Library / Geometry, coordinates, surveying and trigonometry / GEO-12
Geometry, coordinates, surveying and trigonometry
Circumradius of a triangle from sides and area
Radius of the circle through all three corners, from the three sides and the area.
- Ref
- GEO-12
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- R [cm]
The circle through all three corners of a triangle has radius R = abc/(4A), where a, b, c are the sides and A is the area. It follows from the sine rule, a/sin A = 2R, and A = ½ b c sin A. In a right triangle R is half the hypotenuse. The sides must satisfy the strict triangle inequalities.
Radius of the circle through all three corners, from the three sides and the area. The sides must satisfy the strict triangle inequalities, and near-straight triangles have very large circumradii.
Calculator
- Circumcircle diameter, 2R5 cm
- Area from Heron’s formula for these sides6 cm²
- Longest side over the diameter, the sine of the biggest angle1
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A triangle with sides 3, 4 and 5 cm and area 6 cm² has a circumradius of 2.5 cm. A right triangle has R equal to half its hypotenuse.
- If the area given does not match Heron’s value for the sides, the result is the radius of a different triangle, so check the area first.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| A Triangle area | 0.5 to 30 cm² | 0.5 | 30 | 29.5 | +10% | −9.09 % |
| a Side a | 1.5 to 8.5 cm | 1.25 | 7.083 | 5.833 | +10% | +10 % |
| b Side b | 2.5 to 7.5 cm | 1.563 | 4.688 | 3.125 | +10% | +10 % |
| c Side c | 1.5 to 6.5 cm | 0.75 | 3.25 | 2.5 | +10% | +10 % |
Sweep of Triangle area
| Triangle area (cm²) | Circumradius (cm) |
|---|---|
| 0.5 | 30 |
| 4.188 | 3.582 |
| 7.875 | 1.905 |
| 11.56 | 1.297 |
| 15.25 | 0.9836 |
| 18.94 | 0.7921 |
| 22.63 | 0.663 |
| 26.31 | 0.5701 |
| 30 | 0.5 |
Sweep one variable
Variables and how each one contributes
Circumradius
What the equation gives you. Everything else feeds it.
Side a
The length of side a, in cm. A longer side raises the circumradius, since the area is held fixed. Raising it by 10 % (to 3.3) moves the result by +10 %, from 2.5 to 2.75 cm.
Side b
The length of side b, in cm. A longer side raises the circumradius, since the area is held fixed. Raising it by 10 % (to 4.4) moves the result by +10 %, from 2.5 to 2.75 cm.
Side c
The length of side c, in cm. A longer side raises the circumradius, since the area is held fixed. Raising it by 10 % (to 5.5) moves the result by +10 %, from 2.5 to 2.75 cm.
Triangle area
The area of the triangle, in cm². It must agree with the three sides. A larger area gives a smaller circumradius in inverse proportion. Raising it by 10 % (to 6.6) moves the result by −9.09 %, from 2.5 to 2.273 cm.
Worked example with the values above
- Product of the sides: abc = 3 × 4 × 5 = 60.
- Four times the area: 4A = 24.
- R = 60 / 24 = 2.5 cm.
Assumptions and limits
- A non-degenerate triangle: the three sides satisfy the strict triangle inequality.
- All lengths and the area use one consistent unit.
Common mistakes
- Using an area that does not match the three sides, which gives a circumradius for a triangle that does not exist.
- Mixing up which side is bisected, or which side the median goes to.
- Using the perimeter where the semi-perimeter is needed.
Statistics and uncertainty
The inputs are exact lengths and areas. When a triangle is nearly flat, the area becomes very small and the circumradius grows quickly, so a small error in the area gives a large error in R.
Where it is used
- Fitting the largest circle in a triangle and the circle through three points.
- Centre lines, balance points and bracing positions of triangular frames and plates.
Maths you need
- Triangle inequality: any two sides add to more than the third
- Heron’s formula: area from three sides
- Midpoint: the point halfway along a segment