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Thread: s= r x theta

  1. #1
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    s= r x theta

    Ok, i am confusing myself over something that i thought was really basic and i understood!
    Is the arc length formula, s= r*theta , for theta in radians a derived result or a definition?
    And is there a very clear way of seeing what it is? Does it come from a definition of 1 radian? If so, how does it develop into the arc length formula!
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  2. #2
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    Re: s= r x theta

    Quote Originally Posted by rodders View Post
    Ok, i am confusing myself over something that i thought was really basic and i understood!
    Is the arc length formula, s= r*theta , for theta in radians a derived result or a definition?
    And is there a very clear way of seeing what it is? Does it come from a definition of 1 radian? If so, how does it develop into the arc length formula!
    The definition of 1 radian is the measure of the central angle that subtends an arc of length equal to the radius.
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  3. #3
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    Re: s= r x theta

    The circumference of a circle with radius $r$ is $2\pi r$

    there are $2\pi$ radians in a circle.

    Thus there is an arc length of $\dfrac{\theta}{2\pi} \cdot (2 \pi r) = r \theta$

    The important definition is that there are $2\pi$ radians in a circle.

    So you would say that 1 radian corresponds to the angular measure where the subtended arc length equals the radius.
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  4. #4
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    Re: s= r x theta

    Ah ok..

    I guess S = rxtheta is equivalent to C= rx 2pi ? so when theta=2pi , S=C
    Is a correct way of looking at it too?
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  5. #5
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    Re: s= r x theta

    Yes, it is. For the entire circle, the "arc-length" is just the circumference and the angle, in radians, is $2\pi$. "$S= r\theta$" becomes "$C= 2\pi r$"
    Thanks from rodders
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