3D Shapes Summary

Geometry · Axiom Academy

SUMMARY Unit 7 Summary: 3D Shapes & Solid Geometry Everything you measured in three dimensions — volume, surface area, and the elegant one-third rule that ties the pointed solids to the straight ones. Volume measures the space inside a solid (cubic units); surface area measures the skin that wraps the outside (square units). Every "straight-sided" solid follows one pattern: volume = base area height . A prism is V = Bh ; a cylinder is the same idea with a circular base, . Every "pointed" solid holds exactly one-third of the straight solid on the same base and height — the in (pyramid) and (cone). The sphere stands alone: and surface area — and surprisingly, the surface area is exactly the rate at which volume grows with r . Surface area is just addition. Unfold a solid into its net and add up the flat faces — lateral surface (the sides) plus the base(s). A prism has two identical parallel bases joined by flat sides. Its volume is the base area B times the height h — you are simply stacking copies of the base. When to use: any solid with a constant cross-section (box, triangular prism, hexagonal column). Watch out for: B is the area of the base, not a side length — compute it first. A cylinder is a prism with a circular base, so the same V = Bh becomes . Unroll its curved side and it flattens into a rectangle of width and height h . When to use: cans, pipes, tanks — anything round with straight walls. Watch out for: total surface area adds the two circular caps: .

This is the written version of the interactive lesson above. See the full Geometry course.