What is the formula for calculating the circumference of a circle on a sphere?

In summary, the circumference of a circle of radius r, whether it is on a sphere or not, is 2\pi r. This can be seen by fixing one end of a string with length r at a point in 2d-space and making a full circle with the other end. If the circumference is different from 2\pi r, it indicates that the 2d-space is curved intrinsically. When r == PI*R, the circumference created by the circle is zero. This can be calculated using the formula 2\pi R sin(r/R). The formula also shows that if r > R, the circumference will be imaginary.
  • #1
jobyts
227
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What's the circumference of a circle of radius r, on a sphere of radius R?
 
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  • #2
The circumference of a circle of radius r is [itex]2\pi r[/itex], whether it is on a sphere or not.
 
  • #3
HallsofIvy said:
The circumference of a circle of radius r is [itex]2\pi r[/itex], whether it is on a sphere or not.

The following i the context I was talking about:
From https://www.physicsforums.com/showthread.php?t=311787 post #6,
Imagine you fix one end of a string with the length r at a point in 2d-space, and make a full circle with the other end. If you then find that the circumference of that circle is different from 2*PI*r, you conclude that the 2d-space is curved intrinsically.

If r == PI*R, the circumference created by the string will be zero.
 
  • #4
So you aren't talking about radius in the usual sense- you are having the "radius" bending around the sphere. And the "center" of the circle is also on the sphere, not inside it.

We can, without loss of generality, assume that the "center" of the circle is at the top of the circle, the "north pole". Draw a line from the north pole to the center of the sphere, then to a point on the circle. Let the angle made be [itex]\theta[/itex] ([itex]\theta[/itex] is the "co-latitude"). Dropping a perpendicular from the point on the circle to the line from north pole to center of sphere, we have a right triangle with angle [itex]\theta[/itex], hypotenuse of length R, and "opposite side" the (real!) radius of the circle, which I will call r' since we are using r for the "phony" radius. Then we have [itex]sin(\theta)= r'/R[/itex] or [itex]r'= R sin(/theta)[/itex].

Now, the circumference is "really" [itex]2\pi r'= 2\pi R sin(\theta)[/itex]. We only need to calculate [itex]\theta[/itex] in terms of r. The spherical "distance" is measured along a great circle and a whole great circle on a sphere of radius R has circumference [itex]2\piR[/itex] and corresponds to an angle of [itex]2\pi[/itex]. We can set up the proportion
[tex]\frac{2\pi}{2\pi R}= \frac{1}{R}= \frac{\theta}{r}[/tex]
so [itex]\theta= r/R[/itex]

That means our circumference formula becomes
[tex]2\pi r'= 2\pi R sin(\theta)= 2\pi R sin(r/R)[/itex]
 
  • #5
HallsofIvy said:
That means our circumference formula becomes
[tex]2\pi r'= 2\pi R sin(\theta)= 2\pi R sin(r/R)[/itex]

If you input values such that r > R, would you get an answer that is imaginary?
 
  • #6
HallsofIvy said:
That means our circumference formula becomes
[tex]2\pi r'= 2\pi R sin(\theta)= 2\pi R sin(r/R)[/itex]

Did I miss something?

If r == PI*R, shouldn't the circumference created by the circle be zero?
 
  • #7
It should, and it is, according to that formula. ([itex]\sin \pi = 0[/itex].)
 

FAQ: What is the formula for calculating the circumference of a circle on a sphere?

What is the formula for finding the circumference of a circle?

The formula for finding the circumference of a circle is C = 2πr, where C represents the circumference, π is a mathematical constant approximately equal to 3.14, and r is the radius of the circle.

How do you find the circumference of a circle if you only know the diameter?

If you only know the diameter of a circle, you can find the circumference by using the formula C = πd, where C represents the circumference and d is the diameter.

Can the circumference of a circle be negative?

No, the circumference of a circle cannot be negative. It is a measurement of the distance around the circle, and distance cannot be negative.

What units are used to measure the circumference of a circle?

The circumference of a circle is typically measured in units of length, such as inches, centimeters, or meters. The units used will depend on the size of the circle and the desired level of precision.

Why is the circumference of a circle important in geometry?

The circumference of a circle is important in geometry because it is used to calculate other properties of circles, such as area and volume. It is also a fundamental concept in many geometric theorems and formulas.

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