Question

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

Answer 1

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.

Answer 2

ok, thank you!

Answer 3

could you explain the cancellations?

Answer 4

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).

Answer 5

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

s = rφ
2πr = rφ

Answer 6

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.