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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.
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.
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.
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°.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.
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##.
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.
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.
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).
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.
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.