Surface Area of Hemisphere – Formula, and Examples

Total Surface Area & Lateral Surface Area of Hemisphere in 3-D Plane

Definition:

The surface area of a solid refers to the area covered by the outer surface of the three-dimensional figure. The surface area is broadly classified into the Total Surface Area (TSA) and Lateral Surface Area (LSA).
When it comes to TSA, it is the entire area covered by the outside surface of the solid, whereas LSA is the area covered by the outer surface of the solid excluding the area of the bases.

Hemisphere is a solid which is obtained when a sphere is divided into two halves. A hemisphere has two kinds of surface areas that are TSA and LSA. The curved surface area of a hemisphere is half of the curved surface area of a sphere.

Formula:

The total surface area of hemisphere = 3 \pi r^{2}
The curved surface area of hemisphere = 2 \pi r^{2}

Calculation:

For calculating the TSA and LSA, different observations are made concerning the type of surface of the three-dimensional figure. We have two types of surfaces in the hemisphere: a curved surface and a circular surface at the base. While calculating the total surface area of a hemisphere, we consider the sum of the area of the curved surface and the flat surface. Whereas, in the case of CSA of a hemisphere, we only consider the curved surface area of the hemisphere.

In case of a hemisphere with radius ‘r’, we can now formulate,

TSA of hemisphere = area of the curved surface + area of the circular base

⇒ TSA = \frac{1}{2}x curved surface area of sphere + \pi r^{2}

⇒ TSA = \frac{1}{2}x 4 \pi r^{2}+ \pi r^{2}

⇒ TSA = 2 \pi \mathrm{r}^{2}+\pi \mathrm{r}^{2}

\therefore T S A=3 \pi r^{2}

CSA of hemisphere = \frac{1}{2} x curved surface area of a sphere

\Rightarrow \mathrm{CSA}=\frac{1}{2}\times 4 \pi \mathrm{r}^{2} 

\therefore \mathrm{CSA}=2 \pi r^{2}

Solved Examples:

Q1) Calculate the curved surface area of a hemisphere with a radius of 7cm?

A:  Curved surface area of hemisphere with radius 7 \mathrm{~cm}=2 \times \pi \times(7)^{2}=308 \mathrm{~cm}^{2}

Q2) Calculate the total surface area of a hemisphere with a radius of 14cm?

A: Total surface area of a hemisphere with radius 14 \mathrm{~cm}=3 \times \pi \times(14)^{2}=1848 \mathrm{~cm}^{2}.

Q3) What is the radius of a hemisphere with TSA 462 \mathrm{~cm}^{2}? 

A: TSA of hemisphere =3 \pi r^{2}

\Rightarrow \mathrm{r}=\sqrt{49}=7\mathrm{~cm}

APPLICATIONS:

1. The surface area helps us determine the quantity of paint for various hemispherical buildings, monuments, etc.
2. It is used for determining the size of labels to be put on different products.
3. It also finds use in determining the amount of wrap needed for covering hemispherical surfaces.