How do you find the area of a circle when given the area of a sector?

Question

When given the area of a sector of a circle, how can you find the area of the entire circle?

Answer 1

Hi Dani,

Let's begin by defining a sector of a circle - it's a portion (fraction) of the circle expressed as a ratio:  number of degrees in arc of sector / number of degrees in circle (360 degrees).  

For example, if the arc marked out by our sector is 90 degrees that means our sector is 90/360 or one quarter (1/4) the area of the whole circle.  

So if we are given the area of a sector then we can calculate the area of the whole circle by multiplying the area of the sector by the reverse ratio (number of degrees in circle / number of degrees in arc of sector).

For an example. let's use again our sector whose arc is 90 degree sector and say its area is 2π.  Then the area of the circle would be 2π(360/90) = 2π(4) = 8π.  Conversely, if we are told the area of a circle is 8π and are asked to find the area of a sector whose arc is 90 degrees then we would multiply the area of the circle by the ratio (number of degrees of arc in sector / number of degrees of whole circle):   8π(90/360) = 8π(1/4) = 2π.

I trust this will clarify for you the relationship between the area of a sector and the area of circle.

Thanks for submitting your question.

Regards, Jordan.     

Answer 2

The problem will undoubtedly give you more information about the sector. A sector of a circle is like a slice of a pie, and its area will depend on the angle between the 2 "sides" of the pie.

If they tell you that the area of a 60 degree sector is 14 square cm., think about what fraction of the "whole pie" this "slice" is. A circle is a 360 degree arc, so 60 degrees is (60/360) or (1/6) of a whole circle. You would have to multiply the sector area by 6 to get the area of the whole circle.

They might tell you that a 9 degree sector has an area of 38 square cm., this is (9/360) of a whole circle, so you need to multiply the sector area by (360/9) = 40. So 38(40) = 1520 square cm. for the whole circle.

Let's say they tell you that an x degree sector has an area of 20 square cm., this is (x/360) of a whole circle, so you need to multiply the sector area by (360/x). 20(360/x) = 720/x square cm. for the circle.