Question

The curved surface area of a hemisphere with radius ‘r’ is ________

Answer 1

Hint: First of all, try to recollect all the areas related to the hemisphere. Now we know that the hemisphere is half of the sphere and the curved surface area of the hemisphere is only the area of its curved part or outer surface. So, we get the surface area of the sphere and its half would be the curved surface area of the hemisphere.

Complete step-by-step answer:

In this question, we have to find the curved surface area of a hemisphere of radius “r”. Let us first see the different areas for the hemisphere. We know that a hemisphere is formed when the plane cuts the sphere into two equal halves or we can also say that a sphere is a combination of the two hemispheres. The hemisphere of radius “r” is as shown below.

Now let us see the different areas of the hemisphere.
Total Surface Area: The area of the curved surface and the area of the circle (base) is called the total surface area.
Curved Surface Area: The area of the outer surface of the hemisphere is called the curved surface area.
From the surface area of a sphere, we can easily calculate the surface area of the hemisphere. Since the hemisphere is half of the sphere. So we get,
CSA of the hemisphere = \[\dfrac{1}{2}\] surface area of the sphere
\[CSA=\left( \dfrac{1}{2} \right)4\pi {{r}^{2}}\]
\[CSA=2\pi {{r}^{2}}\]
So we get the curved surface area of a hemisphere as \[2\pi {{r}^{2}}\] square units.

Note: In this question, many students make this mistake of taking the half of the total surface area of the sphere which is \[2\pi {{r}^{2}}\] and considering it as the total surface area of the hemisphere which is wrong. They must note that when we cut a sphere in two hemispheres, then apart from the curved part of the hemisphere, we get one more part and that is the circular base whose area is \[\pi {{r}^{2}}\]. So we get the total surface area of a hemisphere as \[3\pi {{r}^{2}}\] whereas curved surface area of a hemisphere as \[2\pi {{r}^{2}}\].