How to calculate the overlapping area of two circles

How to calculate the overlapping area of two circles (oblique line area)?It can be calculated using simple basic formulas.
As long as you know the area of the sector, the area of the triangle and the trigonometric function, you can easily solve this problem!

Overlap area calculation instructions

Two circles with a radius of 5cm, when the distance between the centers of the two circles is 8cm, what is the overlapping area of the two circles?

After seeing this problem, you may not know where to start. If you draw a few lines on it, it seems that it is somewhat related to the area of the sector.

When the areas of the four sectors are added, the overlapping area of the two circles is covered.
As long as the area of the diamond is subtracted, it is the overlapping area of the two circles we want to calculate!

Therefore, when we find the angle θ, we can calculate the area of the sector, and when we find h, we can calculate the area of the rhombus (or the area of four triangles).

To find θ
The distance between the centers of the two circles is 8, and half of the distance is 4. We can use the cosine function to find the θ angle.
Cosθ = 4/5, θ=36.87.

To find h
After calculating θ, you can use the sine function to calculate h=5*sin(36.87) = 3, or use the Pythagorean theorem 52=42+h2, h=3.

Sector area = r*r*π*θ/360=553.1436.87/360=8.04 (cm2)
Area of four sectors = 48.04=32.18 (cm2)
Rhombus area = half of the product of two diagonals = 8(32)/2=24 (cm2), it can also be decomposed into four triangles for calculation=4(4*3/2)=24 (cm2)

The overlapping area of two circles = the area of four sectors-the area of rhombus=32.18-24=8.18 (cm2)

Overlapping area application

Why count this?
In spraying systems (such as production etching operations), the nozzles are mostly round, and if the left and right nozzles are closer, there will be an overlapped area. You can use the above method to calculate this.
In addition, in the lighting system, the illumination is also similar to the circular shape. When the left and right lights are closer, there will be an overlapped area. You can use the above method to calculate this.

Calculator for overlapping area of two circles

Please link to the following to calculate the overlap area

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