Intuition for arc length, angle and radius formula

In summary, the arc length of a circle is equal to the product of the central angle and the radius, as stated by the formula s = r * theta. This can be understood by recognizing that 1 radian is defined as the central angle that subtends an arc of length equal to the radius, and therefore any other central angle can be expressed in terms of the radius and 1 radian.
  • #1
gokuls
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I don't understand the intuition/proof of why arc length = arc angle * radius. This may be partially because I don't fully understand the concept of radians, but anyhow please help.
 
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  • #2
gokuls said:
I don't understand the intuition/proof of why arc length = arc angle * radius. This may be partially because I don't fully understand the concept of radians, but anyhow please help.


Well, what's the arc length of a complete circle? Just [itex]\,2\pi r\,\,,\,r=\,[/itex] the circle's radius, and since [itex]\,2\pi\,\, radians=360^\circ[/itex] , we

just "adjust" the angle of the arc...

DonAntonio
 
  • #3
gokuls said:
I don't understand the intuition/proof of why arc length = arc angle * radius. This may be partially because I don't fully understand the concept of radians, but anyhow please help.

You have the definition of a radian: 1 radian is measure of the central angle of a circle that subtends an arc of length equal to the radius of the circle.

Now say you have an arc of length ##s## subtended by a central angle ##\theta## measured in radians. Then, since the arc length is proportional to the central angle and a central angle of ##\theta= 1## radian subtends an arc of length ##r##,$$
\frac s \theta = \frac r 1$$which gives ##s = r\theta##.
 
  • #4
LCKurtz said:
You have the definition of a radian: 1 radian is measure of the central angle of a circle that subtends an arc of length equal to the radius of the circle.
To help you (the OP, not LCKurtz) understand this a bit better, sketch a unit circle with its center at the origin. Draw a radius out from the center to the point (1, 0). Draw another radius from the center so that the radius makes an angle of 60° with the x-axis. Connect the two points where the two radii intersect the circle. You should have an equilateral triangle, with all three sides of length 1 and all three angles of measure 60°.
The length along the circle between the two intersection points is a bit larger than 1, since the straight line distance is equal to 1.

If you move the 2nd radius slightly clockwise the right amount, you'll get an arc length that is exactly 1. The central angle between the first radius and the repositioned radius will be a few degrees less than 60°, approximately 57.3°. That's the degree measure of 1 radian.
LCKurtz said:
Now say you have an arc of length ##s## subtended by a central angle ##\theta## measured in radians. Then, since the arc length is proportional to the central angle and a central angle of ##\theta= 1## radian subtends an arc of length ##r##,$$
\frac s \theta = \frac r 1$$which gives ##s = r\theta##.
 
  • #5


The intuition behind the formula for arc length is based on the relationship between the angle and the radius of a circle. To understand this, we first need to understand the concept of radians.

Radians are a unit of measurement for angles, just like degrees. However, instead of dividing a circle into 360 equal parts (degrees), radians divide a circle into 2π (approximately 6.28) equal parts. This means that one radian is equal to the angle subtended by an arc that is equal in length to the radius of the circle.

Now, let's consider a circle with a radius of r and an arc with an angle of θ. The arc length, denoted by s, is the distance along the circumference of the circle between the two endpoints of the arc. We can see that as the angle θ increases, the arc length s also increases.

Using the concept of radians, we can see that the arc length s is equal to the radius r multiplied by the angle θ in radians. This is because as mentioned earlier, one radian is equal to the length of the radius. So if we have an angle of 1 radian, the corresponding arc length will also be equal to 1 radius. Therefore, for any given angle in radians, the arc length will be equal to the radius multiplied by that angle.

In formula form, this can be written as s = r * θ. And since we know that the angle θ is measured in radians, we can rewrite it as s = r * (θ in radians).

Therefore, the formula for arc length s = r * θ is derived from the relationship between the angle and radius of a circle, and the concept of radians. I hope this explanation helps to clarify the intuition behind this formula.
 

What is the formula for finding the arc length?

The formula for finding the arc length is L = rθ, where L is the arc length, r is the radius of the circle, and θ is the central angle in radians.

How do I find the central angle if I know the arc length and radius?

To find the central angle, you can rearrange the arc length formula to θ = L/r. Plug in the values for the arc length and radius, and you will get the central angle in radians.

What is the formula for finding the angle of a sector in degrees?

The formula for finding the angle of a sector in degrees is θ = (L/r) * 180/π, where L is the arc length, r is the radius, and π is the constant pi (approximately 3.14).

How do I calculate the radius if I know the arc length and central angle?

To calculate the radius, you can rearrange the arc length formula to r = L/θ. Plug in the values for the arc length and central angle, and you will get the radius of the circle.

Can I use these formulas for any type of circle?

Yes, these formulas can be used for any type of circle, as long as you use the corresponding units (radians or degrees) for the central angle and arc length.

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