calculator derives side lengths, area, circumcenter, and the circumradius from those coordinates.
6. Does unit choice matter?
Yes. The output keeps the same length unit as your inputs. If your sides are in centimeters, the circumradius will also be in centimeters.
7. Why is the graph useful?
The graph helps you compare key values quickly. It gives a visual check for size relationships between the radius, diameter, and related measures.
8. What can I export from this page?
You can export the calculated result table as CSV or PDF. This helps with homework records, reports, and quick sharing.