Posted by Amy~ on Tuesday, May 24, 2011 at 9:22pm.
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.
ok, thank you!
could you explain the cancellations?
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).
I'm still a bit confused on how you get from s = rφ to arc length /circumference = φ/360
s = rφ
2πr = rφ
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.