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
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.
To find the angle corresponding to a given arc length in a circle, you can use the formula:
arc length (s) = radius (r) × angle (θ)
In your case, you are given the arc length (s) as 65 m and the radius (r) as 2300 m. You want to find the angle (θ).
To find θ, you can rearrange the formula:
θ = s / r
Substituting the given values:
θ = 65 / 2300
Simplifying, you would get the value of θ in radians. However, if you want the angle in degrees, you need to convert it. To convert radians to degrees, multiply by the conversion factor 180/π.
θ (in degrees) = (65 / 2300) × (180/π)
Now, let's calculate this:
θ (in degrees) ≈ (65 / 2300) × (180/3.14159)
≈ 1.59957
So, the angle corresponding to an arc length of 65 m in a circle with a radius of 2300 m is approximately 1.6 degrees.
Please note that your calculation of multiplying 2π and the radius is incorrect. The formula s = 2πr gives the circumference, not the arc length. To find the angle, you need to divide the arc length by the radius, not the circumference.
Dividing by 360 allows you to convert the angle from degrees to radians by multiplying it by (π/180). Multiplying by 65 is the final step to obtain the angle in degrees.