Question

Is the surface area of a hemisphere half the surface area of a sphere or not?

Answer 1

Hint: Surface area of the hemisphere and sphere is simply its outer area. There are two types of surface areas, one is total surface area and the other one is curved surface area. Although the hemisphere is half of the sphere visually we’ll see what is the relation between their surface areas.

Formulae used:
The surface area of a sphere $ = 4\pi {r^2}$
The total surface area of the hemisphere $ = 3\pi {r^2}$
The curved surface area of the hemisphere $ = 2\pi {r^2}$

Complete step by step answer:
Given a hemisphere and a sphere with the same radius, say $r$ , we have to tell if the surface area of the hemisphere is half of the surface area of the sphere.
Since a sphere has only a curved surface, therefore, the curved surface area of a sphere is the same as the total surface area of the sphere.
And, we know that, the surface of a sphere $ = 4\pi {r^2}$ .
Now, if we cut a sphere from the middle, we will obtain a hemisphere. ‘Hemi’ simply means ‘half’.
Then, half the surface area of the sphere will give the curved surface area of the hemisphere i.e., $2\pi {r^2}$ .
And, the total surface area will be the curved surface area and the surface area of the bottom part, which is a circle.
And, we know that, area of a circle $ = \pi {r^2}$ .
Then, the total surface area of the hemisphere will become $2\pi {r^2} + \pi {r^2} = 3\pi {r^2}$ .
So, the total surface area of a hemisphere is not half of the surface area of a sphere.
Hence, only the curved surface area of a hemisphere is half of the surface area of a sphere.

Note:Sphere is a three-dimensional solid object, so when we cut it into half then the bottom of the hemisphere obtained will be a circle and not hollow.
Every three-dimensional object has two types of surface areas i.e., curved surface area and total surface area, except for sphere.
For simplicity, take examples from real life, so that you can imagine it better. An example of the sphere is a ball and an example for a hemisphere is a bowl.