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Circumradius

Circumradius

The radius of a plane figure's circumcircle.

 

 

See also

Circumcenter

Key Formula

R=abc4KR = \frac{abc}{4K}
Where:
  • RR = Circumradius of the triangle
  • a,b,ca, b, c = Side lengths of the triangle
  • KK = Area of the triangle

Worked Example

Problem: Find the circumradius of a triangle with side lengths a = 3, b = 4, and c = 5.
Step 1: Identify the type of triangle. Since 3² + 4² = 9 + 16 = 25 = 5², this is a right triangle.
32+42=523^2 + 4^2 = 5^2
Step 2: Calculate the area of the triangle. For a right triangle, the two legs are the base and height.
K=12(3)(4)=6K = \frac{1}{2}(3)(4) = 6
Step 3: Apply the circumradius formula.
R=abc4K=(3)(4)(5)4(6)=6024=52R = \frac{abc}{4K} = \frac{(3)(4)(5)}{4(6)} = \frac{60}{24} = \frac{5}{2}
Step 4: Verify: for any right triangle, the circumradius equals half the hypotenuse, and indeed 5/2 = 2.5, which is half of 5.
R=c2=52=2.5R = \frac{c}{2} = \frac{5}{2} = 2.5
Answer: The circumradius is R = 2.5.

Another Example

Problem: Find the circumradius of a regular hexagon with side length 7.
Step 1: Recall the key property of a regular hexagon: the circumradius equals the side length. A regular hexagon is made of six equilateral triangles whose side length equals the distance from the center to each vertex.
R=sR = s
Step 2: Substitute the given side length.
R=7R = 7
Answer: The circumradius of the regular hexagon is R = 7.

Frequently Asked Questions

How do you find the circumradius of a triangle?
Use the formula R = abc / (4K), where a, b, c are the side lengths and K is the area. You can find the area using Heron's formula if you know all three sides. For a right triangle, the circumradius is simply half the hypotenuse. For an equilateral triangle with side length s, the circumradius is s√3 / 3.
What is the difference between circumradius and inradius?
The circumradius (R) is the radius of the circle that passes through all vertices of the polygon (the circumscircle), measured from the circumcenter to a vertex. The inradius (r) is the radius of the largest circle that fits inside the polygon (the incircle), measured from the incenter to a side. For any triangle, R ≥ 2r, with equality only for an equilateral triangle.

vs.

The circumradius R is the radius of the circumscribed circle passing through all vertices, while the inradius r is the radius of the inscribed circle tangent to all sides. The circumradius is always greater than or equal to twice the inradius (Euler's inequality: R ≥ 2r). They are measured from different centers — the circumcenter and the incenter — which generally do not coincide.

Why It Matters

The circumradius appears throughout geometry and trigonometry. The law of sines directly involves the circumradius: a/sin A = b/sin B = c/sin C = 2R. It is also essential in competition mathematics, coordinate geometry, and engineering applications such as designing parts that must fit inside or around circular boundaries.

Common Mistakes

Mistake: Confusing the circumradius with the inradius, or assuming they are equal.
Correction: The circumradius goes from the center to a vertex (circumscribed circle), while the inradius goes from the center to a side (inscribed circle). They are equal only in very special cases, such as never for a triangle. Always check which circle — outer or inner — the problem refers to.
Mistake: Assuming every polygon has a circumradius.
Correction: Only cyclic polygons (those whose vertices all lie on a single circle) have a circumradius. All triangles are cyclic, but not all quadrilaterals or other polygons are. For example, a non-rectangular parallelogram has no circumscribed circle and therefore no circumradius.

Related Terms