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March 28, 2015

Posted by **Amy~** on Tuesday, May 24, 2011 at 9:22pm.

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 -
**MathMate**, Tuesday, May 24, 2011 at 9:53pmThe 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 -
**Amy~**, Tuesday, May 24, 2011 at 9:57pmok, thank you!

- Math -
**Amy~**, Tuesday, May 24, 2011 at 9:59pmcould you explain the cancellations?

- Math -
**MathMate**, Tuesday, May 24, 2011 at 10:15pmThe 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 -
**Amy~**, Tuesday, May 24, 2011 at 10:29pmI'm still a bit confused on how you get from s = rφ to arc length /circumference = φ/360

s = rφ

2πr = rφ

- Math -
**MathMate**, Wednesday, May 25, 2011 at 12:04ams=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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