Why is R exactly half the hypotenuse in a right triangle?

Thales: an angle inscribed in a semicircle is a right angle, so the hypotenuse is a diameter. With sides 3, 4 and 5 the radius is 2.5.

How does this connect to the law of sines?

The extended law of sines states 2R = a / sin A = b / sin B = c / sin C, so one side and its opposite angle also fix the radius.

Where does the center move as the triangle changes?

It follows the perpendicular bisectors of the sides and can leave the triangle entirely, yet it is always the same distance from all three vertices.

⭕ Circumscribed Circle Calculator

Calculate the circumscribed circle radius and area of a triangle.

Side a Side b Side c

What is a Circumscribed Circle?

A circle passing through all three vertices of a triangle. The circumcenter (O) is the intersection of the perpendicular bisectors of the three sides.

Circumscribed Circle Radius Formula

R = abc / (4S)

Here S is the triangle area, calculated using Heron's formula:

s = (a + b + c) / 2

S = √(s(s−a)(s−b)(s−c))

Relationship with the Law of Sines

a / sin A = 2R

The diameter of the circumscribed circle equals each side divided by the sine of its opposite angle.

Position of the Circumcenter

Acute triangle: circumcenter is inside the triangle

Right triangle: circumcenter is at the midpoint of the hypotenuse (R = hypotenuse/2)

Obtuse triangle: circumcenter is outside the triangle

Worked examples

Sides 3, 4 and 5 give a triangle area of 6, so R = (3 × 4 × 5) / (4 × 6) = 2.5, a circle area of π × 6.25 ≈ 19.635 and a circumference of about 15.708.

An equilateral triangle of side 6 has an area of 15.5885 and R = 6 / √3 ≈ 3.4641, which makes the circle area 12π ≈ 37.6991.

What the radius formula assumes

R = abc ÷ 4K needs a genuine triangle, so side triples that break the triangle inequality return an error rather than a radius. The three points must not be collinear — a straight line has no finite circumscribed circle. The center lies inside an acute triangle, on the hypotenuse of a right one and outside an obtuse one.

Frequently asked questions

Why is R exactly half the hypotenuse in a right triangle?: Thales: an angle inscribed in a semicircle is a right angle, so the hypotenuse is a diameter. With sides 3, 4 and 5 the radius is 2.5.
How does this connect to the law of sines?: The extended law of sines states 2R = a / sin A = b / sin B = c / sin C, so one side and its opposite angle also fix the radius.
Where does the center move as the triangle changes?: It follows the perpendicular bisectors of the sides and can leave the triangle entirely, yet it is always the same distance from all three vertices.