Video: Identifying a Concave Polygon

Which of the following shapes is a concave polygon? [A] Shape A [B] Shape B [C] Shape C [D] Shape D [E] Shape E

02:21

Video Transcript

Which of the following shapes is a concave polygon?

Let’s begin by reminding ourselves what it means for a polygon to be concave. We say that a polygon is concave if one or more of its interior angles is greater than 180 degrees. Now, this results in it looking a little bit like some of the vertices of the shape point inwards towards its center. Now, of course, we’re dealing purely with polygons. And so, our shapes themselves must have only straight sides. And this means we can instantly disregard option (A) as being a concave polygon. It has two straight sides and one curved side. It does, however, have one angle greater than 180 degrees. But we disregard it because it’s not a polygon.

And what about shape (B)? It is indeed a polygon. It’s a two-dimensional shape made up of a number of straight sides. But actually, we identify each of the interior angles as being less than 180 degrees. And so, we disregard option (B) as well.

We now look at option (C). It is a polygon; again, it’s made up of a number of straight sides. This angle here is less than 180, as is this one, and this one. But we have two angles here which are greater than 180 degrees. In fact, we can see they are 270 degrees. This is less than 180, as is this, but this angle is also greater than 180 degrees. And so, the answer must be (C). (C) is a concave polygon. But let’s check option (D) and (E).

We might recall that in a concave polygon, one or more of its diagonals will lie outside of the shape itself. And remember, we construct the diagonals by joining nonadjacent vertices with a straight line. The diagonals of shape (D) are shown. All of these lie within the polygon itself. And so, it cannot be a concave polygon. And we disregard (D) as we expected. Similarly, joining the nonadjacent vertices on shape (E) with straight lines, and we see that all of those diagonals lie inside the polygon. So (E) cannot be concave either. And so, we’re able to confirm that the answer is (C). Shape (C) is a concave polygon.

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