# Find the exact shaded area of the region in 4 overlapping circles

• MHB
• Zekes
In summary, In this article, the author is looking for a proof that each part of the circle which is in an intersection is 1/4 the size of the whole circle's circumference. The author finds an equation to get the area of the shaded region. However, the author does not understand how the solution got to there, and how to do Part 1. The author asks for help from the reader.
Zekes
So, say you got 4 circles intersecting this way:

View attachment 9061

Now, I am looking for two things:

1. A proof that each part of the circle which is in an intersection is 1/4 the size of the whole circle's circumference

• The exact area of the non-shaded region.

Now, in my search to finding the answer to this, I stumbled upon this Circle-Circle Intersection -- from Wolfram MathWorld. The only problem? I have no idea what this article is trying to say, and how it can help me. I did find the equation to get the area of the shaded region ( it's $$\displaystyle A=2(\pi-2)$$ ) which I can use in Part 2 but I still don't understand how the solution got to there, and how to do Part 1. Please help me in learning what is trying to be said here in simpler terms! Thanks!

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Hi Zekes.

Let the radius of each circle be $r$; let P, Q, R, S be the centres of the cirlces and O, A, B, C, D the points of intersections marked as follows:

View attachment 9063

Now PQRS is a square with diagonal $2r$ and so the length of each side is $r\sqrt2$, i.e. $|\mathrm{PQ}|=r\sqrt2$. So the “thickness” of each light-blue lens-shaped region of overlap between circles is $(2-\sqrt2)r$. if T is the point of intersection of the line segments AO and PQ, then

$$|\mathrm{PT}|\ =\ r-\frac{2-\sqrt2}2r\ =\ \frac r{\sqrt2}.$$

Hence $\angle\mathrm{APT}\ =\ \arccos\frac1{\sqrt2}\ =\ 45^\circ$; i.e. $\angle\mathrm{APO}=90^\circ$. That is to say, each circular arc drawn from O is one-quarter the circumference of each circle.

Now:

• area of quadrant APO = $\dfrac{\pi r^2}4$
;
• are of triangle APO = $\dfrac12r^2$
;
• therefore area of each light-blue shaded region = $2\times\left(\dfrac{\pi r^2}4-\dfrac12r^2\right)=\dfrac{(\pi-2)r^2}2$
;
• therefore area of non-shaded region in each circle = $\pi r^2-2\times\dfrac{(\pi-2)r^2}2=2r^2$
;
• therefore total non-shaded area = $4\times2r^2=8r^2$.

Interesting to note that the final answer does not contain $\pi$.

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## 1. What is the formula for finding the exact shaded area of the region in 4 overlapping circles?

The formula for finding the exact shaded area of the region in 4 overlapping circles is A = πr² - 2πr² + 4(√3)r², where r is the radius of each circle.

## 2. How do you determine the radius of each circle in order to find the shaded area?

To determine the radius of each circle, you can measure the distance from the center of the circle to any point on the circumference. Alternatively, if you know the diameter of the circle, you can divide it by 2 to find the radius.

## 3. Can you provide an example calculation for finding the exact shaded area?

For example, if each circle has a radius of 5 units, the calculation would be A = π(5)² - 2π(5)² + 4(√3)(5)² = 25π - 50π + 100√3 ≈ 65.39 units².

## 4. What is the significance of the √3 in the formula for finding the shaded area?

The √3 in the formula represents the equilateral triangle formed by the overlapping circles. This is necessary to accurately calculate the shaded area in the region where all four circles overlap.

## 5. Are there any limitations to using this formula for finding the exact shaded area?

Yes, this formula assumes that the circles are perfectly overlapping and that the shaded region is symmetrical. If the circles are not perfectly overlapping or the shaded region is not symmetrical, this formula will not give an accurate result.

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