We will now start to look at circles and spheres, circular objects in the two-dimensional and three-dimensional planes.First of all, let’s look at a circle and some of the points/properties:


When the length of the arc = length of the radius, the angle between the radii creating the arc is 1 radian, or 57.296o


2π radians = 360o



“C” is the letter that we usually used to denote the circumference of the circle, we usually use a “r” to indicate the radius and a “d” to indicate the diameter of the circle.

You may be given problems which are quite straightforward, for example you will be given the radius of the circle and asked to calculate the circumference. 

In these matters you will almost always be given a value for pi, more often than not 3.14. It’s just a matter of plugging in the numbers to get the answer, but as always remember your units.


Q. Calculate the circumference of a circle with a radius of 4 cm



Q. Calculate the area of a circle with the same radius as above



As you can see these calculations are quite straightforward, but make sure that you get the right units. In the first case we are dealing with a distance and in the second case we are dealing with a two-dimensional area.


You might also be given a different task. You may be given the circumference all the area a circle and from it and asked to calculate the radius, or the diameter. This usually requires you to know how to “transpose” the equations, in other words make what you’re looking for the subject of the equation.



Q. Calculate the diameter of a circle which has a circumference of 106.76 cm



now the equation has been converted to the correct format, we simply plug in the values:



In fact, this question was engineered to give an exact answer for 34 cm, the reason for the longer decimal places is probably down to the way the computer interpreted pi, taking it to more decimal places than we usually do (3.14)


Of course we don't have to necessarily be talking about complete circles, we could be talking about semicircles and things known as "sectors" of a circle. A sector of a circle is simply a fraction of the area of the circle, so we could say that the sector was all of the circle, half of the circle, a third of the circle, a quarter of the circle and so on. It is very easy to calculate the area of our "fraction" if we are given certain information about the sector:



Consider the sector shown above. This is just over a quarter of a complete circle, we are given a value for the radius so of course from this we could calculate the area of the full-circle, but we don't want that. We want to know the area of this particular sector, and the fact that we have been given an angle between 2 radii is a huge clue as to how we are going to go about doing this.


Hopefully you understand that a circle is divided into degrees, and that there are 360° in a full circle. Half a circle would therefore be 180°, and I'm sure you can see that the fraction 180÷360 is in fact a half.


Similarly, a quarter of a circle would be calculated using the fraction 90÷360:



You wouldn't normally be given the value 270°, simply because you shouldn't really need it. Looking at the diagrams above you should be able to see that the values of X and Y added together will always come to 360°, so if X (as in this case is 90°) then automatically Y must equal 270°


If you are therefore given a sector, then simply by knowing the angle between the 2 radii and the value of the radius you should be able to calculate the area of the sector.



If we now go back to our sector as shown above, having been given the angle of 102° and the radius of 8 cm we can calculate the area of this sector thus:




Similarly, we can calculate the length of the arc by applying the "fraction" method to the formula for the circumference of a circle. In this case the formula would be:

As complicated as this may appear, the fractional part of X divided by X+Y is simply returning a fraction which is then multiplied by the formula for the whole circumference (or area depending on what problem you're solving).


Where 50.24cm would have been the circumference of the full circle, and 102÷360 represents the fraction of it that we wanted.


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