Lesson Video: Convex and Concave Polygons Mathematics

Video Transcript

In this video, we’ll learn how to classify polygons as convex or concave. We recall that the word polygon comes from the Greeks. The poly- part means many, whilst the -gon part means angle. A polygon then is a shape with many angles. But in fact, we formalize this a little bit more. And we say that a polygon is a two-dimensional shape with three or more straight sides. Now, of course, polygons can be classified in a number of ways, for instance, by their number of sides or whether their angles are all equal. But a farther way to classify polygons is as convex or concave.

So, what does this mean? We say that a polygon which is convex has all interior angles less than 180 degrees, whereas a concave polygon has one or more interior angles that is greater than 180 degrees. Another property of a convex polygon is that if we draw a single straight line through it, it will intersect the sides or vertices exactly twice, as seen in the picture, whereas if you draw a single straight line through a concave polygon, depending on where we draw it, it could intersect the polygon in more than two places. For example, in this diagram, the straight line intersects our polygon once, twice, three, and four times.

If we add diagonals to each of our polygons, that is, straight lines that join nonadjacent vertices, we see that all of the diagonals lie inside a convex polygon, whereas some of the diagonals of a concave polygon lie outside the polygon itself. It’s also useful to know that a regular polygon is always convex by definition. And so, now we have some of these definitions, let’s look at identifying some concave and convex shapes.

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.

Which of the following polygons is convex?

We have five options here. So, let’s remind ourselves what it means for a polygon, which is a two-dimensional shape with straight sides, to be convex. In a convex polygon, all of the interior angles must be less than 180 degrees. Another property is found by adding in the diagonals to the shape, that is, joining the nonadjacent vertices with straight lines. When we do, we find that all of the diagonals of a convex polygon lie completely within inside the shape. And in fact, this means we can instantly disregard option (A). If, for example, we join this vertex to this vertex, we see that the diagonal lies outside of the shape.

We have a similar concern over shape (C). If we join the two vertices shown, the diagonal lies outside the shape. For shape (D), this diagonal lies outside the shape. And for shape (E), we have another diagonal that lies outside the polygon itself. And so, shapes (A), (C), (D), and (E) all have at least one diagonal that doesn’t lie within the polygon. And so, they cannot be convex. So, by the process of elimination, that must leave us with option (B).

But we’ll check in two ways. Firstly, we’ll double check that each of the diagonals lies within the polygon. When we do, we get this sort of star shape. And we see that every single diagonal does indeed lie within the polygon itself. But we might also check each of the individual angles. This angle is less than 180. This angle is less than 180, as are the remaining three. And so, the polygon that’s convex is (B).

We’ll now look at defining a single shape to be convex or concave.

Determine whether the following polygon is convex or concave.

Now we have a number of ways to identify whether a polygon is convex or concave. Beginning with this simple definition, we say a polygon is convex if all of its interior angles are less than 180 degrees. And it will be said to be concave if at least one angle is greater than 180 degrees. And so, we could go through our shape and look at all of the interior angles. We’d want to find at least one that’s greater than 180 degrees, in other words, a reflex interior angle.

But there is another way. And that is to construct the diagonals of the shape. To do so, we join nonadjacent vertices. So, for example, we have a diagonal here, one here, another here, and we’re going to continue by adding all of the diagonals to our shape. When we’re done, each vertex has three diagonals coming off of it. For a convex polygon, all of these diagonals should lie inside the shape itself, whereas with a concave polygon, at least one of them will lie outside.

Now, actually, if we look at our shape, we see that no matter the color of the line we’ve drawn — whether the diagonal is a green line, a yellow line, or a pink line — they all lie within the shape itself. And so, we can say that this shape is convex.

We’re now going to consider how we will decide whether a polygon that we’re described rather than drawn is concave or convex.

In a pentagon 𝐴𝐵𝐶𝐷𝐸, the measure of angle 𝐴 is equal to 110 degrees, the measure of angle 𝐷 is equal to 62 degrees, the measure of angle 𝐸 is equal to 89 degrees, and the measure of angle 𝐵 is equal to the measure of angle 𝐶. Decide whether 𝐴𝐵𝐶𝐷𝐸 is convex or concave.

Let’s recall what it means for a polygon to be convex or concave. A polygon is said to be convex if all of its interior angles are less than 180 degrees. What happens here is it looks like the vertices are all pointing outwards away from its center. And then, if at least one of its interior angles is greater than 180 degrees, we say that the polygon is concave. And then, it looks like some of the vertices are pointing inwards towards the center.

Now, we don’t have a diagram here, but we have been given some information about the polygon and some of the sizes of its angles. It’s a pentagon, so of course it has five sides and, thus, five angles. Our job is going to be to work out whether each of these angles is greater or less than 180 degrees. And since we’ve only been given the size of three of the angles, let’s work out the size of the remaining two.

We know that the interior angles in a pentagon sum to 540 degrees. That’s a fact that we can state. We also know that the measure of angle 𝐵 is equal to the measure of angle 𝐶, so let’s define each of these to be equal to 𝑥 degrees. We can then say that the sum of our five angles is 540. And so, angle 𝐴, which is 110, plus angle 𝐵 and angle 𝐶, which are both 𝑥, plus angle 𝐷, which is 62, and plus angle 𝐸, which is 89, equals 540.

Let’s simplify the left-hand side of this equation. 𝑥 plus 𝑥 is two 𝑥. And 110 plus 62 plus 89 is 261. And so, we see our equation is now two 𝑥 plus 261 equals 540. Let’s solve by subtracting 261 from both sides to give us two 𝑥 equals 279. And then, we divide through by two. And so, we find 𝑥 is equal to 139.5. We now know the size of all five angles. Angle 𝐴 is 110 degrees, angles 𝐵 and 𝐶 are 139.5, angle 𝐷 is 62, and angle 𝐸 is 89 degrees. None of these angles are greater than 180. All of them are in fact less than that value. And so, we’re able to say that the pentagon 𝐴𝐵𝐶𝐷𝐸 is, in fact, convex.

We’re just going to consider one final question.

True or false? A convex polygon is regular.

Remember, we say a polygon is convex if all of its interior angles are less than 180 degrees. We can also say that a polygon is regular if all of its angles are equal and all of its sides are equal. Now, in fact, we are actually able to say that all regular polygons are in fact convex. But how are we able to be sure of this? Well, we might recall that the interior angles of an 𝑛-sided polygon are found by calculating 180 times 𝑛 minus two.

Now, we know if it’s a regular polygon, we know that each of the 𝑛 angles must be equal. And so, we can say that the size of one interior angle in this polygon is 180 times 𝑛 minus two over 𝑛. Let’s manipulate this expression a little bit. By distributing the parentheses, in other words, multiplying 180 by 𝑛 and 180 by negative two, we can rewrite it as 180𝑛 minus 360 over 𝑛. And then, we can split this fraction up to get 180𝑛 over 𝑛 minus 360 over 𝑛. And then, we simplify the first part by dividing through by 𝑛. And so, the size of one interior angle in an 𝑛-sided regular polygon is 180 minus 360 over 𝑛.

Now, of course, we know that polygons consist of at least three straight sides. And so, 𝑛 will always be greater than two. In fact, we can also say that 𝑛 will always be an integer greater than two. This means that 360 divided by 𝑛 will always be greater than zero. And so, when we work out 180 minus a number greater than zero, that will always be less than 180. And so, any angle in a regular polygon, any interior angle, will be less than 180 degrees.

So, we know that regular polygons are convex. But can we say the converse is true? We’ll, no, let’s take any triangle. In fact, let’s take a scalene triangle. We know that in a scalene triangle no angles are the same. But we also know that the sum of the interior angles in a triangle is 180 degrees. And so, if we define the interior angles to be 𝑎, 𝑏, and 𝑐, we can say that 𝑎 plus 𝑏 plus 𝑐 equals 180 degrees. Now, of course, by definition, neither 𝑎, 𝑏, nor 𝑐 can be equal to zero. They must all be greater than zero. And so, we can say that, individually, each of the angles has to be less than 180 degrees so that their sum is 180. 𝑎 is less than 180, 𝑏 is less than 180, and 𝑐 is less than 180.

And so, we found a single case of a convex polygon which is not regular. This is sufficient enough for us to be able to say that this statement cannot be universally true. And so, it’s false. A convex polygon is not always regular.

We’ll now recap some of the key points from this lesson. In this video, we learned that we call a polygon convex if all of its interior angles are less than 180 degrees. We saw that if we draw a single straight line through a convex polygon, it will intersect the polygon exactly twice. It divides the polygon, in fact, into two pieces. And we also saw that any diagonal, in other words, any straight line joining nonadjacent vertices, will lie inside a convex polygon.

And then, we can say pretty much the opposite to be true for a concave polygon. These will have at least one interior angle that’s greater than 180 degrees. What this means is that if we draw a single straight line through a concave polygon, depending on where we draw it, it could intersect the sides more than twice. And we’ll find if we draw all of the diagonals that at least one will lie outside of the polygon. Finally, we saw that, by definition, a regular polygon must be convex, but the opposite cannot always be said. We cannot say that all convex polygons are regular.