From inputs to Triangle Circumradius output
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
The reasoning underneath Triangle Circumradius
The circumradius satisfies R=abc/(4A), where A is triangle area.
Start the Triangle Circumradius cross-check with Side a. Apply the original Side b and recompute Circumradius. Treat a different Side c as new Triangle Circumradius data. That prevents its Circumradius from being attributed to the earlier Triangle Circumradius setup.
Limits of the chosen Triangle Circumradius model
Invalid or nearly collinear side sets are rejected; as area approaches zero, circumradius grows without bound.
Perturbing one Triangle Circumradius input
The Triangle Circumradius meaning depends on Side a. The Triangle Circumradius meaning also depends on Side b. Carry those Triangle Circumradius roles into any later Triangle Circumradius work.
Preserving the Triangle Circumradius setup
Use Heron’s formula for area, multiply the sides, and divide by four times the area. To continue from Triangle Circumradius, try inradius comparison.
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.
It describes the triangle’s circumscribed circle and supports chord, navigation, and cyclic-geometry calculations. A related application of Triangle Circumradius is area.
A second verification of Triangle Circumradius
Sketch Side a before running Triangle Circumradius. Place Side b on the Triangle Circumradius sketch and confirm its unit.
Test a symmetric Triangle Circumradius case. In that Triangle Circumradius case, compare Side b with the expected Triangle Circumradius scale.
Before using Circumradius downstream, substitute a simple Side a into the same Triangle Circumradius relation. Preserve Side b so the comparison remains fair. The resulting Circumradius provides a benchmark for detecting a misplaced sign, decimal, or input order.
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.