Surface Area of a Hemisphere

A hemisphere is an interesting three-dimensional figure with many real-life applications. It is half of the other most common figure of the universe, a sphere. A hemisphere possesses many important and worth-mentioning properties, which help in solving various theoretical and real-life problems. One of these properties is the surface area of a hemisphere, which we will be exploring here. 

1.0What is a Hemisphere?

The word hemisphere is a compound of the Greek words “hemi”, meaning half, and “sphere”, meaning a ball, which ultimately means a half ball. It can be visualised as the upper or lower half of a sphere cut by a plane passing through its centre. The shape is a symmetrical figure with two components, which include: 

• Curved Surface – This is the outer portion of the hemisphere, creating the dome-shaped part.
• Flat Circular Base – The flat, circular surface that is the base of the hemisphere.

The radius (r) of the hemisphere is the distance from the centre of the flat circular base to any point on the curved surface. The surface area of a hemisphere encompasses both the curved surface and the circular base. See the following diagram of a hemisphere to get a better understanding of these components. 

2.0Surface Area of a Hemisphere Formula

The surface area of the hemisphere is a function of the radius of the hemisphere and consists of two parts: the curved surface area (CSA) and the total surface area (TSA). These equations of surface area of a hemisphere are derived from the geometry of circles and spheres.

Curved Surface Area (CSA) of a Hemisphere 

The curved surface area of a hemisphere represents the surface area of the hemisphere without its base. In other words, it is the area of the outer dome-like structure of the hemisphere, which doesn’t include its flat circular base. The formula for the curved surface area can be mathematically represented as: 

$\text{ Curved Surface Area of a Hemisphere }=2\pi r^2$

Here, r is the radius of the hemisphere. By the formula, you may notice that it is half of the surface area of a sphere; this is because a hemisphere is essentially half of a sphere. 

Total Surface Area (TSA) of a Hemisphere:

The curved surface area and the area of the circular flat base together are the total surface area of the hemisphere. The base is a mere circle, and its area can be found from the formula of the area of a circle, which is r2. This additional area gives the following formula for the hemisphere: 

$\text{ Total Surface Area of Hemisphere }=3\pi r^2$

Here, note that the total surface area of a hemisphere accounts for both the external curved surface and the flat bottom of the hemisphere.

3.0How to Calculate the Surface Area of a Hemisphere: 

Although it possesses an easy formula to calculate the surface area of a hemisphere, it is required to have a complete understanding to use the above-mentioned formulas: 

• Identify the Radius: The first step for solving questions related to the surface area of the hemisphere is to identify the radius of the given hemisphere. Convert this radius to the units required in the problem, if any.  
• Use the Required Formula: Use the formula as per the requirement of the question. It is interesting to note that in a hollow hemisphere, the curved surface area is always used, while the total surface area is used for a solid hemisphere. 

Units of the Surface Area of a Hemisphere: 

The units of the surface area of a hemisphere are always the square of the units of the radius in a given hemisphere. For instance, if the radius is in centimetres (cm), the surface area will be in square centimetres (cm²).

4.0Solved Examples of Surface Area of a Hemisphere: 

Problem 1: Find the radius of a hemisphere with a curved surface area of 154 cm².

Solution: Given that the curved surface area of the hemisphere is 154 cm² 

$\begin{array}{c}\text{ Curved Surface Area of a Hemisphere }=2\pi r^2\\ 154=2\pi r^2\\ 154=2\times \frac{22}{7}\times r^2\\ r^2=\frac{154\times 7}{2\times 22}=\frac{49}{2}\\ r=\sqrt{\frac{49}{2}}=\frac{7}{\sqrt{2}}\mathrm{cm}\end{array}$

Problem 2: Find the total surface area of a hemisphere where the diameter is 14 cm.

Solution: Given that, 

Diameter of hemisphere = 14cm 

Radius of hemisphere = 7 cm 

$\begin{array}{c}\text{ Total Surflace Area of Hemisphere }=3\pi r^2\\ \text{ TSA of Hemisphere }=3\times \frac{22}{7}\times 14\\ \text{ TSA of Hemisphere }=3\times 22\times 2\\ \text{ TSA of Hemisphere }=132{\mathrm{cm}}^2\end{array}$

Problem 3: A construction company is building a dome-shaped roof for a new sports stadium. The roof is in the shape of a hemisphere, and the radius of the dome is 21 meters. The company needs to cover the entire outer surface of the dome with material. What is the total surface area of the hemisphere roof that needs to be covered with material? Also, find the total cost of this material, such that it costs Rs 50 per m². 

Solution: Given that the dome of sports will be covered from the outside only, we will calculate the curved surface of this dome: 

$\begin{array}{c}\text{ Curved Surface Area of a Hemisphere }=2\pi r^2\\ \text{ Curved Surface Area of the dome }=2\times \frac{22}{7}\times (21{)}^2\\ \text{ Curved Surface Area of the dome }=2\times 22\times 21\times 3\\ \text{ Curved Surface Area of the dome }=2,772{\mathrm{m}}^2\end{array}$

Now, the cost of covering this dome will be $=2,772\times 50=\text{ Rs }1,38,600$