• Knowing the parts of a circle and how they are related
• Meeting the famous and mysterious number π
• Understanding the meaning of the formula πr2
Geometry is full of beautiful shapes, but it has often been said that the simplest and most bewitching of them all is the circle. Certainly this is among the most useful shapes for humans, and has been since the invention of the wheel many thousands of years ago.
There is much than can be said about the geometry of circles. In fact, some of the most famous formulae in science are about circles. In this chapter we will explore these mathematical marvels.
To begin at the beginning: what is a circle? The ancient Greek geometer Euclid defined it this way: first pick a spot on the ground. That will be the circle’s center. Now choose a fixed distance, say 5 feet. Then mark every point on the ground which is exactly 5 feet from the center. The shape that emerges is a circle.
Circles have their own little lexicon of terms that you need to get to grips with. To begin with, the radius of a circle is the distance between the center and the edge of the circle. (That was 5 feet in the example above.)
IF YOU HAVE A COMPASS, THEN HAVE A GO AT QUIZ 1.
A compass is a useful tool for drawing circles (also known as a pair of compasses: that’s the tool with a pin and pencil, not the one for finding North!). If you have a compass, then the distance you set between the pin and the pencil will be the radius of the resulting circle.
Another key word is diameter. This is the distance across the circle, from one side to the other, passing through the middle. A little thought should confirm that a diameter can be split into two radii, meeting in the middle. This gives us our first formula for the circle: if d is the diameter of a circle and r is its radius, then the number d is r doubled. Or more concisely, d = 2 × r. Omitting the multiplication sign, as usual, we have:
d = 2r
So a radius of 5cm corresponds to a diameter of 10cm, a diameter of 6 miles corresponds to a radius of 3 miles, and so on.
So far, so easy. But there is another significant distance which we might be interested in: the length of the circle itself, that is to say the circumference. If I have a circular pond in my garden, with a fountain at the center, and a radius of 1 meter, how long is the wall around the outside of the pond?
This question, of finding the circumference of a circle when we know the radius and diameter, is a remarkably deep one, and has baffled thinkers around the world since the dawn of civilization. The ancient Babylonians, almost four thousand years ago, thought that if you multiplied the diameter by 3.125 (that is, 
) you would get the length of the circumference. Egyptian thinkers around 1650 BC believed that the value should be 
(which is around 3.160). In China around AD 500 Zu Chongzhi settled on a figure of 
, while al-Khwarizmi in ninth-century Baghdad believed it should be 
.
All these geometers did at least agree on one thing: there is some number which works for all circles, however large or small. When you multiply the circle’s diameter by this mystery number, you get the circumference. The only difficulty was in identifying this number exactly.
It was in 1706, at the hands of Welsh mathematician William Jones, that this elusive number finally received the name by which it is now universally known: π. Pronounced “pi,” this symbol is the Greek letter “p” (probably chosen to stand for “periphery”).
In the centuries that followed, we have learned a great deal about π. In particular, we now understand why the geometers of old struggled with it so much. The number π is an example of what is today known as an irrational number. This means that it can never be written exactly as a fraction of two whole numbers. (“Irrational” here means “not a ratio”; it has nothing to do with rationality in the sense of being sensible, intelligent or logical.)
This immediately means that all the old values attributed to π must be wrong because they were fractions (although some were excellent estimates, and were near enough for practical purposes). What about a decimal? We can certainly start writing out the value of π:
3.1415926535897932384626433832795 …
The interesting thing is that this sequence of numbers will continue forever, never ending, and never getting caught in a repetitive loop (unlike the recurring decimals we came across when converting fractions to decimals). It simply keeps going, ever unpredictable. Hence we can never write down the value exactly, except, of course, under its name: “π.” (The drive to calculate ever more digits of π has now reached the trillion mark, and, like π itself, is set to continue indefinitely.)
Interesting though the history is, the mystery of π is now largely solved. In particular, it is now easy to calculate the circumference of a circle from its diameter: you simply multiply by π. So we get the next formula for a circle: c = π × d, where c is the circumference and d is the diameter. Equivalently, because the diameter is twice the radius (r), we might say c = 2 × r × π, or omitting the multiplication signs, and reordering:
c = 2πr
For most of us, the correct tool for using this formula is the 
button on a calculator. Though not exact, this will give as good an approximation of π as we will ever need. (In some calculators you may have to press SHIFT or 2nd FN and then another button to access the π function.) So, to return to the example of the pond above, if my circular pond has a radius of 1 meter, then its circumference is 2 × π × 1. Typing in 
to my calculator produces an answer of 6.28 meters (when rounded to two decimal places; see Decimals).
Turning this round, if we know that a circle has a circumference of 10 meters, how can we calculate its radius? In other words, we have to find the value of r so that 2 × π × r = 10. To solve this, we just need to divide 10 by 2 × π (see Equations if you have forgotten why this works). On my calculator I do this by typing 
but other calculators may work differently. This brings up an answer of 1.59 meters, to two decimal places. (The brackets are needed to make sure that I divide 10 by twice π, rather than dividing by 2 and then multiplying the answer by π.)
TRY SOME CALCULATIONS YOURSELF IN QUIZ 2.
So now you know how the radius, diameter and circumference of a circle are all related.
We might also want to know the area of a circle, that is, the amount of space it occupies. For some shapes, such as a square, finding the area is easy: if the square’s side is 3cm long, then its area is 3 × 3 = 9cm2. Once again, though, the circle is less straightforward, and again the number π takes center stage.
If a circle has radius r = 3cm, then what is its area? If we build a little square on the radius of the circle, we know that its area is r × r, or r2 for short. That comes out as 3 × 3 = 9cm2 in the above example.
So, the question is, how many times does this little square fit inside the circle? The answer—for any circle, large or small—is again π. So the area of the circle is given by π × r × r. Calling the area A, this gives us one of the most famous of all formulae:
A = πr2
In the example above, where the circle has radius 3cm, the area is π × 9 = 28.27cm2, to two decimal places.
CALCULATE THE AREA OF SOME CIRCLES IN QUIZ 3.
We might want to work backward. If we know the area of a circle, how can we work out its dimensions? Suppose a circle has an area of 4cm2. If we call its radius r, it must be that πr2 = 4. This is now an equation, where our job is to find out r. (See Equations for more discussion of this.) We begin by dividing both sides by π, to get 
. We could now put this into the calculator, but let’s first see how to finish the calculation off. We now know what r2 is. To find out r from this information, we need to take the square root (see Roots and logs) of the number we have just found. So the exact answer is 
There are various ways to get a value for this, depending on your calculator. One way would be 
. Again the brackets are crucial. On an older calculator, the best approach might be 
followed by 
(or maybe 
, where 
is the button which recalls the answer to the previous calculation). Whatever method you use, the answer should come out as 1.13cm, to two decimal places.
TRY QUIZ 4, AND THE MORE CHALLENGING QUIZ 5.
Sum up Circles are among the most beautiful and useful shapes. They are also associated with some of the most beautiful and useful of all mathematics!
1 Use your compass to draw circles with these sizes. Then, for each circle, estimate the length of the circumference (that is, the distance around the outside).
a A circle with a radius of 2cm
b A circle with a diameter of 2cm
c A circle with a radius of 4cm
d A circle with a diameter of 4cm
e A circle with a radius of 1cm
2 Calculate these lengths. Give your answers to two decimal places.
a The circumference of a circle with diameter 5cm
b The circumference of a circle with radius 5cm
c The radius of a circle with diameter 7 miles
d The diameter of a circle with circumference 50km
e The radius of a circle with circumference 4.4mm
3 Calculate these areas. Give your answers to two decimal places.
a The area of a circle with radius 5cm
b The area of a circle with diameter 5cm
c The area of a circle with radius 2.13mm
d The area of a circle with diameter 2.13mm
e The area of a circle with circumference 2.13mm
a What is the radius of a circle with area 5 square miles?
b What is the diameter of a circle with area 13 mm2?
c What is the circumference of a circle with area 5.3km2?
d A circle fits inside a square exactly (touching but not crossing all four sides). If the area of the circle is 8cm2, what is the area of the square?
e Overnight, a pattern of flattened crops appears in a farmer’s field, consisting of four non-overlapping circles, all the same size. If the total area of flattened crops is 700m2, what is the radius of each circle?
5 Concentric circles are circles with the same center.
a A picture consists of two concentric circles with radii 3cm and 4cm respectively. What is the area of each?
b The circular strip between the two circles in part a is painted red. What is the area of the red stripe?
c A circle fits inside a square exactly. The width of the square is 5cm. What are the areas of the square and the circle?
d In part c what is the total area of the parts of the square outside the circle?
e Inside a circle of radius 6cm is a triangle, whose base is a diameter of the circle, and whose height is a radius. What is the total area of the parts of the circle outside the triangle?
