Question

Find the length of the arc in terms of \[\pi \] that subtends an angle of \[{{30}^{o}}\] at the center of a circle of radius 4 cm.

Answer 1

Hint: First of all convert \[{{30}^{o}}\] in radians by multiplying it by \[\dfrac{\pi }{180}\]. Then use the formula for the length of arc which subtends angle at the center, \[\theta =R\theta \] where R is the radius of the circle and \[\theta \] is in radians.

Here, we are given a circle of radius 4 cm in which an arc subtends an angle of \[{{30}^{o}}\] at the center. We have to find the length of this arc in terms of \[\pi \].

First of all, we must know that an arc of a circle is a portion of the circumference of the circle. The length of the arc is simply the length of its portion of circumference. Also, the circumference itself can be considered a full circle arc length.

We have two types of arc between any two points in a circle. One is the major arc and other is the minor arc. We know that the total angle at the center of the circle is \[2\pi \]. Here, the shorter arc made by AB which subtends angle \[\theta \] at the center of the circle is a minor arc. Then, the longer arc made by AB which subtends angle \[\left( 2\pi -\theta \right)\] at the center of the circle is the major arc.

Here, as we can see, \[\left( 2\pi -\theta \right)>\theta \].

Now, we know the circumference of the circle \[=2\pi R\], where R is the radius of the circle.
We know that circumference is also an arc of full circle arc length. Therefore, we can say that the length of the arc made by \[2\pi \] angle = \[2\pi R\].

By dividing \[2\pi \] on both sides, we get the length of the arc made by unit angle = R.
By multiplying \[\theta \] [in radian] on both sides, we get, length of the arc made by \[\theta \] angle \[=R\theta ....\left( i \right)\].
Now, we have to find the length of the arc which subtends an angle of \[{{30}^{o}}\] at the center.

We can show it diagrammatically as follows,

Here, we have to find the length of minor arc AB. As we have to find the length in terms of \[\pi \], we will first convert \[{{30}^{o}}\] into radians.

We know that, to convert any angle from degree to radians, we must multiply it by \[\dfrac{\pi }{180}\],
So, we get \[{{30}^{o}}=30\times \dfrac{\pi }{180}\text{radians}\]
By simplifying, we get
\[{{30}^{o}}=\dfrac{\pi }{6}\text{radians}\]
Now, to find the length of arc subtended by \[\dfrac{\pi }{6}\] angle,
We will put \[\theta =\dfrac{\pi }{6}\] and R = 4 cm in equation (i), so we get,
Length of the arc made by \[\dfrac{\pi }{6}\] radian angle \[=\dfrac{\pi }{6}\times 4cm\]
\[=\dfrac{2\pi }{3}cm\]
Therefore, we get the length of the arc which subtends an angle of \[{{30}^{o}}\] at the center of the circle in terms of \[\pi \] as \[\dfrac{2\pi }{3}cm\].

Note: Students can also find the length of the arc directly by remembering the following formulas.
Length of arc = \[R\theta \] when \[\theta \] is in radian and length of arc \[=\dfrac{R\theta \pi }{{{180}^{o}}}\] when \[\theta \] is in degrees.
Also, students must properly read if the major or minor arc is asked in question and use \[\theta \] accordingly.