Question Video: Finding a Formula Linking the Area and Circumference of a Circle

Video Transcript

Using the formulae for the circumference and area of a circle, eliminate the variable 𝑟 to find a formula that allows you to calculate the area of a circle from its circumference.

Let’s begin by recalling our formulae for this circumference and area of a circle. The circumference, 𝑐, is equal to 𝜋 multiplied by the diameter, 𝑑. As the diameter is double the length of the radius, this can also be written as 𝑐 is equal to two 𝜋𝑟. We will call this equation one. The area of a circle is equal to 𝜋𝑟 squared. We will call 𝐴 equals 𝜋𝑟 squared equation two.

We are asked to eliminate the variable 𝑟 to find a formula that allows us to calculate the area from the circumference. One way to do this is to rearrange equation one to make 𝑟 the subject. Dividing both sides of equation one by two 𝜋 gives us 𝑐 over two 𝜋 is equal to 𝑟. We can then substitute this into equation two. We will replace 𝑟 with 𝑐 over two 𝜋. This gives us 𝐴 is equal to 𝜋 multiplied by 𝑐 over two 𝜋 squared.

We recall that when squaring a fraction, we can square the numerator and denominator separately. In this case, we need to square 𝑐 and two 𝜋. 𝑐 multiplied by 𝑐 is 𝑐 squared and two 𝜋 multiplied by two 𝜋 is four 𝜋 squared. We now have 𝐴 is equal to 𝜋 multiplied by 𝑐 squared divided by four 𝜋 squared. We can divide the numerator and denominator by 𝜋.

𝐴 is, therefore, equal to 𝑐 squared over four 𝜋. This formula allows us to calculate the area of a circle, 𝐴, when given its circumference, 𝑐. We can also rearrange this formula to make the circumference, 𝑐, the subject. This would allow us to calculate the circumference of a circle from its area.