Estimating square numbers

Question

A large dice has a side length of 9.2 cm. Estimate the surface area of the cube.

What I did:

6× $9^2$ = 6 × 81 = 6 × 80 = 480

But the answer says that $9.2^2$ is 85 as an estimate.

How do I get that?

Thank You

Answer 1

Surface area of dice $=6a^2=6\cdot (9.2)^2=6(\frac{92}{10})^2=6\cdot \frac{92\cdot92}{100}=\frac{50784}{100}=507.84\approx508$ sq. units

However, if you divide this by 6 you will get roughly $85$ sq. units but that is the surface area of one side of dice.

Answer 2

The estimation for one square face could be like done this: $$ 9.2^2 = (9+0.2)^2 = 9^2 + 2\cdot 9\cdot 0.2 + 0.2^2 \approx 81 + 10\cdot 0.4 + 0 = 81 + 4 = 85 $$

Answer 3

You can use the binomial expansion : $(a+b)^{2} = a^{2} + 2ab + b^{2}$ or maybe $(a-b)^{2} = a^{2} + 2a(-b) + (-b)^{2} = a^{2}-2ab+b^{2}$ , so $9.2^{2}$ can be written as $(9+0.2)^{2}$ or $(10-0.8)^{2}$.

But wait ! what is the value of $0.2^{2}$ or $0.8^{2}$ ? that's easy, for most decimal between 0 and 1 (in example $0.5$ or $0.1$, but in case of maybe difficult repeating decimal like $0.11111111$ and so on so on this wont work) you can write them as a number divided by $10$

so then : $0.2^{2}$ = $(\frac{2}{10})^{2} = \frac{2^{2}}{10^{2}} = \frac{4}{100} = 0.4$. You can also do the same for $0.8$, but i recommend having decimal as small as possible so the calculation would be easy