Is it possible to calculate this geometrical relationship between circles?

[Sameh soliman]

TL;DR Summary

Is it possible to calculate that geometrical relationship

A large cirlcle with radius 50 m contains a smaller circle with radius 7.4 m that is tangent to its surface internally. Is it possible to calculate what number of the small circle the larger circle can contain iside it in which all are tangent to its surface ... but without using trig. Functions

 

[PAllen]

Why do you not want to use trig functions? They are ideal for circles. Do you also want to play tennis with a baseball instead of tennis ball?

 

[anuttarasammyak]

A prescription is : Draw two lines from the center of large circle of radius R so that they are tangible to the small circle of radius r. How much is the angle between the lines?

With trigonometric, it is $$2\ sin^{-1}\ \frac{r}{R-r}$$

Similar but another approach is, thinking N-regular polygon inside and tangent to the large circle, find N that satisfies
$$sin \frac{\pi}{N+1}< \frac{r}{R-r} < sin \frac{\pi}{N}$$.
Mostly we can replace the most LHS and RHS by their first term of Taylor expansion
$$\frac{\pi}{N+1}< \frac{r}{R-r} < \frac{\pi}{N}$$
where no trigonometric appear. For OP's case the middle term is 0.1737... so
$$N<18.08...<N+1$$
So N=18. I hope Taylor expansion approximation works at least for the OP's case.

 

[Sameh soliman]

A prescription is : Draw two lines from the center of large circle of radius R so that they are tangible to the small circle of radius r. How much is the angle between the lines?

With trigonometric, it is $$2\ sin^{-1}\ \frac{r}{R-r}$$

Similar but another approach is, thinking N-regular polygon inside and tangent to the large circle, find N that satisfies
$$sin \frac{\pi}{N+1}< \frac{r}{R-r} < sin \frac{\pi}{N}$$.
Mostly we can replace the most LHS and RHS by their first term of Taylor expansion
$$\frac{\pi}{N+1}< \frac{r}{R-r} < \frac{\pi}{N}$$
where no trigonometric appear. For OP's case the middle term is 0.1737... so
$$N<18.08...<N+1$$
So N=18. I hope Taylor expansion approximation works at least for the OP's case.

Thanks for the great work... but is that the closest that we can get ?.. i mean it's the right answer because N is equal 18 but what if in another case N has a fraction ... then by this method we can never know it's accurate value

 

[anuttarasammyak]

N is a number of sides of polygon, so is an integer, with no fraction. If you are keen on accuracy, estimate second and higher terms of Taylor expansion or, as the straightest way , use trigonometry.

 

[Sameh soliman]

N is a number of sides of polygon, so is an integer, with no fraction. If you are keen on accuracy, estimate second and higher terms of Taylor expansion or, as the straightest way , use trigonometry.

I meant by N has a fraction that of number of circles not the number of polygon sides ... anyway thanks for the great work

 

[mathman]

A different approach: Draw a circle, (A) concentric with the big circle, through the center of the small circle. How many small circles can fit tangent to each other centered on A? A very good approximation is $\frac{42.6\pi }{7.4}=18.08$. It can be made slightly more accurate by estimating a straight line distance between centers.

 

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