Geometry Basics at the 700-800 Level

At this level the topic is almost entirely solid geometry inside a word problem. The formulas are given on the reference sheet; the difficulty is the layer wrapped around them — a density, a purchase, a rate, an already-elapsed time, a hidden face.

Five archetypes cover the bank. The scaling trio: multiply every length by $k$ and areas scale by $k^{2}$, volumes by $k^{3}$ — nearly every 'by what percent' item is this rule in disguise, and percent increase is the ratio minus one. Composite and subtracted solids, where the leftover piece is as likely to be asked for as the piece you built. Surface area with a hidden or missing face. Rates, elapsed time, and unit conversion — 'how much longer' means subtract what was already spent, and cubic conversions cube the factor ($1$ ft³ $= 1728$ in³, not $144$). Similar-solid partial fill: a cone filled to $\frac{3}{4}$ of its height holds $\left(\frac{3}{4}\right)^{3} = \frac{27}{64}$ of capacity, not $\frac{3}{4}$.

Two habits carry the topic: circle the piece the question actually wants, and apply the last multiplier — the density, the price, the rate — because the number just before that final multiplication is always a choice.