Chapter 14
Solid Geometry
A drawing of a box fits on a sheet of paper, but the box itself needs room beyond the page. A single picture can hide parts of the object or make its shape hard to judge. How can we reason precisely about something we cannot see all at once? Solid geometry takes familiar questions about shape beyond the flat page, giving us a language for objects with depth.
What You'll Explore
8 lessons.
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What can a flat drawing tell you about a solid, and what might it hide? You'll explore how views and nets describe objects in space, where even familiar lines can raise unfamiliar questions.
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Imagine slicing a solid or turning a flat shape around a fixed line: what shapes might appear? You'll investigate connections between flat figures and objects in space, asking how the direction of a cut or the choice of a rotation axis affects the picture.
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Could a prism lean sideways without changing how much it holds? Comparing the space inside prisms and cylinders leads to another question: how much material would be needed to cover their surfaces?
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How much can a pointed container hold compared with one that does not taper? Pyramids and cones invite a closer look at this comparison, including which measurements matter and what geometric argument could settle the question.
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Wrapping a pointed solid raises a different question from filling it: how much material covers the outside? You'll explore the surfaces of regular pyramids and right cones, asking what a net can reveal about the measurements needed for that covering.
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A ball has no flat faces to unfold, so how could you measure its surface or the space inside? You'll investigate spheres and hemispheres, looking for ways to compare their curved shapes with solids whose measurements are more familiar.
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Some objects look like familiar solids joined together, while others have spaces cut out of them. You'll investigate their volume and exposed surface, asking which parts of the drawing matter for the quantity you want to measure.
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When a model is enlarged, should its covering and its capacity grow in the same way? Scaling in three dimensions raises questions about how different measurements respond to a change in the size of similar solids.