Question

The total surface area of a hemisphere of a radius r is given by
(a)$2\pi {{r}^{2}}$
(b) $\pi {{r}^{2}}$
(c) $3\pi {{r}^{2}}$
(d) $4\pi {{r}^{2}}$

Answer 1

Hint: For solving this above problem, we should know the formula for curved surface area of a hemisphere. This is given by $2\pi {{r}^{2}}$. Here, r is the radius of the hemisphere. We then add this to the area of the cross section of the bottom surface to get the total surface area.

Complete step-by-step answer:

Before we begin to find the total surface area of the hemisphere, we first try to understand about the basics of total surface area. Basically, total surface area means the summation of all the areas of the surfaces of a solid shape that one can see. Thus, for example, in case of a cube, one can see that there are six square faces. Thus, total surface area of the cube would be the summation of the area of all those square faces. Thus, in the case of cube, total surface area would be –
${{a}^{2}}+{{a}^{2}}+{{a}^{2}}+{{a}^{2}}+{{a}^{2}}+{{a}^{2}}=6{{a}^{2}}$
Where, a is the side length of a cube. Now, with the basic terminology of the cube, we begin to solve for the total surface area of the hemisphere. We have,
Curved Surface area of a hemisphere is $2\pi {{r}^{2}}$ (where r is the radius of the hemisphere). However, this doesn’t include the area of the bottom circular area. Area of the circular face is given by $\pi {{r}^{2}}$. Thus, the total surface area is given by-
Total surface area = Curved Surface area + Area of cross section
$2\pi {{r}^{2}}$+$\pi {{r}^{2}}=3\pi {{r}^{2}}$
Hence, the correct answer is (c) $3\pi {{r}^{2}}$.

Note: Another way to solve this question is knowing the value of surface area of the sphere. This is given by $4\pi {{r}^{2}}$. Since, the hemisphere is basically half of a sphere, we can divide this by 2 to get $2\pi {{r}^{2}}$. This would give us the curved surface. We then add the cross-section area ($\pi {{r}^{2}}$) to get the total surface area ($2\pi {{r}^{2}}$+$\pi {{r}^{2}}=3\pi {{r}^{2}}$).