The trick: someone says ‘what’s 7.5 squared?’ and - mentally squaring in a flash - you say: 56.25.

Squaring halves

Squaring halves is really easy if you know your times tables. Here’s the method:

  1. Take your number and find the whole numbers immediately above and below. If you’re trying to square 7.5, that would be 7 and 8; if you’re squaring 11.5, it would be 11 and 12.
  2. Multiply these numbers together (56 for $7 \times 8$; 132 for $11 \times 12$).
  3. Add on 0.25. That gives $7.5^2 = 56.25$ and $11.5^2 = 132.25$.

Squaring fives

You can also use this to square any number that ends in 5. It’s the same idea:

  1. Find the ten above and the ten below (so 25 is between 20 and 30)
  2. Multiply those together (600 - it’s always going to be …00)
  3. Add 25. $25^2 = 625$.

Why does this work? Well, it’s the old ‘difference of two squares’ trick. Let me write it this way:

\((x + 0.5)(x - 0.5) = x^2 - 0.25\) \((x + 0.5)(x - 0.5) + 0.25 = x^2\)

… and that’s all there is to it!

Squaring in reverse

You can use this trick backwards to get a better guess for square roots - for example, if you spot that 110 is $11 \times 10$, you can say that its square root must be a little less than 10.5, because $10.5^2 = 110.25$.