Why does a circles angles equal up to 2pi? Wouldn't it be 1?

In summary, the conversation discusses the concept of radians and their relationship to degrees in measuring angles in a circle. The number of radians in a full circle is 2π, and this unit is used as a natural way to divide up a circle. The term "subtended" is defined as the arc on the circumference between the two lines that make up an angle at the center of a circle. This term is often used in the context of measuring angles in radians.
  • #1
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Why does a circles angles equal up to 2pi? Wouldn't it be 1pi? Please help.

My geometry teacher (last year - and now he's gone) drew a circle for me and defined all the points in terms of pi (is there an ASCII character for pi, by-the-way, on the keyboard?). Where the 90 degree point would be he put 1/2 pi, were the 180 degree point was he put just 'pi', at 270 he put 3/2 pi, and 360 he put 2 pi. What the heck?! I thought pi was the ratio of the diameter to the circumference. If that is true then shouldn't it be: at 90 degrees 1/4 pi, at 180 deg. 1/2 pi, 270 deg. 3/4 pi, and 360 deg. just 'pi'? Can you explain this or direct me to a site that can? Please, I'm going crazy trying to figure it out.
 
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  • #2
It's 2Pi radians. One radian is equal to 57.2958 degrees.
Therefore, 2pi radians would be 2 x 57.2958 x 3.1416 = 360 degrees
 
  • #3


Originally posted by Cyberice
My geometry teacher (last year - and now he's gone) drew a circle for me and defined all the points in terms of pi (is there an ASCII character for pi, by-the-way, on the keyboard?). Where the 90 degree point would be he put 1/2 pi, were the 180 degree point was he put just 'pi', at 270 he put 3/2 pi, and 360 he put 2 pi. What the heck?! I thought pi was the ratio of the diameter to the circumference. If that is true then shouldn't it be: at 90 degrees 1/4 pi, at 180 deg. 1/2 pi, 270 deg. 3/4 pi, and 360 deg. just 'pi'? Can you explain this or direct me to a site that can? Please, I'm going crazy trying to figure it out.

What your geometry teacher was doing was marking out the circles in Radians.

1 Radian is the angle that an arc the same length as the radius of the circle will make.

And since, as you pointed out, the ratio of diameter to circumference is [pi], and the radius is 1/2 the diameter, there are 2[pi] radians to a circle.

Thus a quarter of the way around the circle is 1/2 [pi] radians, half way [pi] radians, three quarters, 1 1/2 [pi] radians and all the way around 2[pi] radians.

Radians are the natural way to divide up a circle. ( the number of degrees were just an arbitary choice)
 
  • #4
Thank you! I was wondering about that and it really bothered me. BTW, would you happen to know any sites that could explain it also (just for the heck of it)?
 
  • #5
Janus' reply made me ask myself why are there 2 Pi radians in a full circle (360 degrees)?

My answer is so that the angle times the radius gives the arc length.

Doesn't that mean 2 Pi radians is a calculation convenience?
 
  • #6
The number 2π is forced on us, it is the circumfrence divided by the radius. Now, Janus says this in his post but I will repeat it. Take the radius, mark that length off on the arc of the circumference, the angle defined is 1 radian. This definition means that there must be 2π radians in a circle. Repeat that to yourself until it starts to make sense.

Ah..., no, I really do NOT miss Donde!
 
  • #7
To add, the use of this natural units system makes a lot of calculations possible. Eg. calculus with trignometric functions, which is generally based on the result that: x -> 0 : sinx -> x, provided x is in radians.
 
  • #8
Using a numberline helped me out tremendously.
 
  • #9
Euler is the first person who started to use radian in trigonometry. The circumference of a unit circle is 2pi, therefore he defined 360 degrees be 2pi radian.

A radian is the measure of an angle subtended at the centre of a circle by an arc equal in length to the radian.
 
  • #10
(URGENT) What does subtended mean?

[?] [?] [?]

What does subtended mean? I forgot and I have a test tomorrow
 
  • #11
Subtended means spanned on the circumference between the two lines making the angle. So the arc subtended by an angle at the center of a circle means you draw the two radii that make the angle at the center of the circle, and then the smaller arc of the circumference between those two radii is the arc subtended.
 
  • #12
Thank-you:smile: It makes it clearer although I already had my test
 

1. Why does a circle's angles equal up to 2pi?

This is because the circle is a two-dimensional shape and has no sides or edges. Therefore, it can be divided into an infinite number of lines that form angles, all of which add up to 360 degrees or 2pi radians.

2. How does 2pi relate to radians?

Radians are a unit of measurement for angles, just like degrees. However, unlike degrees which are based on a system of 360, radians are based on a system of 2pi. This is because 2pi radians is equivalent to a full rotation or 360 degrees.

3. Why is 2pi used instead of 1 for circle angles?

Using 2pi as the unit for measuring circle angles is more convenient and consistent for mathematical calculations. It is also closely related to the concept of radians, which is commonly used in trigonometry and calculus.

4. Can a circle's angles equal anything other than 2pi?

No, a circle's angles will always add up to 2pi, regardless of its size or radius. This is a fundamental property of circles and is a result of their symmetry and lack of edges or corners.

5. How does the concept of pi relate to a circle's angles?

The number pi (π) is defined as the ratio of a circle's circumference to its diameter. Since a circle's circumference is equal to 2pi times its radius, pi can also be seen as the ratio of the circle's circumference to its radius. This ratio is used in many calculations involving circles, including measuring angles.

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