Video: Identifying Symmetry in Three-Dimensional Figures

This solid is formed of eight cubes. How many planes of symmetry does it have, if any? Does it have an axis of symmetry?

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Video Transcript

This solid is formed of eight cubes. How many planes of symmetry does it have, if any?

To start, let’s have a quick reminder of what a plane of symmetry is. A plane of symmetry cuts a shape into two halves, which are a mirror image of each other. When discussing two-dimensional shapes, we talk about axes of symmetry. So, for example, a rectangle has two axes of symmetry. The words, plane of symmetry, will be used when we have a three-dimensional shape. If we think in terms of slicing through our eight cubes, then if we slice it in the following way, we would have a mirror image. For example, the cube at the top on the front is a mirror image of the cube at the top on the back.

Since we sliced the other cubes exactly in two, then the cube part at the front is a mirror image of the cube part at the back. And so we found a plane of symmetry. Let’s see if there are any more. Let’s see if we slice our shape in this way. Will it be a plane of symmetry? No, it wouldn’t be. We would need, for example, squares on the bottom to create a mirror image of one another. And we would also need additional squares on the top, so that we had two mirror images. So this is not a plane of symmetry.

Let’s now try slicing our three-dimensional shape in the following direction. Now, we we need to check if we have a mirror image on either side of this plane. We can see that the cube on the left and on the right are mirror images. If we cut each cube exactly in half, then the part on the left will be a mirror image of the part on the right, which means that this is also a plane of symmetry. And since there are no other planes that will create a plane of symmetry, then there are two planes of symmetry in total for this solid.

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