Basic 3D geometry

Three-dimensional figures are handled with a small toolkit of formulas, so the work is mostly about choosing the right one and supplying the missing measurement.

For a cube of edge $s$, the six identical square faces give a surface area of $6s^2$, and the volume is $s^3$. For a rectangular box with length $l$, breadth $b$ and width $w$, there are three pairs of rectangular faces, so the surface area is $2lb + 2bw + 2lw$ and the volume is $lbw$.

For curved solids: a cylinder of radius $r$ and height $h$ has volume $\pi r^2 h$; a sphere of radius $r$ has surface area $4\pi r^2$; and a pyramid has volume $\frac{1}{3} \times \text{base area} \times h$.

A very common two-step pattern is to back out a length from a given quantity and then use it. If a cube has surface area $54$, then $6s^2 = 54$ gives $s = 3$, and its volume is $3^3 = 27$. Recognising which formula links what you are given to what you are asked is the whole skill.

A cube has surface area $96$ square inches.
From $6s^2 = 96$ we get $s^2 = 16$, so $s = 4$ inches.
Its volume is then $s^3 = 4^3 = 64$ cubic inches.

A sphere has diameter $14$, so its radius is $7$. Using surface area $4\pi r^2$ with $\pi = \frac{22}{7}$: $4 \times \frac{22}{7} \times 7^2 = 616$ square units.

A cylinder has radius $4$ and height $10$. Its volume is $\pi r^2 h = \pi \times 4^2 \times 10 = 160\pi$.