Question

A piece of wire $22cm$ long is bent into the form of an arc of a circle subtending an angle of ${60^ \circ }$ at its centre. Find the radius of the circle. $\left[ {{\text{Use }}\pi = \dfrac{{22}}{7}} \right]$

Answer 1

Hint:
According to the definition of an arc, the length of an arc can be calculated by the product of radius and angle subtended by it on centre. Now convert the given angle from ${60^ \circ }$ degrees to radian. Substitute the values of angle and the given arc length. Solve for the unknown value of radius.

Complete step by step solution:
Here in this problem, we have a piece of wire of length $22{\text{cm}}$ that is bent in the form of an arc of a circle that subtends an angle of ${60^ \circ }$ at its centre point. With this information, we need to find the radius of the circle made by the arc.
As we know that an arc is a portion of the circumference of the circle and if we join the endpoints of an arc to the centre of that circle, then it subtends an angle which is less than ${360^ \circ }$ .
And the length of an arc can be given by the product of the radius of the circle whose part it is and the angle it subtends at the centre in radians. For an arc of the length $'l'$ , radius $'r'$ and the angle $'\theta '$ in radians, we have the relation:
$ \Rightarrow l = r \times \theta $
According to the question, the angle subtended by the arc is ${60^ \circ }$ . Now we need to convert this angle from degrees to radian. We know that the measure of $\pi $ radians is equal to ${180^ \circ }$. So, we can find the measure of the required angle as follows:
$ \Rightarrow {180^ \circ } = \pi \Rightarrow {1^ \circ } = \dfrac{\pi }{{{{180}^ \circ }}} \Rightarrow {1^ \circ } \times 60 = \dfrac{\pi }{{{{180}^ \circ }}} \times 60 \Rightarrow {60^ \circ } = \dfrac{\pi }{3}$
Now we can use the above relation to relate the arc length, radius and angle
$ \Rightarrow 22 = r \times \dfrac{\pi }{3} \Rightarrow r = \dfrac{{22 \times 3}}{\pi }$
We also have given $\pi = \dfrac{{22}}{7}$ . Now we can use it to find the value of the radius
$ \Rightarrow r = \dfrac{{22 \times 3}}{\pi } = \dfrac{{22 \times 3}}{{\dfrac{{22}}{7}}} = \dfrac{{22}}{{22}} \times 7 \times 3 = 21$

Therefore, we get the radius of the circle is $21{\text{ cm}}$.

Note:
In questions like this, we need to utilize fundamental definitions and formulas. Notice that an arc length can also be found with an angle and area or perimeter. An arc length for an angle of ${\text{18}}{{\text{0}}^{\text{o}}}{\text{ or }}\pi $ will be the perimeter of semi-circle and arc length for an angle of $2\pi {\text{ or 3}}{60^ \circ }$ will become the formula for the perimeter of a circle.