An arc of a circle subtends an angle of 54 at the centre. If the arc is 9cm long, calculate the circumference of the circle.
54° is 3/20 of a full circle. SO, the whole circumference
3/20 c = 9
c = 60
To calculate the circumference of the circle, we need to find the radius (r) of the circle first.
Given that the arc length (s) is 9 cm and it subtends an angle (θ) of 54 degrees at the center of the circle, we can use the formula:
s = rθ
Rearranging the formula to solve for the radius (r):
r = s/θ
Substituting the values:
r = 9 cm / 54 degrees
Now, we need to convert the angle from degrees to radians since the formula requires angles to be in radians.
To convert degrees to radians, we use the conversion factor:
1 radian = π/180 degrees
So, 54 degrees can be expressed as:
54 degrees × (π/180 degrees)
Next, we substitute the angle in radians into the formula to find the radius:
r = 9 cm / (54 degrees × (π/180 degrees))
Simplifying further:
r = 9 cm / (54 × π/180)
r = 9 cm × (180/54) × (1/π)
r = 30 cm / π
Now that we have the radius, we can find the circumference (C) using the formula:
C = 2πr
Substituting the radius back into the formula:
C = 2π × (30 cm / π)
C = 60 cm
Therefore, the circumference of the circle is 60 cm.