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## 'Efficient Cutting' printed from http://nrich.maths.org/

Ysanne
and Joanna from Central Newcastle High School sent us their
thoughts:
Roughly half of the paper should be used to make the circles and
roughly half to make the rectangle as the height of the cylinder
and the area of the circle are equally important in creating the
volume.

We decided to use the side of length $29.6$ as the circumference of
the cylinder and the length of the rectangle. As this was the
circumference of the cylinder, and therefore the circles, we
divided $29.6$ by $\pi$ to find the appropriate diameter for the
circles. This came to $9.42$, so the remaining paper left for the
rectangle meant that the height of the cylinder would be 11.58. To
find the volume, we found the area of the circle and multiplied by
the height of the cylinder: $\pi \times 4.71^2 \times 11.58 \approx
807.5$ cubic centimetres as our volume.

Lyman
from Nanjing International School sent us two different
arrangements for the circles and rectangles to try to maximise the
area - click here
to see
them.
It is
possible to improve on this volume:
Let the width of the paper fix the height of the cylinder, then
h=21

If $r$ is the radius, then the length of the rectangle is $2 \pi r$
so if we fit the rectangle plus the circle along the $29.6$cm
length we have $$2r + 2 \pi r = 29.6$$

$$r(2+2\pi) = 29.6$$

$$r = \frac{29.6}{2+2\pi}$$

So the volume of the cylinder is:

$$\pi r^2 h = \pi \times \frac{29.6^2}{(2+2\pi)^2}
\times 21 \approx 842$$