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Does every arbitrary geometric shape have an apothem? This is not entirely clear to me.

(I have heard that the apothem of a geometric shape is a generalization of the notion of the radius of a circle.)

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2 Answers

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An apothem is usually defined for regular polygons, where the idea of "center" is entirely unambiguous. For more irregular shapes, how do we define a center?

Even in cases where the center is clear, it's not hard to construct a polygon that has no segment from the center perpendicular to a side. On the $xy$-plane, join the sequence of points: $(3,0),(1,1),(0,3),(-1,1),(-3,0),(-1,-1),(0,-3),(1,-1),(3,0)$. This is an octagon shaped like a four-pointed star. The origin is its center, but you can't drop a perpendicular from there to any side.

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Apothem is radius of inradius of a polygon. Every polygon can't have a incircle in it.
Therefore every polygon can't have a apothem.

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