Question

Find the area of the corresponding major sector of a circle with radius 7 cm and angle $120{}^\circ $.

Answer 1

Hint: The angle contained in a circle is $360{}^\circ $ and its area is $\pi {{r}^{2}}$ where $r$ is the radius of the circle. Then find the area for \[1{}^\circ \] which will be $\dfrac{\pi {{r}^{2}}}{360}$. Finally find the area for the sector that contains an angle of $120{}^\circ $. After finding it, subtract it from the total area to get the desired answer.

Complete step-by-step solution -

In the question we have been asked to find the area of the major sector of a circle with a radius of 7 cm and angle $120{}^\circ $. A circular sector of circle sector is the portion of a disk enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger is known as the major sector. Now, we know that the angle in a circle is $2\pi $ or $2\times 180{}^\circ $ or $360{}^\circ $. The area of a circle can be found by using the formula, $\pi {{r}^{2}}$, where $r$ is the radius of the circle and $\pi =\dfrac{22}{7}$. So, the area of the circle with a radius of 7 cm is, $\pi \times {{\left( 7 \right)}^{2}}=\dfrac{22}{7}\times {{7}^{2}}=154\text{c}{{\text{m}}^{2}}$
So, if the central angle is $360{}^\circ $, then the area contained in the circle is $154\text{c}{{\text{m}}^{2}}$.
If the central angle becomes $1{}^\circ $, then the area contained in the circle would become $\dfrac{154}{360}\text{c}{{\text{m}}^{2}}$.
Now the angle given in the question is $120{}^\circ $, so the area of the sector would be, $\dfrac{120}{360}\times 154=\dfrac{154}{3}\text{c}{{\text{m}}^{2}}$.
So, the area corresponding to angle $120{}^\circ $ is $\dfrac{154}{3}\text{c}{{\text{m}}^{2}}$, but it will be considered as the minor sector. We know that the area of the major sector plus the area of the minor sector is equal to the area of the circle. We know that the area of the circle is $154\text{c}{{\text{m}}^{2}}$ and the area of the minor sector is $\dfrac{154}{3}\text{c}{{\text{m}}^{2}}$, so we can find the area of the major sector by subtracting the area of the minor sector from the total area of the circle.
Hence, the area of the corresponding major sector of the circle is $\left( 154-\dfrac{154}{3} \right)c{{m}^{2}}=\dfrac{308}{3}c{{m}^{2}}$.

Note: We can solve this problem using the formula of area of sector with angle $\theta $, which is $\dfrac{\theta }{360}\times \pi {{r}^{2}}$, where $r$ is the radius of the circle. Here students leave the question after finding the value of the minor sector and consider it as an answer which is truly wrong so they need to know the difference between minor sector and major sector.