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## 'An Unusual Shape' printed from http://nrich.maths.org/

### Why do this problem?

This problem offers the opportunity to practise calculating areas
of circles and fractions of a circle in the context of an
optimisation task.

### Possible approach

Start by showing the
diagram from the problem, and ask learners to think on their own
for a few moments about where the goat might have been tethered to
yield the area shown.

Next, allow them to
discuss their ideas with their partner, and finally share
convincing arguments with the whole class.

Having determined where
the hook is, the challenge is to work out the area available to the
goat and then to consider different positions of the hook in order
to find the maximum possible area.

This is a fantastic
opportunity to talk about the benefits of factorising, working in
terms of $\pi$, and only using the calculator at the very end of
the computation.

Once learners have established where to fix the hook for maximum
goat nutrition (!), move on to other lengths of rope. Suggest that
pairs of learners work with different lengths of rope, and finish
by sharing their findings with the rest of the class.

### Key questions

What can you say about
the radius of each part of a circle?

How does this help you to
pinpoint where the hook must be?

How does the space
available to the goat change if the hook is moved?

### Possible extension

What happens if the rope
is longer than the sum of the sides of the shed?

Investigate what happens
to the area available for sheds of different dimensions, or sheds
which are not rectangular.

### Possible support

The activity can be modelled by building the frame of the shed from
multilink cubes, and learners could use string to work out the
shapes of the regions that could be made available to the goat,
when fixing the hook at different points.