### Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

### Great Squares

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

### Square Areas

Can you work out the area of the inner square and give an explanation of how you did it?

# Square Pegs

##### Stage: 3 Challenge Level:

Different interpretations as to how to decide on better fit' all suggest that it is the round peg. Some contributors compared the area left over' but, since the holes are not the same shape, this does not compare like with like.

For the square peg: if the diameter of the hole is $2r$ then the side of the square is $r\sqrt{2}$. The fraction of the area of the cross section of the hole taken up by the peg is $(r\sqrt{2})^2/\pi{r^2} = 2/\pi$.
For the round peg: if the diameter of the peg is $2r$ then the side of the square is $2r$. The fraction of the area of the cross section of the hole taken up by the peg is $\frac{\pi{r^2}}{4r^2} = \frac{\pi}{4}$.

We know that $\pi^2 > 8$ so it follows that the round peg is a better fit as it takes up more of the hole because $\frac{\pi}{4} > \frac{2}{\pi}$.