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