Circumscribed and Inscribed

The terms circumscribed and inscribed refer, respectively, to geometric figures that have been drawn around the outside of or within some other geometric figure. For example, imagine that a circle is drawn around a triangle so that the circle passes through all three vertices of the triangle. Then the circle is said to be circumscribed around the triangle, and the triangle is said to be inscribed within the circle.

Many combinations of figures could be substituted for the triangle and circle described above. For example, a circle circumscribed about any kind of polygon is one that passes through all of the vertices of the polygon. Then the polygon itself is said to be inscribed within the circle. Conversely, a polygon can be circumscribed around a circle if all of the sides of the polygon are tangent to the circle. Then the circle is inscribed within the polygon.

Three-dimensional figures can be circumscribed around and inscribed within each other also. For example, a cone can be circumscribed around a pyramid if the vertices of the cone and pyramid coincide with each other, and the base of the cone circumscribes the base of the pyramid. In such a case, the pyramid is inscribed within the cone. As another example, a sphere can be inscribed within a cylinder if all parts of the cylinder are tangent to the sphere's surface. Then the cylinder is circumscribed around the sphere.