The circumference of a circle is 65?. In terms of pi, what is the area of the circle?

Accepted Solution

Answer:1056.25Ο€ square unitsStep-by-step explanation:A few formulas an definitions which will help us:(1) [tex]\pi=\frac{c}{d}[/tex], where c is the circumference of a circle and d is its diameter(2) [tex]A=\pi r^2[/tex], where A is the area of a circle with radius r. To put it in terms of d, remember that a circle's diameter is simply twice its radius, or mathematically, (3) [tex]d=2r \rightarrow r=\frac{d}{2}[/tex]. We can rearrange equation (1) to put d in terms of Ο€ and c, giving us (4) [tex]d = \frac{c}{\pi}[/tex], and we can make a few substitutions in (2) using (3) and (4) to get use the area in terms of the circumference and Ο€:[tex]A=\pi r^2\\=\pi\left(\frac{d}{2}\right)^2\\=\pi\left(\frac{d^2}{4}\right)\\=\pi\left(\frac{(c/\pi)^2}{4}\right)\\=\pi\left(\frac{c^2/\pi^2}{4}\right)\\=\pi\left(\frac{c^2}{4\pi^2}\right)\\\\=\frac{\pi c^2}{4\pi^2}\\ =\frac{c^2}{4\pi}[/tex]We can now substitute c for our circumference, 65, to get our answer in terms of Ο€:[tex]A=\dfrac{65^2}{4\pi}=\dfrac{4225}{4\pi}=1056.25\pi[/tex]