Identifying symmetrical figures | Math | 4th grade | Khan Academy

Which shapes are symmetrical? To answer this, we need to know what it means for a shape to be symmetrical. A shape is symmetrical if it has at least one line of symmetry. A line of symmetry, and now that answer is only helpful if we know what a line of symmetry is.

So let's talk about it. A line of symmetry is a line where we can fold the image and have both halves match exactly. Let's look at an example. Let's draw, let's maybe draw a circle, and then we can put a line on that circle. Let's draw a line maybe somewhere like this. This line is a line of symmetry if we can take one side of the line and fold it onto the other and have them match exactly.

So let's take one side, doesn't matter which one, let's say the top. The top side, and if we were going to fold this top side down onto the bottom, would it match exactly what is shown under here? Let's see, it would probably look something like this. And does that match exactly? No, definitely not. So this is not a line of symmetry.

Let's try another line, maybe if we drew a line and we'll try to get as close down the center as we can, here like this. Try to be as close to the center as possible. And here, if we took one side again, it doesn't matter which side, let's say over here, let's say the left side, and we folded this left side onto the right side, would it match exactly?

And if our line truly was in the center of the circle, then yes, it would. Which means that this line is a line of symmetry. And because we can draw this line of symmetry on our circle, it means that our circle is symmetrical. Shapes are symmetrical if they have at least one line of symmetry, and circles have many, many, many lines of symmetry. There were many places we could have drawn a line and folded it so that it worked so the two halves matched exactly.

But here's one, and as soon as we find one, we know we have a symmetrical shape. So let's go back to the shapes we were given. We can start with the triangle. If we draw a line, maybe a vertical line, let's try to draw it as close to the middle as possible, something like this. And we fold, let's take one side. If we fold this side over, these two lines might match up nicely. But this line here is going to create something more like this, which does not match, which is what's shown over here. So that's not a line of symmetry.

And anywhere else, vertical, say same thing, we're not going to have it lined up. So let's try maybe a horizontal line. Is there anywhere horizontally we could draw a line? And again, I think we're going to see the same thing that the top and the bottom of the line are not going to match up exactly. So maybe one last thing we could try is a diagonal line, something like this. Maybe this could be our line of symmetry. If we fold this bottom side, this might line up pretty nice here, and then this side is going to do something like this.

So it's close, it's the closest we've gotten, but still does not match exactly. For it to be a line of symmetry, it needs to match exactly. So we weren't able to find a way to draw a vertical line, or a horizontal line, or even a diagonal line. So this shape has no lines of symmetry, so we can say it is not symmetrical.

Moving on to the rectangle, let's try here. Here again, this time we may try a horizontal line. We can draw one right here, and if that line truly is in the middle, which is what I've tried for, then this side should match up nicely to this one across the top should match across the bottom, and these sides, if I was right at the halfway point, should fold over each other also. So it has a line of symmetry, so it is symmetrical. It has more than one line of symmetry; it has another one in the middle right here. But once we found one, we know that it's symmetrical.

And finally, let's look, we have a pentagon here. Again, trying a line in the middle in some way is usually a good place to start. We can try to draw a line right; if this is right down the center here, then if we folded this side, it should line up nicely to this side. This side and this side would overlap, and these two would match exactly.

So again, it has one line of symmetry, so it is symmetrical. And just like the rectangle, this one had quite a few lines of symmetry. Here's another line of symmetry; here's another line of symmetry; here's one more line of symmetry. And so it has quite a few. It has, it looks like one, two, three, four lines of symmetry. But as long as it has one, it is symmetrical.

So, of the shapes we were given, the rectangle and the pentagon were symmetrical.