Squares, cubes and the real world

Why have humans spent thousands of years thinking about $n^2$ and $n^3$? Because the moment you move from a flat world to one that has size, weight, or growth, squares and cubes appear by themselves. This subtopic gathers a few of the most useful places they show up.

Concept

Area uses squares. Any flat shape's area scales with the square of its linear size. Double a poster's height and width and you need $4$ times the paper, not $2$. The area of a square of side $a$ is $a^2$; the area of a circle of radius $r$ is $\pi r^2$. Both are squares of a length.

Volume uses cubes. Any solid's volume scales with the cube of its linear size. Double the side of an ice cube and it now contains $8$ times the water. The volume of a cube of side $a$ is $a^3$. The volume of a sphere of radius $r$ is $\frac{4}{3}\pi r^3$.

This square-cube law is everywhere. It is why a baby elephant cannot have the body shape of a baby mouse: scaled up, the volume (and therefore weight) grows faster than the leg cross-section (which is an area). It is also why pizza-by-area is much cheaper than pizza-by-diameter , a $16$-inch pizza has roughly $\big(\tfrac{16}{12}\big)^2 \approx 1.78$ times the area of a $12$-inch one.

Pythagoras' theorem is built on squares. For a right triangle with legs $a, b$ and hypotenuse $c$,

$a^2 + b^2 = c^2.$

That is, the square on the longest side is the sum of the squares on the other two. Squares are not just numerical here , they are real square shapes you can draw, cut, and rearrange.

Special cube identities turn up in everyday algebra:

$(a+b)^3 = a^3 + 3a^2 b + 3ab^2 + b^3, \qquad (a-b)^3 = a^3 - 3a^2 b + 3ab^2 - b^3.$

These let us cube numbers like $103$ in our head: $103^3 = (100+3)^3 = 10^6 + 3(100^2)(3) + 3(100)(9) + 27 = 1{,}092{,}727$.

Another useful identity is

$a^3 + b^3 = (a + b)(a^2 - ab + b^2).$

It is the secret behind Ramanujan's $1729 = 1^3 + 12^3 = 9^3 + 10^3$.

Worked examples

Example 1. A square photo of side $6 \text{ cm}$ is enlarged so each side is $4$ times as long. By what factor does the area increase?

• Linear scale factor $= 4$.
• Area scale factor $= 4^2 = 16$.
• Original area $= 36 \text{ cm}^2$, new area $= 576 \text{ cm}^2$.

Example 2. A cubical tank's edge is tripled. How much more water does it hold?

• Volume factor $= 3^3 = 27$.
• It holds $27$ times the original water.

Example 3. A right triangle has legs of $5$ cm and $12$ cm. Find its hypotenuse.

• $c^2 = 5^2 + 12^2 = 25 + 144 = 169$.
• $c = \sqrt{169} = 13$ cm.

Example 4. Use $(a+b)^3 = a^3 + 3a^2 b + 3ab^2 + b^3$ to compute $102^3$.

• Let $a = 100, b = 2$. Then $102^3 = 100^3 + 3(100)^2(2) + 3(100)(4) + 8 = 1{,}000{,}000 + 60{,}000 + 1{,}200 + 8 = 1{,}061{,}208.$

Try it yourself

1. A square's side is doubled. By what factor does its area grow? Its perimeter?
2. A cubical box has edge $10$ cm. If the edge is increased by $50\%$, find the new volume.
3. Find the hypotenuse of a right triangle with legs $8$ and $15$.
4. Compute $99^2$ using $(100-1)^2 = 100^2 - 2(100) + 1$.
5. Compute $101^3$ using the cube identity.
6. The areas of two similar squares are in the ratio $9 : 25$. Find the ratio of their sides.
7. A small ice cube has edge $2$ cm. A big one has edge $6$ cm. Compare their volumes.
8. Show that $7^3 + 1^3 = (7+1)(7^2 - 7 + 1) = 8 \times 43 = 344$.

Activity / Insight

The pizza experiment. Find prices for three pizza sizes from any restaurant. Compute the area of each (treat them as circles, $\pi r^2$). Make a table of price per square centimetre. You will almost always find the largest pizza is the best value , because area grows as the square of the diameter while price grows only roughly linearly.

A surprising connection. The sum of the first $n$ cubes equals the square of the sum of the first $n$ counting numbers:

$1^3 + 2^3 + 3^3 + \dots + n^3 = (1 + 2 + 3 + \dots + n)^2.$

Try $n = 4$: $1 + 8 + 27 + 64 = 100 = 10^2 = (1+2+3+4)^2$. Squares and cubes are deeply linked.