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Circumscribed and Inscribed Circles and Polygons - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

A circle is circumscribed about a polygon if the polygon's vertices are on the circle. For triangles, the center of this circle is the circumcenter. A circle is inscribed a polygon if the sides of the polygon are tangential to the circle. For triangles, the center of this circle is the incenter. Circumscribed and inscribed circles show up a lot in area problems.

Two terms that get confused in Geometry are the words circumscribed and inscribed. But what's the difference? Well let's take a look at a couple of examples.
This first example we have a triangle that is inside of a circle. Well, the way that we could say this is either the polygon is inscribed in the circle, that's the general way. But since we have a triangle I could say the triangle is inscribed in the circle. Notice that the triangle's within the triangle so that's how you can say it's inscribed. The reverse of that is to say the circle is circumscribed about the polygon. So if you're wondering this is formed by the point of concurrency called the circumcenter which is formed by the three perpendicular bisectors.
Let's look at inscribed. With an inscribed circle, we have the circle that's within the polygon. Notice that here we have a triangle and a circle inside of it, and here we have the quadrilateral with a circle inside of it. So if I look at this quadrilateral, there's two ways that I can describe this picture.
The first is the quadrilateral is circumscribed about the circle. Notice that when we use the word circumscribe we're using about. We could also say that this circle is inscribed inside the quadrilateral. So two different ways to describe the same picture.
Inscribe you should think is within its inside, and whenever you hear the word circumscribing you should be thinking about or outside.

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