AI Tutor
If a city orders paint to coat an elevated water tower, but accidentally calculates the tower's interior capacity instead of its exterior skin, they might end up with 50,00050,000 gallons of paint when they only needed 4040.
Surface area measures the two-dimensional exterior boundary covering an object in square units, while volume measures the three-dimensional space enclosed inside it in cubic units.
Real-world structures are rarely single, simple solids β€” they are usually combinations of prisms, cylinders, cones, and spheres.
How do we combine formulas when two 3D shapes fuse together into one?

The Hidden Face Rule

A composite solid is a 3D figure made by joining two or more distinct geometric shapes.
While total volume is always the simple sum of all component volumes, total surface area only includes exposed outer faces β€” never the shared interior boundaries where the shapes connect.
Let's put these formulas to work on a full real-world construction problem.

Worked Example: The Grain Silo

A farm builds a grain silo consisting of a cylindrical main body with radius r=6Β ftr = 6\text{ ft} and height h=20Β fth = 20\text{ ft}, capped with a hemispherical dome of radius r=6Β ftr = 6\text{ ft}.
First, we find the maximum grain capacity (volume). The volume of a cylinder is Vcyl=Ο€r2hV_{\text{cyl}} = \pi r^2 h, and a hemisphere is half a sphere, Vhemi=23Ο€r3V_{\text{hemi}} = \frac{2}{3}\pi r^3.