Surface area and volume
Distinguish external area from occupied volume and connect units and formulas.
Meaning
Surface area sums the areas of a solid's surfaces; volume measures occupied space. A prism or cylinder has volume base area times perpendicular height; a pyramid or cone with the same base area and height has one third that volume. Choose one cube vertex as a common apex and the three faces not containing it as bases. These form three congruent pyramids that fill the cube. Each pyramid has the cube's base area and height, so its volume is one third of the corresponding prism. Scaling each of the width, depth, and height by the same respective factors in both solids preserves that volume ratio. Pyramids or cones with the same base area and height also have equal cross-sectional areas when cut parallel to their bases at every equal fraction of the height from the apex. Such a parallel section with length scale t has t² times the base area. Accumulating equal areas through the same height gives equal volumes, extending the formula base area×height÷3 across base shapes.
Example
Practice for this concept
Scope and references
Topics
Surface areas and volumes of prisms, pyramids, cylinders, cones, and spheres
Learning goal
Distinguish external area from occupied volume and connect units and formulas.
Helpful background
Explore faces, nets, and area and volume units.
Surface area and volume
These choice exercises use stated geometric conditions. They do not establish construction, drawing, spatial manipulation, written proof, or mastery of the whole geometry area.