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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.

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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.
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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,

seo images

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.
Last updated date: 18th Sep 2023
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