Video Transcript
Two 3D shapes lie between two parallel planes. Any other plane which is parallel to the two planes intersects both shapes in regions of the same area. What can you deduce about the shapes?
Alright, so here we’re told we have two three-dimensional shapes. And we’re told that these shapes exist between two parallel planes. We’re then told that any other plane parallel to these two planes intersects these shapes in regions of the same area. So say we have a third plane like this parallel to our first two. The two-dimensional regions of our three-dimensional shapes that this plane intersects have the same area. It’s hard to tell that from this two-dimensional sketch. But we can imagine these two areas shaded in pink as being equal.
Our problem statement tells us that any plane parallel to the first two will have the same result. It will intersect both shapes in regions of equal area. Based on this, we want to know what we can deduce about these two shapes.
Imagine we have a plane parallel to our first two and we move it across these shapes. As we do, at every instant in time, the regions in these two three-dimensional shapes that the plane intersects have the same area. Once this plane passes completely through the two shapes then, if we add up all of the cross-sectional areas of the first shape, that sum will equal the total cross-sectional area of the second shape. That tells us that these two shapes must have the same volume.
This conclusion, based on what we’ve been told in the scenario, is a confirmation of a general mathematical principle called Cavalieri’s principle. And here we’ve seen this principle apply to these two three-dimensional shapes.
