3D Solids — Volume, Surface Area, and Geometry in Space

Applied mathematics begins where geometry meets the physical world. Engineers designing a grain silo, packaging designers minimizing cardboard, and architects planning a glass atrium all rely on the same core toolkit: formulas for volume and surface area, plus the ability to measure distances and angles inside three-dimensional objects.

Volume and surface area of standard solids

The IB formula booklet provides the essential results, but understanding when to use each one is what matters in a modeling context. For a right cone of base radius $r$ and vertical height $h$, the volume is $V = \frac{1}{3}\pi r^2 h$ and the curved surface area is $A = \pi r l$, where $l = \sqrt{r^2 + h^2}$ is the slant height. For a sphere, $V = \frac{4}{3}\pi r^3$ and $A = 4\pi r^2$. A pyramid of base area $B$ h