Geometry – Calculating the Total Area Belonging to Only One of Four Unit Circles

Question

Please forgive the crudeness of this diagram.

(I took an image from some psychobabble website and tried to delete the larger circle that's not relevant to my question).

Let's say these are four unit circles joined together such that each circle shares some area with two other circles.

Obviously the total area not shared with other circles is four times the area of this (again please forgive the crude diagram)

Or I could calculate the total are of a single "petal" and multiply that by $4$. But I have truly forgotten all the calculus and trigonometry I was taught more than half a century ago.

Am I on the right track? Is there a better way than either of the two ideas I've had so far?

P.S. Not sure if osculating circle tag applies.

Answer 1

For one "petal" only, consider this:

$$\text{area of petal} = \text{area of square} = \left(\;\sqrt{2}\;r\;\right)^2 = 2 r^2$$

For the entire figure:

$$\text{total area} \;=\; 4 r^2 + 4\cdot \frac{1}{2} \pi r^2 \;=\; 4 r^2 + 2 \pi r^2$$