An investigation on how to enclose a maximal area given a certain length of fencing. In this case we use 100 metres.

How to calculate area of any triangle.

1. Print off one copy of (A3) sheet for each pupil, or pair if preferred.

2. To introduce, perhaps you might ask your class to consider how to enclose the maximum possible area given 100 metres of fencing. In this conversation establish there are two main things to consider for each shape. Firstly, the number of sides it has, and then how to distribute the 100 metres of fencing to its various lengths.

3. Pupils will first consider only quadrilaterals on the first side of the pupil sheet. Pupils should realise that for a given number of sides, the regular polygon with that number of sides encloses the largerst area.

4. Pupils can go onto the next page when they are ready where they will investigate how the number of sides affects the enclosed area. You may wish to demonstrate on the board (slide 4) how to calculate the area enclosed by splitting the polygon up into triangles, and using $\frac{1}{2} a b \sin{\left ( c \right )}$ formula.

Pupils should reach the conclusion that more sides results in greater area and therefore a circle (infinite sides) encloses the maximum possible area.

*Solutions are given on the presentation, and worksheets are structured so that pupils can work through independently.