◼️Squares & Cubes

A square room 5 meters on a side: how many floor tiles? 5 tiles per row, 5 rows — $5 \times 5 = 25$. Multiplying a number by itself is so useful that math gave it its own name: squaring.

Squares: the area of a square

You counted unit squares in area: a square of side $a$ has area $a \times a$, written $a^2$ and read "a squared". Squaring is the busiest member of the exponents family: $a^2$ just means "two copies of a, multiplied".

The first 12 squares are worth memorizing:

$1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144$

These are $1^2$ through $12^2$. Knowing them on sight lets you spot perfect squares instantly. Neighbouring squares also differ by odd numbers: $7^2 = 49$ while $8^2 = 49 + 15 = 64$ — and the gap is exactly $7 + 8 = 15$.

9 × 3 = ?

✓

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Cubes: the volume of a cube

Move squaring into 3D: a cube of edge $a$ has volume $a \times a \times a$, written $a^3$ and read "a cubed".

$2^3 = 8, \quad 3^3 = 27, \quad 4^3 = 64, \quad 5^3 = 125$

You can count your way there too: $5^3$ is one layer of $5 \times 5 = 25$ small cubes, stacked 5 layers deep. Fun fact: $64$ is both $8^2$ and $4^3$ — numbers that are perfect squares and perfect cubes at once are rare.

Running it backwards: square roots

Squaring turns a side length into an area; the square root turns an area back into a side length: $\sqrt{144} = 12$, because $12^2 = 144$. The symbol is called the radical sign (it looks like $\sqrt{\phantom{00}}$).

• $\sqrt{81} = 9$, $\sqrt{100} = 10$, $\sqrt{121} = 11$;
• Memorize the squares and these answer themselves.

Cubes have an inverse too — the cube root: $\sqrt[3]{125} = 5$, because $5^3 = 125$.

Check yourself

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Quick quiz

↺ Try again

1. 1. 7² = ?

   144942

2. 2. Which of these is a perfect square?

   506472

3. 3. √81 = ?

   92740.5