Math

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Given info:
arc length = 65 m
formula for arc length: s=rø
s = 2pi r
radius of circle: 2300 m

so to find the angle
would I put in 2pi(2300)= rounded to 14451

Then would I multiply: 65(360/14451)

if that is right, why would I divide by 360 and multiply by 65? Could someone explain

  • Math -

    The simplest way to find the angle is to use the formula:
    s=rφ
    where φ is in radians.
    Solve for φ and convert to degrees:
    φ = (s/r) *(180/π)
    = 65/2300 * (180/π)
    = 1.619°

    What is done above is to divide 65m by the circumference of the circle, and the convert the fraction of 360° to degrees. That is why some of the numbers cancel out in the end.

  • Math -

    ok, thank you!

  • Math -

    could you explain the cancellations?

  • Math -

    The circumference is
    2πr = 2π*2300 = 4600π
    the arc length is 65
    arc-length/circumference = φ/360°
    φ=(65/4600π)*360°
    =1.619° (same as the other method).

    Your procedure is to calculate them numerically to get 14451m (rounded) as the circumference.
    then you multiply the fraction
    (65/14451) by 360° (angle of a complete turn).
    Really there is no cancellation required.

    So
    "Then would I multiply: 65(360/14451)"
    is already the correct answer.

    "if that is right, why would I divide by 360 and multiply by 65? Could someone explain "
    No you don't need to divide by 360 and multiply by 65 (unless you want to get back the circumference).

  • Math -

    I'm still a bit confused on how you get from s = rφ to arc length /circumference = φ/360

    s = rφ
    2πr = rφ

  • Math -

    s=rφ where φ is in radians.
    To change from radians to degrees, we multiply by 180°/π, since π radians equal 180°.

    For an angle a whole circle, it is 2π radians or 360°.
    So 2πr is the circumference, where φ=2π. (we know this formula from elementary school, but did not know from where it came).

    So
    s = rφ
    2πr = rφ
    are the one and same formula, the first one for an angle of φ, and the second for φ=2π (or a complete circle).

    Hope I made that clear. If not, feel free to post.

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