Circumscribed Circle of a Triangle — Definition, Formula & Examples

The circumscribed circle (or circumcircle) of a triangle is the unique circle that passes through all three vertices of the triangle. Its center, called the circumcenter, is equidistant from each vertex.

Given a triangle $\triangle ABC$ , the circumscribed circle is the circle of radius $R$ centered at the point $O$ such that $OA = OB = OC = R$ , where $O$ is the intersection of the perpendicular bisectors of the triangle's sides.

Key Formula

R = \frac{abc}{4K}

Where:

$R$ = Circumradius — the radius of the circumscribed circle
$a, b, c$ = The lengths of the three sides of the triangle
$K$ = The area of the triangle

How It Works

To construct the circumscribed circle, find the perpendicular bisector of at least two sides of the triangle. The point where these bisectors intersect is the circumcenter

O

. Measure the distance from

O

to any vertex — that distance is the circumradius

R

. Draw a circle centered at

O

with radius

R

, and it will pass through all three vertices. For an acute triangle, the circumcenter lies inside the triangle; for a right triangle, it lies on the hypotenuse; for an obtuse triangle, it lies outside.

Worked Example

Problem: Find the circumradius of a triangle with sides

a = 3

b = 4

, and

c = 5

Identify the triangle type: Check whether this is a right triangle:

3^2 + 4^2 = 9 + 16 = 25 = 5^2

. It is a right triangle.

a^2 + b^2 = c^2

Find the area: For a right triangle with legs 3 and 4, the area is:

K = \frac{1}{2}(3)(4) = 6

Apply the circumradius formula: Substitute the side lengths and area into

R = \frac{abc}{4K}

R = \frac{(3)(4)(5)}{4(6)} = \frac{60}{24} = 2.5

Answer: The circumradius is

R = 2.5

. Notice this equals half the hypotenuse, which is always true for right triangles.

Why It Matters

The circumscribed circle appears in coordinate geometry proofs, triangle congruence arguments, and problems involving cyclic quadrilaterals. Engineers and surveyors use circumcircles when fitting circular arcs through three known points.

Common Mistakes

Mistake: Confusing the circumcenter with the centroid or incenter.

Correction: The circumcenter is the intersection of perpendicular bisectors and is equidistant from the vertices. The centroid is the intersection of medians, and the incenter is the intersection of angle bisectors (equidistant from the sides, not vertices).