Seeing the method in action
For sides 5,7,9 and area about 17.412, R≈4.523.
Math calculator
Triangle Circumradius Calculator
Find the radius of the circle through all three triangle vertices. Beside circumradius, the output shows the operation used to obtain it.
Triangle Circumradius inputs
Build the expression
Situations that fit the model
It describes the triangle’s circumscribed circle and supports chord, navigation, and cyclic-geometry calculations. A related application of Triangle Circumradius is area.
Why this relationship works
The circumradius satisfies R=abc/(4A), where A is triangle area.
For this page, triangle circumradius is interpreted under the stated convention and input order.
Steps without the calculator
Use Heron’s formula for area, multiply the sides, and divide by four times the area. To continue from Triangle Circumradius, try inradius comparison.
Boundaries and common traps
Invalid or nearly collinear side sets are rejected; as area approaches zero, circumradius grows without bound.
Review the sign, scale, and unit of circumradius after entering side a, side b and side c.
A closer look at result behavior
The cleanest input audit is to restate the problem using side a, side b and side c. If that sentence sounds wrong, correct the assignment before asking for circumradius.
How the result responds
Scaling every side by a factor scales R equally, while preserving all angles. The pattern also provides a quick estimate of whether a revised result is plausible.
Inradius measures tangent clearance inside; circumradius passes through the vertices. The wording of the problem should decide which operation is appropriate.
Recording the answer
The formula and result serve different readers: the formula shows what was done, and the rounded circumradius communicates scale. Keep both when the work needs review.
A screenshot is unnecessary when the shown formula and the supplied inputs (side a, side b and side c) are saved in plain text. Include the rounding rule if the reported circumradius is approximate.
How to audit the Triangle Circumradius result
Use the extended sine rule as an independent check when an angle is known: a/sin A should equal 2R. The circumradius can exceed every side in a very obtuse triangle, so size alone is not a reliable rejection test.
For Triangle Circumradius, a useful audit records the formula, the supplied quantities, and one relationship that the answer must satisfy. Carry extra digits through that relationship and round only the final comparison. This separates numerical display differences from a genuine setup error.
Questions about Triangle Circumradius
Where is the circumcenter?
At the intersection of perpendicular bisectors.
Can it lie outside the triangle?
Yes for an obtuse triangle.
What is R for a right triangle?
Half the hypotenuse.