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Exact answers: multiples of pi

When performing calculations which involve circles we will usually be required to use a value for pi.
PAULA KELLY: Another example of an irrational number is pi. Pi is an irrational number. It can’t be written as a fraction. So we never get an exact value for pi. In most cases, working to two decimal places is sufficient. So using the approximation that pi equals 3.14 is good enough. You could use the approximation of pi to be 22 divided by 7. But it wouldn’t give you an accurate answer. The only way to give an exact answer is to leave your answer as a multiple of pi. When working with pi, students are often asked to give their answers as a multiple of pi instead of rounding.
If we use that pi is worth 3.14– so the area of any circle can be found by multiplying pi by the radius squared. For this circle, we find the area by multiplying pi and 6 squared. In this case, our area is 113.04 squared centimetres. So 113.0 to one decimal place squared centimetres. However, if we use the value of pi in our calculator, the area comes to 113.1 squared centimetres to one decimal place. There’s a slight difference in answers. The exact answer is 36 pi squared centimetres. A running track is 400 metres long. Each straight is 100 metres long. Each semicircle is also 100 metres long. What’s the perpendicular distance between each of the straights.
If we do this twice, so once using pi all the way through, we get a distance of 63.66 metres to two decimal places. If we’re using 3.14, we get a distance of 63.69 metres to two decimal places. The general rule about rounding is only to round your final answer. Rounding intermediate answers can produce large rounding errors in your final answers when those intermediate answers are used in subsequent calculations. If you’d like to look further at different kinds of errors when using measurements and calculations, we recommend the resources from the STEM Centre website.
When performing calculations that involve circles we will usually be required to use a value for \(\pi\) (pi).
The exact value of pi cannot be found as it an irrational number, a number which goes on forever. We usually use a rounded value of 3.14. If we are asked to provide an exact answer for a solution involving pi, we are required to leave our answer as a multiple of pi.
In this video, Paula explains the solutions to two problems involving pi.

Problem worksheet

Now complete questions 7, 8 and 9 from this week’s worksheet.

Teaching resource

This set of resources on exact answers give pupils the opportunity to calculate exactly with fractions, surds and multiples of \(\pi\).
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Maths Subject Knowledge: Understanding Numbers

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