Area and volume

The SAT gives you most of these formulas, so the work is choosing the right one and substituting carefully — not memorising.

Notice the pattern. A prism or cylinder is base area times height, $V = Bh$. A cone or pyramid is exactly one third of the prism that would contain it, $V = \frac{1}{3}Bh$. The sphere stands apart with $V = \frac{4}{3}\pi r^3$.

So the real question is usually what $B$ is. For a cylinder the base is a circle, giving $B = \pi r^2$; for a rectangular prism it is a rectangle, giving $B = lw$.

Surface area is the total area of every face, which for a prism means unfolding it: two bases plus the sides. Questions often ask for one and offer the other as a trap.

The most tested idea here is scaling. If every length of a solid is multiplied by $k$, areas multiply by $k^2$ and volumes by $k^3$. Doubling the radius of a sphere multiplies its volume by $8$, not $2$.

Finally, watch the units. Lengths in centimetres give areas in $\text{cm}^2$ and volumes in $\text{cm}^3$, and a question that mixes metres with centimetres is testing whether you converted first.

A cylinder has radius $3$ and height $10$. Find its volume.
The base is a circle, so $B = \pi(3)^2 = 9\pi$

A sphere's radius is tripled. What happens to its volume?
Volume scales with the cube of the length factor so the volume becomes $27$ times larger. Answer choices will offer $3$ and $9$ for the students who scaled lengths or areas instead.