Geometry Lab reference
Volume & surface area
What fits inside, and what covers the outside.
Cylinder: V = πr²h · S = 2πr(r + h)
The idea
Volume measures three-dimensional space in cubic units. Surface area measures the total outside covering in square units. A rectangular prism has volume whd and surface area 2(wh + wd + hd). A sphere has volume 4πr³/3 and surface area 4πr².
Work through an example
A cube with edge 3 cm has volume 27 cm³ and surface area 54 cm². Double the edge to 6 cm and its volume becomes 216 cm³, eight times larger, while the surface area becomes 216 cm², four times larger.
What to watch for
A cone’s curved surface area uses its slant height ℓ = √(r² + h²), not its vertical height h. The total surface area is πr(r + ℓ), including the base.
Make the formula move.
Explore this idea with real inputs.