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Many people must have worked really hard on this one, there were lots of detailed answers. Well done! Some people found some solutions like Alison's.
Area: 24 cm2
Size | Perimeter |
4 cm x 6 cm | 20 cm |
3 cm x 8 cm | 22 cm |
2 cm x 12 cm | 28 cm |
1 cm x 24 cm | 50 cm |
A key idea is that you can use fractions or decimals to get more solutions. Alison found one more for us, and Patrick found a few extra too.
1.5 cm x 16 cm | 35 cm |
0.5 cm x 48 cm | 97 cm |
0.25 cm x 96 cm | 192.5 cm |
Notice that we can get an odd number for the perimeter in more than one way, and even find a fractional perimeter. Thomas and Zaki made a great observation.
There are infinitely many perimeters you can make with an area of 24 as you
can keep on multiplying the length by 2 and dividing the width by 2 so the
perimeter keeps getting bigger.
Raadiyah realised that you could divide by numbers other than 2 to find the following.
0.000001 cm x 24000000 cm | 48000000.000002 cm |
0.0000001 cm x 240000000 cm | 480000000.0000002 cm |
Amy had a great idea for finding a small perimeter using square roots. The square root of 24 is not even expressible as a fraction or a finite decimal, it is irrational!
To get the smallest possible perimeter it needs to be a square so I found the square root of 24 which was approximately 4.898979486.
$\sqrt{24}$ cm x $\sqrt{24}$ cm | $4\times \sqrt{24}$ cm |
4.898979486 cm x 4.898979486cm | 19.59591794 cm |
If you know how to plot tricky graphs and how to find the solution to simultaneous equations you might like to think about how to make any possible perimeter.