Circumcircle

Circumcircle
Circumscribed Circle

A circle that passes through all vertices of a plane figure and contains the entire figure in its interior.

Note: All triangles have circumcircles and so do all regular polygons. Most other polygons do not.

A circle passing through all five vertices of a pentagon, with the label "circumcircle of a pentagon.

Key Formula

R = \frac{abc}{4K}

Where:

$R$ = Circumradius — the radius of the circumcircle
$a, b, c$ = The side lengths of the triangle
$K$ = The area of the triangle

Worked Example

Problem: Find the circumradius of a triangle with sides 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 with the hypotenuse c = 5.

3^2 + 4^2 = 5^2

Step 2: Compute the area of the triangle. For a right triangle, the two legs serve as base and height.

K = \frac{1}{2} \cdot 3 \cdot 4 = 6

Step 3: Apply the circumradius formula.

R = \frac{abc}{4K} = \frac{3 \cdot 4 \cdot 5}{4 \cdot 6} = \frac{60}{24} = 2.5

Step 4: Verify: For any right triangle, the circumradius equals half the hypotenuse. Indeed, 5/2 = 2.5. ✓

R = \frac{c}{2} = \frac{5}{2} = 2.5

Answer: The circumradius is R = 2.5, so the circumcircle has radius 2.5 and is centered at the midpoint of the hypotenuse.

Another Example

This example uses a triangle where all sides are equal, demonstrating how the general formula simplifies for a regular polygon. It also shows how to rationalize a denominator.

Problem: Find the circumradius of an equilateral triangle with side length 6.

Step 1: Compute the area of an equilateral triangle with side s = 6 using the standard formula.

K = \frac{\sqrt{3}}{4} s^2 = \frac{\sqrt{3}}{4} \cdot 36 = 9\sqrt{3}

Step 2: All three sides are equal, so a = b = c = 6. Substitute into the circumradius formula.

R = \frac{abc}{4K} = \frac{6 \cdot 6 \cdot 6}{4 \cdot 9\sqrt{3}} = \frac{216}{36\sqrt{3}}

Step 3: Simplify the fraction.

R = \frac{6}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3} \approx 3.46

Step 4: Verify using the direct formula for an equilateral triangle: R = s/√3. Indeed, 6/√3 = 2√3. ✓

R = \frac{s}{\sqrt{3}} = \frac{6}{\sqrt{3}} = 2\sqrt{3}

Answer: The circumradius of the equilateral triangle is 2√3 ≈ 3.46.

Frequently Asked Questions

What is the difference between a circumcircle and an inscribed circle?

A circumcircle passes through all the vertices of a polygon and encloses the figure. An inscribed circle (incircle) is tangent to every side of the polygon and sits inside it. The circumcircle is always larger than or equal to the incircle for the same polygon. Their centers — the circumcenter and incenter — are generally at different points.

Do all polygons have a circumcircle?

No. Every triangle and every regular polygon has a circumcircle, but most irregular quadrilaterals and other polygons do not. A polygon that has a circumcircle is called a cyclic polygon. For a quadrilateral, a circumcircle exists if and only if the opposite angles sum to 180°.

How do you find the center of a circumcircle?

For a triangle, the circumcenter is the point where the perpendicular bisectors of all three sides intersect. You can find it by constructing the perpendicular bisector of any two sides and locating their intersection. This point is equidistant from all three vertices, and that common distance is the circumradius.

Circumcircle (circumscribed circle) vs. Inscribed circle (incircle)

	Circumcircle (circumscribed circle)	Inscribed circle (incircle)
Definition	Circle passing through all vertices of the polygon	Circle tangent to all sides of the polygon, lying inside it
Center name	Circumcenter (intersection of perpendicular bisectors)	Incenter (intersection of angle bisectors)
Radius formula (triangle)	R = abc / (4K)	r = K / s, where s is the semi-perimeter
Position relative to figure	Surrounds the polygon	Contained within the polygon
Existence for triangles	Always exists	Always exists
Relative size	Always at least as large as the incircle	Always at most as large as the circumcircle

Why It Matters

Circumcircles appear throughout geometry courses when you study triangle centers, cyclic quadrilaterals, and the extended law of sines (a/sin A = 2R). They also arise in real-world applications such as determining the smallest circular boundary that encloses a set of points, which is used in engineering, navigation, and computer graphics. Understanding circumcircles deepens your grasp of how angles and distances relate within polygons.

Common Mistakes

Mistake: Assuming every polygon has a circumcircle.

Correction: Only triangles and regular polygons are guaranteed to have one. An arbitrary quadrilateral or irregular polygon usually does not. A polygon must be cyclic — meaning all its vertices lie on a single circle — for a circumcircle to exist.

Mistake: Confusing the circumcenter with the centroid or incenter.

Correction: The circumcenter is the intersection of the perpendicular bisectors of the sides, the incenter is the intersection of the angle bisectors, and the centroid is the intersection of the medians. In most triangles these three points are all different. Only in an equilateral triangle do they coincide.

Related Terms

Circumcenter — The center of the circumcircle
Circumradius — The radius of the circumcircle
Inscribed Circle — Circle tangent to all sides, the dual concept
Circumscribable — Property of having a circumcircle
Circle — The underlying geometric shape
Triangle — Always possesses a circumcircle
Regular Polygon — Always possesses a circumcircle
Vertex — Points that the circumcircle passes through