Modeling with Geometry: Shapes, Density and Design

Replacing an object with a shape

No real object is a geometric solid. A tree trunk tapers, a silo has a lip, a can has a rim. Measuring any of them exactly is a job for a scanner.

Modeling replaces the object with a shape close enough to answer the question.

Object	Model	Question it answers
Tree trunk	cylinder	how much lumber it holds
Grain silo	cylinder with a hemisphere on top	how much grain it stores
Soup can	cylinder	how much metal it takes
Camping tent	triangular prism	how much air is inside
Human torso	cylinder	roughly how much skin area
County	polygon	population per square mile

The model is chosen for the question, not for looks. A trunk modeled as a cylinder answers a volume question well and a bark-texture question not at all.

Density in two dimensions

Density is an amount divided by the space it occupies. In two dimensions that space is an area.

$$\text{population density} = \frac{\text{people}}{\text{area}}$$

A city of $86{,}000$ people covering $24$ square miles has

$$\frac{86{,}000}{24} \approx 3{,}583 \text{ people per square mile}$$

Dividing by area is what makes two places comparable. A county with more people than another may be less crowded, and only the density says so.

Density in three dimensions

In three dimensions the space is a volume, and the quantity is usually mass or energy.

$$\text{density} = \frac{\text{mass}}{\text{volume}} \qquad\Longrightarrow\qquad \text{mass} = \text{density} \times \text{volume}$$

Quantity	Units	Typical use
Mass density	pounds per cubic foot	weight of a load of gravel
Energy density	BTUs per cubic foot	heat from natural gas
Concentration	milligrams per liter	a contaminant in water

The pattern is the same every time: geometry supplies the volume, density supplies the rest.

Why a rough shape still gives a usable number

Treating a tapered trunk as a straight cylinder looks careless. It is not, and the reason is worth stating.

A model’s error in volume tracks its error in shape. A trunk whose radius varies by $10\%$ over its length gives a volume within roughly $20\%$ of the cylinder’s. That is the worst case, because the fat end and the thin end pull in opposite directions and partly cancel.

So the model answers “about how much lumber?” and refuses to answer “exactly how much?” That is the right division of labor. The question that gets asked in practice is the first one, because the second requires equipment the first one makes unnecessary.

Two habits keep a rough model honest.

Bracket it. Compute with the largest plausible measurement and the smallest. If both answers lead to the same decision, the model is good enough.

Say which way it errs. A cylinder around a tapered trunk overestimates, because the cylinder is as wide as the widest point everywhere.

Design problems

The third use of geometry is to choose a shape rather than measure one. A design problem fixes a requirement and asks for the cheapest, lightest or smallest way to meet it.

A standard soda can holds $355$ cm³. The metal used is its total surface area.

$$V = \pi r^2 h = 355 \qquad\Longrightarrow\qquad h = \frac{355}{\pi r^2}$$ $$S = 2\pi r^2 + 2\pi r h = 2\pi r^2 + \frac{710}{r}$$

The first term grows with $r$ and the second shrinks, so somewhere between them the total bottoms out.

Radius (cm)	Height (cm)	Metal (cm²)
$2.5$	$18.08$	$323.3$
$3.0$	$12.56$	$293.2$
$3.5$	$9.22$	$279.8$
$3.84$	$7.67$	$277.5$
$4.0$	$7.06$	$278.0$
$4.5$	$5.58$	$285.0$

• surface area, cm²

The least-metal can has $r \approx 3.84$ and $h \approx 7.67$, and those two numbers are related:

$$h = 2r$$

The cheapest can is exactly as tall as it is wide. That result holds for every volume, not only $355$ cm³.

A real soda can is taller and narrower than that. The shape is not a mistake — it is a different problem. A narrow can fits a hand, holds a label that can be read, and has a thicker top and bottom than its side, which shifts where the minimum falls.

A constraint instead of a cost

Some design problems fix a limit rather than a price. Ramps built to the American accessibility standard rise at most $1$ inch for every $12$ inches of run.

Worked examples

Common mistakes

Practice problems