Video Transcript
What is the order of rotational symmetry of a parallelogram?
Let’s begin by recapping that a parallelogram is a quadrilateral with opposite sides parallel. So we could, for example, draw a parallelogram like this. We can see that it has got four sides, and opposite sides are parallel. We need to work out the order of rotational symmetry of a parallelogram. The order of rotational symmetry of a geometric figure is the number of times you can rotate a figure so it looks like itself over a rotation of 360 degrees.
Let’s imagine that we cut out our parallelogram and begin to rotate it clockwise. After a 90-degree rotation, the image would look like this. We can even highlight the top-right vertex on the original shape and see that it would now appear on the bottom right. After a 180-degree rotation of the original parallelogram, we can see that the shape would look like this. The vertex that was originally highlighted at the top right is now on the bottom left. So does this parallelogram look like it originally did? Yes, it does. It’s effectively upside down, but it still appears to look the same.
If we then rotated the original shape through 360 degrees to make a complete turn, the shape would look like itself as it’s back in the original starting position. Since the shape looked like itself after a 180-degree rotation and then at a 360-degree rotation, we say that the order of rotational symmetry is two.
An important point to make is that we don’t count the original starting position when we’re counting up the order of rotational symmetry. Here, because the shape looked like itself after a 180-degree rotation, that counted as once. And then when it goes back at 360 degrees, we count that position. A shape that had an order of rotational symmetry of order one could also be described as having no rotational symmetry. But here we can give our answer that the order of rotational symmetry of a parallelogram is two.
