Question

In the figure, PQ and AB are respectively the arcs of two concentric circles of radii 7 cm and 3.5 cm and center O. If \[\angle POQ = 30^\circ \], then what is the area of the shaded region?

Use \[\pi = \dfrac{{22}}{7}\]

Answer 1

Hint: The area of a sector of a circle of angle x° is given by \[\dfrac{{x^\circ }}{{360^\circ }}\pi {r^2}\]. Find the area of the sector OPQ and the sector OAB and find the difference between them to get the required area.

Complete step-by-step answer:

A sector of a circle is the portion of the circle joined by two radii and an arc.

The sector where the area is smaller is called the minor sector and the sector where the area is larger is called the major sector.

The arc usually subtends an angle at the center of the circle that is called the angle of the sector.

The area of the sector that subtends an angle of x° is given as follows:

\[A = \dfrac{{x^\circ }}{{360^\circ }}\pi {r^2}..........(1)\]

The area of the shaded region can be found by the difference in the area of the sectors OPQ and OAB.

The area of the sector OAB whose radius is 3.5 cm and angle is 30° is given using equation (1) as follows:

\[{A_1} = \dfrac{{30^\circ }}{{360^\circ }} \times \dfrac{{22}}{7}{(3.5)^2}\]

Simplifying, we get:

\[{A_1} = \dfrac{1}{{12}} \times \dfrac{{22}}{2}(3.5)\]

\[{A_1} = \dfrac{{77}}{{24}}c{m^2}................(2)\]

The area of the sector OPQ whose radius is 7 cm and angle is 30° is given using equation (1) as follows:

\[{A_2} = \dfrac{{30^\circ }}{{360^\circ }} \times \dfrac{{22}}{7}{(7)^2}\]

Simplifying, we get:

\[{A_2} = \dfrac{1}{{12}} \times 22 \times 7\]

\[{A_2} = \dfrac{{77}}{6}c{m^2}................(3)\]

The area of the shaded region is calculated using equation (2) and equation (3).

\[A = {A_2} - {A_1}\]

\[A = \dfrac{{77}}{6} - \dfrac{{77}}{{24}}\]

Simplifying, we get:

\[A = \dfrac{{77 \times 4 - 77}}{{24}}\]

\[A = \dfrac{{77 \times 3}}{{24}}\]

\[A = \dfrac{{77}}{8}\]

\[A = 9.625c{m^2}\]

Hence, the area of the shaded region is 9.625 sq. cm.

Note: You can also construct the entire circle, find the area in between the concentric circle and then calculate the area of the shaded region then we will get the same answer. Students should remember the formula of area of the sector of a circle subtended by angle and should know definitions of the sector, area of the sector and also properties of an area of sector and arc length subtended by the angle which is given by arc length $l=r \times \theta$