Circle: The area of the shaded region

[ashcrimson]

I know the necessary formulas for the circumference and area of both a circle and a rectangle but I just can't seem to know where to start. Please, help would be appreciated.

 

[Dr.Peterson]

You need more information. What are the curves inside the rectangle? They don't look like circular arcs; are they perhaps cycloids (the path of the marked point as the circle rolls)?

Please state the entire problem as given to you, including a description of how the figure is constructed. Also, what is the context of the question -- e.g. can you use calculus, or have you learned some facts about this figure?

 

[ashcrimson]

You need more information. What are the curves inside the rectangle? They don't look like circular arcs; are they perhaps cycloids (the path of the marked point as the circle rolls)?

Please state the entire problem as given to you, including a description of how the figure is constructed. Also, what is the context of the question -- e.g. can you use calculus, or have you learned some facts about this figure?

The leaf-like figure inside the rectangle is an ellipse. I have heard that it is done by subtracting the area of the ellipse from the area of the rectangle, which is again found by finding out the length of the rectangle or the traversed distance of the circle, since the breadth is just 1+1 = 2 metres. I know the process sure, I simply don't understand how to find the length of the rectangle/the traversed distance of the circle. It's usually the circumference but there seems to be a division by 2 to the formula. Like 2πr/2 = Length. I was hoping someone would explain that to me.

 

[Dr.Peterson]

Again, please state the original problem. What are you told about the length of the rectangle? Where does "the traversed distance of the circle" come into the problem? (That sounds to me as if you are talking about a cycloid, not an ellipse.)

 

[pka]

I agree with Prof Peterson that the arc is one-half of a cycloid see the link.
If one enlarges the original image in the OP it says that the radius of the circle is \(\displaystyle 1m\) .
That means that the rectangle is \(\displaystyle (\pi m)\times(2m)\)
Can you now use the link?