How to find the area of a quarter circle given only the perimeter of a quarter circle?

Question

I have been looking for a formula for these types of problems but I still haven't gotten an answer. The question you guys can use can be found here. All I need is the formula but if you want you can include the answer for the question. Also if you find the formula for a semicircle then please include that. Be sure to keep in mind that you need to include $2r$ in your formula.

Also if you are confused please tell me so I know whether I should elaborate.

Answer 1

The perimeter of a full circle is $2\pi r$. The perimeter of a quarter circle is therefore $P = \frac {2\pi r}4 + 2r$. The formula for the area of a full circle is $\pi r^2$. The area of a quarter circle is therefore $A= \frac {\pi r^2}4$.

So solve for $r$ in terms of $P$ and plug that equation into $A$.

I assume you can do that yourself.

Answer 2

First find the radius of the circle using the perimeter. For the quarter circle, the perimeter is made up of two lots of the radius plus a quarter of the circumference, so we have that $$2r + \frac{\pi}{2} r = P$$ giving that $$r = \frac{2P}{4 + \pi}$$ Now that you have the radius, you can use the area of a quarter circle formula to get the answer.

Answer 3

Hint -

$\frac14 \times 2\pi r + 2r = 5.355$

Find value of r.

Then put it in.

Area = $\frac14 \times \pi r^2$