What is pi?
For any circle, \(circumference \div diameter = 3.141592鈥)
This number is so special that it is given its own symbol \(\pi\) (the Greek letter pi).
The value \(\pi\) is a constant, but is called an irrational number as an exact value for \(\pi\) does not exist.
In non-calculator working out, approximate values of \(\pi\) are used, of which \(3.14\) and \(3.142\) are probably the most common.
Other, less accurate approximations of \(\pi\) are \(\frac{22}{7}\) and \(3\).
All scientific calculators have a \(\pi\) button.
You can use this to make your calculations more accurate.
History
The earliest known use of the Greek letter \(\pi\), to represent the ratio of a circle's circumference to its diameter, was by the Welsh mathematician William Jones in his 1706 work 鈥Synopsis Palmariorum Matheseos鈥 or 鈥A New Introduction to the Mathematics鈥.
Circumference
We know that \(circumference \div diameter = \pi\)
It therefore follows that \(circumference = \pi \times diameter\).
This can also be written as:
\(C = \pi d\)
The diameter is twice the length of the radius.
So, an alternative formula for the circumference of a circle is:
\(C = 2 \pi r\)
Circumference of circle slideshow
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Question
A circle has a diameter of \({10~cm}\). Work out its circumference, using \(\pi = 3.14\).
Answer
Using \(C = \pi d\)
\(C = 3.14 \times 10\)
\(C = 31.4~cm\)
Using \(C = 2 \pi r\)
\(C = 2 \times 3.14 \times 5\)
\(C = 31.4~cm\)
Question
Anish and Becky each have a circular pond in their garden.
Anish's pond has a diameter of \({6~m}\).
Becky's pond has a diameter of \({3~m}\).
Anish says that the circumference of his pond is twice the circumference of Becky's pond.
a) Find the circumference of each pond, using \(\pi = 3.14\).
b) Is Anish's statement correct?
Answer
a) The circumference of Anish's pond is \({18.84~m}\) and the circumference of Becky's pond is \({9.42~m}\).
b) From the answers to a) you can see that Anish's statement is correct.
Area of a circle
The formula for working out the area of a circle is:
\(A = \pi r^2\), where \(r\) is the radius of the circle.
\(\pi r^2\) means \(\pi \times r \times r\).
Only the \(r\) is squared.
How to show the area of a circle is 蟺 x r2
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Question
Find the area of the following circles, using \(\pi = 3.14\).
a) a circle of radius \({6~cm}\)
b) a circle of diameter \({10~cm}\)
Answer
a) \(r = 6~cm\), so we calculate:
\(A = 3.14 \times 6 \times 6 = 113.04~cm^2\)
b) The diameter is \({10~cm}\), so the radius is \({5~cm}\).
We calculate: \(A = 3.14 \times 5 \times 5 = 78.5~cm^2\)
Question
The dartboard above has a radius of \({20~cm}\).
The bullseye in the centre of the board has a radius of \({1~cm}\).
By calculating the area of the two circles, work out the area of the dartboard outside of the bullseye, using \(\pi = 3.14\).
Answer
Remember, the area of the large circle is \({3.14}\times{20}\times{20} = {1,256}~cm^{2}\).
The area of the small circle is \(3.14 \times 1 \times 1 = 3.14~cm^2\).
So, the area of the dartboard outside of the bullseye is \({1,256} - {3.14} = {1,252.86}~cm^{2}\).
Key point
When calculating the area of a circle, remember to use the radius, not the diameter.
How many times do the wheels on a scooter go round during a lap of the park?
Test section
Question 1
What is a commonly used value of \(\pi\)?
a) \({3.14}\)
b) \({3.16}\)
c) \({13.5}\)
Answer
\({3.14}\) is the commonly used value of \(\pi\).
So, the correct answer is a).
Question 2
Which of the following is not a formula to find the circumference of a circle?
a) \(\pi{r}^{2}\)
b) \(\pi{d}\)
c) \({2}\pi{r}\)
Answer
The correct answer is a) \(\pi{r}^{2}\).
Question 3
What is the formula to find the area of a circle?
a) \(\pi{d}\)
b) \(\pi{r}^{2}\)
c) \({2}\pi{r}\)
Answer
The correct answer is b) \(\pi{r}^{2}\).
Question 4
What is the circumference of a circle that has a diameter of \({7}~{cm}\)?
Answer
The formula for circumference is: \({circumference}=\pi\times{d}\). so \(\pi\times{7}~{cm}={22.0}~{cm}~(to~1~dp)\).
Question 5
What is the circumference of a circle that has a radius of \({5}~{cm}\)?
Answer
The formula for circumference is: \({C}={2}\times\pi\times{r}\) or \({C}=\pi\times{d}\). \(\pi\times{5}\times{2}={31.4}~{cm}~(to~1~dp)\).
Question 6
What is the diameter of a circle that has a circumference of \({15}~{cm}\)?
Answer
The formula for diameter is \({d}=\frac{C}{\pi}\).
In this case, \({d}=\frac{15}{\pi}={4.8}~{cm}~(to~1~dp)\).
Question 7
What is the area of a circle that has a radius of \({5}~{cm}\)?
Answer
\(\pi\times{r}^{2}\) is the formula for the area of a circle.
\(\pi\times{5}^{2}=\pi\times{25}={78.5}~{cm}^{2}~(to~1~dp)\).
Question 8
What is the area of a circle that has a diameter of \({16}~{cm}\)?
Answer
\({A}=\pi\times{r}^{2}\) is the formula for the area of a circle.
\({d}={16}~{cm}\), so \({r}={8}~{cm}\).
\({A}=\pi\times{8}^{2}=\pi\times{64}={201.1}~{cm}^{2}~(to ~1~dp)\).
Question 9
What is the area of a half circle that has a radius of \({10}~{cm}\)?
Answer
\(\pi\times{r}^{2}\) is the formula for the area of a circle.
Therefore, the area of a half circle \(=\frac{(\pi\times{r}^{2})}{2}\).
\(\frac{(\pi\times{10}^{2})}{2}={157.1}~{cm}^{2}~(to~1~dp)\).
Question 10
What is the radius of a circle that has an area of \({42}~{cm}^{2}\)?
Answer
\({A}=\pi\times{r}^{2}\) is the formula for the area of a circle.
The formula to find the radius is \({r}=\sqrt\frac{A}{\pi}\).
Therefore, \({r}=\sqrt\frac{42}{\pi}\) which is \({3.7}~{cm}~(to~1~dp)\).
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