Radius of a circle?

[Gareth56]

I often read in books of the radius of a circle being expressed in units of degrees. What does this mean? I'm used to writing and measuring the radii of circles in units of length but not angular units.

 

For example what does a circle of radius 22deg mean and how would you determine the radius of a circle in degrees?

 

Thanks

G56

[ecoli]

There are 360 degrees in a circle.

 

if you split a circle into 4 pieces, by starting from the origin (the center) and drawing 4 radii up, down, left and right. the angle formed between each of these perpendicular lines is 90 degrees. you can divide the circle up further, created smaller and smaller angles. By definition, one degree is the angle formed between two lines when you have 360 evenly spaced radii dissecting the circle.

 

[Gareth56]

But that method doesn't give you any sense of the magnitude of the radius of the circle. For example a circle with radius 20m is far larger than a circle with a radius of 20cm.

 

I'm still not clear on how does a radius given in degrees denote the magnitude of the radius.

 

As an example from the astronomy magazine I've just read. It's about Sun Haloes. It explains a haloe as being a "bright circle of apparent radius of 22deg." To me that description doesn't give any description of how large the haloe is.

[Klaynos]

I don't know the answer. But I shall guess that in the case of the astronomy magazine might be the area of the sky taken up by the circle so if you measure a 22deg arc of the sky from the earths surface... 22deg does seem really rather large though!

[ecoli]

But that method doesn't give you any sense of the magnitude of the radius of the circle. For example a circle with radius 20m is far larger than a circle with a radius of 20cm.

 

I'm still not clear on how does a radius given in degrees denote the magnitude of the radius.

 

As an example from the astronomy magazine I've just read. It's about Sun Haloes. It explains a haloe as being a "bright circle of apparent radius of 22deg." To me that description doesn't give any description of how large the haloe is.

 

oh, my apologies. I didn't realize the question you were asking.

 

Perhaps it makes use of the Taylor series?

 

Using trigonometry, you can relate the opposite side of the right triangle (a section of the circumference) to the angle with the adjacent side (the radius) (this is the tangent of a right triangle). If you use a really small angle, and integrate, you can approximate the circumference of a section of circle with just knowing the angle and radius.

 

Does that make sense to you (I don't know how much calculus you have)

 

for example, in this image: http://upload.wikimedia.org/wikipedia/en/8/83/Circle_center_a_b_radius_r.svg

 

Using such a large angle, the opposite angle doesn't tell you anything about the circumference. But if you use a smaller angle, the opposite side comes much closer to the circumference length. If you take a bunch of consecutive small angles and add up all the opposite sides you calculate from the tangent of the angle (knowing the radius, you can solve for this easily). This will give you an approximation of the circumpherence.

[Cap'n Refsmmat]

Gun accuracy is often measured in minutes of arc (1/60th of a degree). It's the same idea. The angle represents the angle between imaginary lines drawn from the observer to both sides of the object being measured.

[Gareth56]

Like this?

 

See attached file.

Radius.doc

[ecoli]

Like this?

 

See attached file.

Yes, though you'd have to bisect the angle, so you can get a right triangle you can do some trigonometry with.

[Gareth56]

So in the above example the radius of that circle would have been 10deg and a diameter of 20deg?

[ecoli]

So in the above example the radius of that circle would have been 10deg and a diameter of 20deg?

Perhaps...

 

I should have mentioned, that this is only speculation on my part. I've never actually head of this method being used to calculate the radius of a star. I was just proposing a way in one could use the angle to measure a radius (or a circumference).

 

I'm not sure if this is the context in which whatever paper you read, is referring to.

[Klaynos]

It's quite common in astrophysics for stars and other things to be given in the amount of sky of which they take up in angle...

[ecoli]

It's quite common in astrophysics for stars and other things to be given in the amount of sky of which they take up in angle...

But the radius of the star to be expressed as an angle? Surely that must be some trig-related calculus, no?

[Klaynos]

you need to know how far away the star is aswell, which is ofter harder, so giving a radius is (well until quite recently) impossible just guesswork.