# If the circumference of a circle is 64 cm what is the length of an arc that measures 30 degrees .

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circumference = 64 cm

We know that the formula for the circumference is:

C = 2*r*pi

==> 64 = 2*pi*r

To find r , we will divide by 2*pi

==> r= 64 / 2*pi = 32/ pi

Then the radius of the circle is:

r = 32/pi

Now we need to find the length of the arc such that the angle = 30 degrees.

Then we now that the arc length is:

a = r*2*pi * 30/ 360

= r*pi*30/ 180

= r*pi / 6

= ( 32/pi) * pi / 6

= 32/ 6 = 16/ 3

** ==> The arc length is 16/3 units**

The circumference of a circle is given by the formula 2pi*r .

The actual length of the cicumference = 64.

Therefore 2pir = 64 cm.

An angle of 30 degrees = pi/6 radians.

Therefore the length of an arc which subtends an angle of x radians at the centre should measure x*radius = xr.

Therefore an arc subtending pi/6 radians should measure pi*r/6.

But 2pir = 64 cm given.

Divide both sides by 12 :

2pir/12 = 64/12.

Therefore pir/6 = 64/12 cm = 5 1/3 cm

Therefore an arc angle of 30 degrees of a circle whose circumference is 64 measure 5 1/3 cm or 5.33cm nearly.

We'll write the formula of the circumference of a circle:

C = 2pi*r

Now, we'll substitute the circumference by the given value:

64 = 2pi*r

We'll divide by 2 both sides:

32 = pi*r

We'll divide by pi:

r = 32/pi

Now, we'll write the length of the arc of 30 degrees:

l = 30*r*pi/180

We'll divide by 30:

l = r*pi/6

l = (32/pi)*(pi/6)

l = 16/3

l = 5.(3)