Inscribed and Circumscribed Circles

Sometimes a circle can be both inscribed and circumscribed with respect to a polygon. They are then called inscribed or circumscribed circles. Shown below is an inscribed and a circumscribed circle with respect to a triangle.

Let us discuss them in detail.

What is an Inscribed Circle

‘Inscribe’ means to draw inside of any figure, just touching it but will not cross the figure. It is thus the opposite of the circumscribed circle.

In geometry, an inscribed circle, also known as the incircle of a polygon is the largest possible circle that can be drawn inside a regular, cyclic polygon. The inscribed circle will touch each of the three sides of the triangle at exactly one point. The center of such a circle is called the incenter. It is the point where the angle bisectors of the triangle meet. The radius of such a circle is called the inradius.

Circle Inscribed in a Triangle

When a circle inscribes a triangle, the triangle is outside of the circle and the circle touches the sides of the triangle at one point on each side. The sides of the triangle are tangent to the circle. Thus, for a polygon, a circle is not inscribed unless each side of the polygon is tangent to the circle.

Thus, all triangles and regular polygons have inscribed circles.

Given below is a circle inscribed in a triangle

Note that each side of the triangle is tangent to the circle, so if we draw a radius from the center of the circle to the point where the circle touches the edge of the triangle, the radius will form a right angle with the edge of the triangle.

How to Inscribe a Circle in a Triangle

Given below is a link showing how to inscribe a circle in a triangle.

What is a Circumscribed Circle

The circumscribed circle of a polygon is a circle that touches all 3 vertices of the polygon. The center of such a circle is called the circumcenter, the point where the perpendicular bisectors of the sides meet. The radius of such a circle is called the circumradius.

Not every polygon though has a circumscribed circle. A polygon that has a circumscribed circle is called a cyclic polygon. It is also called a concyclic polygon since its vertices are concyclic. All triangles and all regular polygons such as square, rectangle, trapezoid, and kite are concyclic.

Circumscribed Circle of a Triangle

Since all triangles are cyclic, they always have a circumscribed circle. When a circle circumscribes a triangle, the triangle is inside the circle, and the triangle touches the circles at each of its vertices.

Given below is a diagram showing a circle circumscribes a triangle.

The center of the circumscribed circle can be obtained by drawing the three perpendicular bisectors of the triangle. The point where the three perpendicular bisectors meet is the center of the circumscribed circle.

How to Circumscribe a Circle in a Triangle

Given below is a link showing how to circumscribe a circle in a triangle.