Final answer:
An inscribed polygon is one with all vertices on a circle's circumference, while a circumscribed polygon contains a circle within it, with its sides tangent to the circle. These polygons have special geometric relationships and properties useful in solving problems.
Step-by-step explanation:
The terms inscribed and circumscribed are used in geometry to describe the relationship between polygons and circles. An inscribed polygon is one that is contained within a circle such that all of its vertices (corners) lie on the circumference of the circle. A classic example of an inscribed polygon is a square whose corners touch the circle at four points.
On the other hand, a circumscribed polygon is one that contains a circle within it, with each side of the polygon tangent to the circle. In other words, the circle touches the polygon at points along the sides, but not the vertices. A regular hexagon that has a circle inside it, where each side is tangent to the circle, is an example of a circumscribed polygon.
These relationships have important properties and are often used to solve problems involving angles, lengths, and geometric proportions that are schematic in nature. For instance, knowing that a polygon is inscribed in a circle can allow you to use particular theorems regarding angles and arcs in circles to find unknown values.