Does the area of this quadrilateral also apply to all polygons?

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Module 2 Week 3 Day 12 Your Turn Explanation Part 1
Does the area of this quadrilateral also apply to all polygons?

That's a good question; sometimes it's hard to tell when a principle from one area of math applies to another area of math. It's really good that you are thinking about this! In the lesson, we saw that the quadrilateral's area is related to its perimeter in a special way because of the circle inscribed inside it. The radius of the circle is equivalent to the altitudes of the four little triangles that make up the perimeter of the quadrilateral.
The quadrilateral below would also have this property, because it has an inscribed circle tangent to four of its sides:
The question is, can any quadrilateral have a circle inscribed within it? It is not always possible to have an inscribed circle which touches every single side of the quadrilateral. Like, for example, this:
Here the circle is only tangent to three of the sides of the quadrilateral, not four, so the area property doesn't hold.I hope this helps! Please let us know if you have any more questions!
Happy Learning!
The Daily Challenge Team