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 $${\displaystyle \frac{\pi }{180}}$$. 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 $$(2\pi -\theta )$$ at the center of the circle is the major arc.

Here, as we can see, $$(2\pi -\theta )>\theta$$.

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 $${\displaystyle \frac{\pi }{180}}$$,
So, we get $${30}^o=30\times {\displaystyle \frac{\pi }{180}}\text{radians}$$
By simplifying, we get
$${30}^o={\displaystyle \frac{\pi }{6}}\text{radians}$$
Now, to find the length of arc subtended by $${\displaystyle \frac{\pi }{6}}$$ angle,
We will put $$\theta ={\displaystyle \frac{\pi }{6}}$$ and R = 4 cm in equation (i), so we get,
Length of the arc made by $${\displaystyle \frac{\pi }{6}}$$ radian angle $$={\displaystyle \frac{\pi }{6}}\times 4cm$$
$$={\displaystyle \frac{2\pi }{3}}cm$$
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 $${\displaystyle \frac{2\pi }{3}}cm$$.

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 $$={\displaystyle \frac{R\theta \pi }{{180}^o}}$$ when $$\theta$$ is in degrees.
Also, students must properly read if the major or minor arc is asked in question and use $$\theta$$ accordingly.