Question

How will you find the length of an arc of a circle with radius \[12\] cm if the arc subtends a central angle of \[30\] degrees?

Answer 1

Hint:The above problem is based on the concept to find arc length of a circle. So here we need to convert \[30\] degrees to a radian measure which is \[2\pi = 360\] degree and in the formula \[l = r \times \theta \] substitute it. In this formula mentioned \[l\] stands for length of an arc and we need to perform required calculations and find the value of \[l\] in cm.

Formula used:
The formula to find the length of an arc of circle is
\[l = \dfrac{\theta }{{{{360}^ \circ }}} \times 2\pi r\]
Where \[\theta \] stands for angles subtended at the center and \[2\pi r\] is the circumference of the circle.

Complete step by step answer:
In the question asked we are asked to find the length of an arc of circle of radius \[12\] cm which subtends an angle of \[30\] degrees. Since we know that,
\[l = \dfrac{\theta }{{{{360}^ \circ }}} \times 2\pi r\]
We can calculate the circumference of circle as
\[2\pi r = (2) \times (3.14) \times (12) \\
\Rightarrow 75.36cm \\ \]
Lastly to find the length of arc and after further simplification we will get
\[l = \dfrac{\theta }{{{{360}^ \circ }}} \times 2\pi r \\
\Rightarrow l = \dfrac{{30}}{{{{360}^ \circ }}} \times 75.36 \\
\Rightarrow l = \dfrac{{75.36}}{{12}} \\
\therefore l = 6.28cm \\ \]
Hence, the length of an arc of circle of radius \[12\] cm subtending an angle of \[30\] degrees is \[6.28cm\].

Note:Keep in mind that arc length is a fractional part of the circumference. While solving such types of problems try making the required calculations in the equation for getting a final solution. Remember not to make any calculation mistakes due to sign conventions and converting degree into radians. Lastly, the value of \[\pi \] as \[3.14\] is always preferred. We can also write the answer for a given problem as \[\dfrac{{12\pi }}{3}\].A portion of circumference of a circle is an arc and the length of an arc is basically the length of its portion of circumference.