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

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

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Video Transcript

Using the formulae for the circumference and area of a circle, eliminate the variable π‘Ÿ 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œ‹ 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 πœ‹π‘Ÿ. We will call this equation one. The area of a circle is equal to πœ‹π‘Ÿ squared. We will call 𝐴 equals πœ‹π‘Ÿ squared equation two.

We are asked to eliminate the variable π‘Ÿ 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 π‘Ÿ the subject. Dividing both sides of equation one by two πœ‹ gives us 𝑐 over two πœ‹ is equal to π‘Ÿ. We can then substitute this into equation two. We will replace π‘Ÿ with 𝑐 over two πœ‹. This gives us 𝐴 is equal to πœ‹ multiplied by 𝑐 over two πœ‹ 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œ‹. 𝑐 multiplied by 𝑐 is 𝑐 squared and two πœ‹ multiplied by two πœ‹ is four πœ‹ squared. We now have 𝐴 is equal to πœ‹ multiplied by 𝑐 squared divided by four πœ‹ squared. We can divide the numerator and denominator by πœ‹.

𝐴 is, therefore, equal to 𝑐 squared over four πœ‹. 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.

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