Question

The measure of an arc of a circle is $ 90^\circ $ and its length is $ 11{\rm{ cm}} $ . What is the diameter of the circle?
A. $ 21{\rm{ cm}} $
B. $ 7{\rm{ cm}} $
C. $ 14{\rm{ cm}} $
D. $ 28{\rm{ cm}} $

Answer 1

Hint: We know that the circumference of the circle is equivalent to the multiplication of diameter and the ratio of arc angle and 360. Using this we will obtain the radius of the circle. Then using the formula that diameter is twice the radius we obtain our result.

Complete step-by-step answer:
The arc angle of a circle is $ 90^\circ $ .
The length is given as $ 11{\text{ cm}} $ .
The length is equal to the circumference of the circle.

The formula to find the circumference of the circle is equal to the pi times diameter and the ratio of arc angle and $ 360 $ . The formula for diameter is equal to twice the radius. Since, we know that the formula for circumference of a circle is,
 $ \Rightarrow {\text{circumference of arc}} = 2\pi r \times \dfrac{{{\text{arc angle}}}}{{360}} $
Substituting the values of arc angle and circumference of the circle we obtain,
 $ \Rightarrow {\text{circumference of arc}} = 2 \times \pi \times r \times \dfrac{{90}}{{360}} $ .
Now, we will substitute the value of circumference of the circle in the above equation.
 $
11 = \dfrac{{90}}{{360}} \times \pi \times d\\
11 = \dfrac{{90}}{{360}} \times 3.14 \times d\\
44 = 3.14 \times d\\
\Rightarrow d = \dfrac{{44}}{{3.14}}
 $
On further solving the above value we will get the diameter of the circle which is,
  $ 14.01{\text{ cm}} $
The diameter of the circle is approximately equal to $ 14.01{\text{ cm}} $ .
Therefore, the diameter of the circle for the given problem is $ 14{\text{ cm}} $ .
So, the correct answer is “Option C”.

Note: In this question, always be careful with the arc angle while taking in the arc length or circumference of the circle because the arc angle is the ratio of arc angle and the total angle which is $ 360^\circ $ . Students generally get confused between the formula of circumference of a circle or area of a circle, so make sure to choose the correct formula.