Question 314082
The circumference of a circle is equal to 2*pi*r


The measurement of the arc of a circle is equal to the degree of the arc divided by 360 times the circumference of the circle.


Since the circumference of the circle is equal to 2 * pi * r, this means that the measurement of the are of the circle is equal to the degree of the arc divided by 360 times 2 * pi * r.


If you let L = the length of the arc and D equal the degree of the arc, then the equations you should be working with are:


L = D/360 * C which is equivalent to L = D/360 * 2 * pi * r


Then all you have to do is solve for r.


In your first problem, you are given:

What is the radius if: 
degree of measurement of arc=30 
length=1/3xy(pi) 


The equation you have to work with is L = D/360 * 2 * pi * r


Substituting in this equation, you get:


1/3 * x * y * pi = 30/360 * 2 * pi * r


Simplify this equation to get:


1/3 * x * y * pi = 1/6 * pi * r


Divide both sides of this equation by pi to get:


1/3 * x * y = 1/6 * r


Multiply both sides of this equation by 6 to get:


2 * x * y = r


That's your answer.


To confirm, substitute for r in the equation for the circumference of the circle to get:


C = 2 * pi * r = 2 * pi * 2 * x * y = 4 * pi * x * y


Since L = D/360 * C, then substitute for C to get:


L = D/360 * 4 * pi * x * y


Substitute for D to get:


L = 30/360 * 4 * pi * x * y


Simplify to get:


L = 4/12 * pi * x * y which simplifies further to L = 1/3 * pi * x * y


Since this is the length of the arc you started with, then you are good.


In your second problem, you are given:

degree of measurement of arc=40
length=8/9(t)(pi)


The equations you need to work with are L = D/360 * C and L = D/360 * 2 * pi * r.


To find the radius, use L = D/360 * 2 * pi * r


Substitute for L to get:


8/9 * t * pi = 40/360 * 2 * pi * r


Simplify to getr:


8/9 * t * pi = 2/9 * pi * r


Divide both sides of this equation by pi to get:


8/9 * t = 2/9 * r


Multiply both sides of this equation by 9/2 to get:


4*t = r


That should be your answer.


If r = 4*t then C = 2 * pi * r = 2 * pi * 4 * t = 8 * pi * t


Since L = D/360 * C, substitute 40 for D and 8 * pi * t for C to get:


L = 40/360 * 8 * t * pi which simplifies to 8/9 * t * pi.


Since this is the length of the arc you started with, then you are good.


In your third problem, you are given:


degree of measurement of arc=18
length=6(y)pi


Use the formula L = D/360 * 2 * pi * r to get:


6 * y * pi = 18/360 * 2 * pi * r


Simplify to get:


6 * y * pi = 1/20 * 2 * pi * r which simplifies further to get:


6 * y * pi = 1/10 * pi * r


Divide both sides of this equation by pi to get:


6 * y = 1/10 * r


Multiply both sides of this equation by 10 to get:


60 * y = r


That's your answer.


Since C = 2 * pi * r, then you get:


C = 2 * pi * 60 * y which simplfies to C = 120 * pi * y


Since L = D/360 * C, then you get:


L = 18/360 * 120 * pi * y which simplifies to L = 120/20 * pi * y which simplifies further to:


L = 6 * pi * y.


Since this is the length of the arc you started with, they you are good.


The key to solving this problem is to use the equations:


C = 2 * pi * r


L = D/360 * C


C is equal to the circumference of the circle.
L is equal to the length of the arc
r is equal to the radius of the circle.
D is equal to the degree of the arc.