Question

In figure, AB and CD are two diameters of a circle (with centre O) perpendicular to each other and OD is the diameter of the smaller circle. If \[OA=14\,cm\], find the area of the shaded region.

Answer 1

Hint: We will first calculate the area of the larger semicircle which will be half of the area of the circle that is \[\dfrac{1}{2}\pi {{r}^{2}}\] and then we will calculate the area of the smaller circle with diameter OD. Now after this we will add these two areas and finally we will subtract the area of the triangle BCA to get the area of the shaded portion.

Complete step-by-step answer:
Now from the figure we can see that the radius of the larger circle, \[OA=R=14\,cm.........(1)\]. And diameter of the smaller circle is \[OD=14\,cm.........(2)\]. And hence the radius of the smaller circle will be half of OD. So radius of smaller circle \[=r=7\,cm\].
And in triangle BCA, height is same as radius of larger circle. So the height of the triangle BCA is \[h=14\,cm......(3)\].
And the base in triangle BCA is twice of OA. So \[b=28\,cm........(4)\].
Now we know that the area of the triangle BCA, \[A=\dfrac{1}{2}\times b\times h.....(5)\]
So now substituting the value of h from equation (3) and value of b from equation (4) in equation (5) we get,
\[A=\dfrac{1}{2}\times 28\times 14=196\,{{m}^{2}}.........(6)\]
Now area of larger semicircle with radius 14 cm \[=\dfrac{1}{2}\times \pi \times {{R}^{2}}=\dfrac{1}{2}\times \dfrac{22}{7}\times {{14}^{2}}=308\,c{{m}^{2}}..........(7)\]
And area of smaller circle with radius 7 cm \[=\pi \times {{r}^{2}}=\dfrac{22}{7}\times {{7}^{2}}=154\,c{{m}^{2}}..........(8)\]
Now the area of the shaded region is the area of smaller circle plus area of larger semicircle minus area of triangle BCA. So using this information we will add values from equation (7) and equation (8) and then we will subtract the value from equation (6).
Hence the area of the shaded region \[=308+154-196=266\,c{{m}^{2}}\].

Note: Here we have to remember the formula of the circle and the formula of the triangle. Also we have to keep in mind that the area of the semicircle is half of the area of the circle. We have to see the figure given in the question properly to understand which areas we need to find. We in a hurry can substitute the diameter in place of radius so we need to be careful.