The height of a section of overlapping circles.

In summary, the diagram shows two overlapping circles with radii of one, where the overlapping section is 1/6th of the total circumference. The goal is to find the length of y, which can be achieved by creating a right triangle with hypotenuse 1 and side opposite the centre of the circle as 1/2. Using trigonometry and the Pythagorean theorem, y can be solved by setting up the equation y/2 = 1 - cos30 = 1 - √3/2 = (2-√3)/2, giving an exact solution of 2-√3.
  • #1
Zekes
4
0
Say I have two identical circles, both of radii of one, overlapping, as shown in the diagram below:

View attachment 9059

In this diagram, x is the circumference of the circles, and the bit of the bottom circle which is drawn blue (the overlapping bit) is 1/6th of the whole circumference.

What I'm looking for is y, which is this:

View attachment 9060

Now, working out x is easy - it's 2 \pi r, thus the overlapping bit is 1/3 \pi r. But how do I proceed in finding y from here? Help is much appreciated! Thanks!
 

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  • #2
Imagine a right triangle from the centre of a circle, the midpoint of the segment y and one of the points of intersection of the circles. This triangle has hypotenuse 1 and side opposite the centre of the circle 1/2. Also, this triangle is 30-60-90. Can you use some basic trig and the Pythagorean theorem to find y?
 
  • #3
Greg said:
Imagine a right triangle from the centre of a circle, the midpoint of the segment y and one of the points of intersection of the circles. This triangle has hypotenuse 1 and side opposite the centre of the circle 1/2. Also, this triangle is 30-60-90. Can you use some basic trig and the Pythagorean theorem to find y?

Although I know of trigonometry and Pythagoras, I don't see how having a point of the triangle at the midpoint of y can help me find the whole of y.

Also using some segment formulas I came up with I plugged in my values to get \(\displaystyle y/2=1(1-\cos ((\pi /3)/2))\) which gives me a formula \(\displaystyle \approx\) 0.133975 however this is not exact and I need an exact answer.
 
Last edited:
  • #4
\(\displaystyle \frac{y}{2}=1-\cos30=1-\frac{\sqrt3}{2}=\frac{2-\sqrt3}{2}\implies y=2-\sqrt3\)
 

Related to The height of a section of overlapping circles.

1. What is the height of a section of overlapping circles?

The height of a section of overlapping circles is the vertical distance from the topmost point of the section to the bottommost point of the section.

2. How is the height of a section of overlapping circles calculated?

The height of a section of overlapping circles can be calculated by finding the difference between the radius of the larger circle and the distance between the centers of the two circles.

3. Does the number of overlapping circles affect the height of the section?

Yes, the number of overlapping circles can affect the height of the section. The more circles that overlap, the smaller the height of the section will be.

4. Can the height of a section of overlapping circles be negative?

No, the height of a section of overlapping circles cannot be negative. It is always a positive value representing the vertical distance between the top and bottom points of the section.

5. How does the diameter of the overlapping circles affect the height of the section?

The diameter of the overlapping circles does not directly affect the height of the section. However, it can indirectly impact the height if it changes the distance between the centers of the circles.

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