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Ever thought that you can fit a circle inside of a polygon? Yes, you can fit a circle inside of a polygon. In fact, it has a special name for it. Mathematicians call it circumscribed polygons. The word circumscribed means that a figure that is drawn inside of another figure touching the boundary but it doesn't exceed. This means that the circle, inside of a polygon, will touch the boundary of the polygon and it won't get out of the polygon's boundary. You might be thinking, is touching of boundary is necessary for circumscribed polygon? Absolutely. The boundary of the polygon should be in contact with the boundary of the circle, if this is not the case then it won't be a circumscribed polygon. Check the below diagram.
We have a circle within a polygon, does the above diagram fulfil all the requirements of a circumscribed polygon? The answer is yes, as you can see the boundary of the circle just touches the polygon, not cutting the polygon. It is the perfect example of a circumscribed polygon. In conclusion, a polygon is circumscribed in a circle if its vertices are outside the circle and its sides are tangent to the circle.
Properties of Circumscribed Polygon
There are a few properties of the circumscribed polygon that makes it special. The centres of polygon and circumscribed polygons are the same. In addition, the circumscribed polygon intersects at the midpoint of each side to the inscribed circle. Not to mention that the centre of the inscribed circle is equidistant from all sides of the circumscribed polygon. Last but not the least, the apothem the circumscribed polygon is the radius of the inscribed circle.
I’m just curious if the area between the polygon and the circumscribed circle has a name.
https://www.superprof.co.uk/resources/academic/maths/geometry/plane/orthocenter-centroid-circumcenter-and-incenter-of-a-triangle.html