Question

Find the total surface area of a solid hemisphere of radius $r$

Answer 1

Step 1: Defining a hemisphere

A hemisphere is a three-dimensional form created by cutting a sphere along a plane that passes through its center. A hemisphere, in other terms, is half of a sphere. It is possible for the hemisphere to be hollow or solid. A hemisphere’s surface area is measured in square units.

A solid body surface area is the area of all of the surfaces together and is usually measured in squared units.

Step 2: Calculate the total surface area of a hollow hemisphere

Let $r$ be the radius of the hemisphere.

We know that the total surface are of a sphere with radius $r$ is $4{\mathrm{\pi r}}^2$

A hollow hemisphere is half of a sphere. Thus its total surface area will be, $\frac{4{\mathrm{\pi r}}^2}{2}=2{\mathrm{\pi r}}^2$

Step 3: Calculate the total surface area of a solid hemisphere

If you have a solid hemisphere, then it will have a base that is a circle of radius $r$

The area of a circle of radius, $r$ is ${\mathrm{\pi r}}^2$

The total surface area of the solid hemisphere$=$total surface area of hollow hemisphere$+$area of the base circle

Thus, the total surface area of the solid hemisphere$=2{\mathrm{\pi r}}^2+{\mathrm{\pi r}}^2=3{\mathrm{\pi r}}^2$

Hence the total surface area of the hollow hemisphere is $2{\mathrm{\pi r}}^2$ and that of a solid hemisphere is $3{\mathrm{\pi r}}^2$.