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