Three-dimensional figures: volume, surface area and diagonals

One idea behind most volumes

A solid whose cross-section stays the same from one end to the other (a box, a cube, a cylinder, any right prism) has

$$\text{volume} = \text{area of the base} \times \text{height}$$

Picture stacking identical thin slices of the base, one on another, until the stack is as tall as the solid. That single idea gives every formula below, so you need to remember the base area, not a list.

Solid	Volume	Surface area
Rectangular solid $\ell \times w \times h$	$\ell w h$	$2(\ell w + \ell h + w h)$
Cube with edge $s$	$s^3$	$6s^2$
Cylinder, radius $r$, height $h$	$\pi r^2 h$	$2\pi r^2 + 2\pi r h$
Right prism	base area $\times$ height	two bases plus the rectangular sides

Surface area: list the faces

Surface area is the total area of everything you could paint on the outside. A box has three pairs of matching rectangles, which is where the $2(\ell w + \ell h + wh)$ comes from. A cylinder has two circular ends, $2\pi r^2$, and a curved side. Unroll the side and it becomes a rectangle whose width is the circumference $2\pi r$ and whose height is $h$, so its area is $2\pi r h$.

Read what a question wants painted or covered. An open-topped tank has only one circular end; a label round a can covers only the curved side (the lateral area). Listing the faces in words before you add saves you from including a face that is not there or missing one that is.

The space diagonal

The longest straight segment inside a box runs from one corner to the opposite corner, through the middle. Find it with two right triangles. The diagonal of the floor is $\sqrt{\ell^2 + w^2}$. That floor diagonal and the vertical edge $h$ are the legs of a second right triangle standing upright inside the box, so

$$d = \sqrt{\ell^2 + w^2 + h^2}$$

For a cube with edge $s$ this is $s\sqrt{3}$. A box measuring $3 \times 4 \times 12$ has a space diagonal of $\sqrt{9 + 16 + 144} = \sqrt{169} = 13$.

Describing a solid in words

Without a picture, a solid is fixed by naming its shape and its measurements: "a cylinder standing on one circular end, with radius 3 and height 10", or "a box with a square base of side 4 and height 9". When a solid is cut, filled or stacked, say what changes and what stays fixed. Pouring water from one container into another keeps the volume of water; the new depth comes from dividing that volume by the new base area.

Scaling

If every length of a solid is multiplied by $k$, its surface area is multiplied by $k^2$ and its volume by $k^3$. Double a cube's edge and its surface area becomes 4 times as large while its volume becomes 8 times as large. Volume ratios grow much faster than length ratios: a cube with three times the edge holds 27 times as much.

Units and fitting

Cubic units convert by the cube of the length factor: 1 metre is 100 centimetres, so $1 \text{ m}^3 = 100^3 = 1{,}000{,}000 \text{ cm}^3$.

When rigid objects are packed into a container, do not divide volumes. Work one dimension at a time: how many fit along the length, along the width and up the height, each rounded down to a whole number, then multiply. Space left over in one direction cannot be combined with space left over in another, because a cube cannot be cut to fill the gaps. Dividing volumes gives only an upper bound that is often not reachable.

Beyond boxes and cylinders

Geometry is one of the four content areas of the Quantitative Reasoning measure. This lesson covers solids with a constant cross-section, where base times height applies. For any solid, the same habits work: name each face or base in words, find its area with the plane formulas, and keep track of which measurements are lengths, areas and volumes.