MHB The height of a section of overlapping circles.

Zekes
Messages
4
Reaction score
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!
 

Attachments

  • rDBQC.png
    rDBQC.png
    3.5 KB · Views: 117
  • kIcDZ.png
    kIcDZ.png
    9 KB · Views: 108
Mathematics news on Phys.org
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?
 
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 $$y/2=1(1-\cos ((\pi /3)/2))$$ which gives me a formula $$\approx$$ 0.133975 however this is not exact and I need an exact answer.
 
Last edited:
$$\frac{y}{2}=1-\cos30=1-\frac{\sqrt3}{2}=\frac{2-\sqrt3}{2}\implies y=2-\sqrt3$$
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
I'm interested to know whether the equation $$1 = 2 - \frac{1}{2 - \frac{1}{2 - \cdots}}$$ is true or not. It can be shown easily that if the continued fraction converges, it cannot converge to anything else than 1. It seems that if the continued fraction converges, the convergence is very slow. The apparent slowness of the convergence makes it difficult to estimate the presence of true convergence numerically. At the moment I don't know whether this converges or not.

Similar threads

Replies
11
Views
7K
Replies
2
Views
2K
Replies
1
Views
1K
Replies
2
Views
4K
Replies
11
Views
3K
Replies
1
Views
2K
Replies
4
Views
2K
Replies
29
Views
4K
Back
Top