Surface Area of a Hemisphere
Last Updated :
18 Apr, 2024
A hemisphere is a 3D shape that is half of a sphere’s volume and surface area. The surface area of a hemisphere comprises both the curved region and the base area combined.
- Hemisphere’s Total Surface Area (TSA) = Curved Surface Area + Base Area = 3πr² square units.
- Curved Surface Area (CSA) = 2πr² square units.
In this article, we will learn in detail about the calculation of volume and surface areas of a hemisphere, its derivation, and some solved examples.
What is a Hemisphere?
A hemisphere is formed when a plane divides a sphere into two equal parts. In other words, a hemisphere is exactly half of a sphere in geometry. It is made up of two parts: “hemi,” meaning half, and “sphere,” which is a three-dimensional mathematical shape for round objects. When a sphere is sliced at the exact centre along its diameter, two equal hemispheres are generated.
Therefore, a hemisphere is a three-dimensional geometric object consisting of half of a sphere with one side flat and the other side shaped like a round bowl.
Sphere and Hemisphere
Hemisphere has a flat bottom surface and a rounded top surface, like a bowl or the inside of a hollow ball that has been cut in half. You can get a hemisphere by slicing a sphere horizontally or vertically through its centre, resulting in two identical halves with the same diameter but different heights. One direction (the slice) will have half the diameter, while the other direction will match the full diameter of the original sphere.
Real-Life Examples of Hemispheres
Examples of hemispheres can be seen in everyday life. For instance, the bowl that we use to eat food from is nothing but a hollow hemisphere. Half-cut shell of the coconut is an example of a hollow hemisphere. When we cut round-shaped fruits like oranges, tamarind, watermelon, etc., the shape of the fruit becomes a solid hemispherical shape. Below shown images are real-life examples of hemispheres.
Real-life Examples of Hemisphere
What is Surface Area of a Hemisphere?
Surface area of a hemisphere is the total surface area of a hemisphere. Hemispherical area is defined by two types of hemispheres, the solid hemisphere, and the hollow hemisphere. Surface area can be found in two ways:
- Curved Surface Area of a Hemisphere (CSA)
- Total Surface Area of a Hemisphere (TSA)
Curved surface area of the hemisphere formula is defined as the area covered by its curved surface. It is equal to half of the total surface area of a sphere. The formula for the curved surface area of the hemisphere equals two times the product of pi and the square of the radius of the hemisphere.
Curved Surface Area of a Hemisphere = 2πr2
where,
- π is constant with the value of 3.14
- r is radius of hemisphere
Derivation of the Formula of Curved Surface Area of a Hemisphere
Curved Surface Area for Hemisphere is the area of all the curved surfaces of the hemisphere which is only half of the spherically curved surface as the base of the hemisphere is a flat surface that is not curved. Thus,
Curved Surface Area of a Hemisphere = 1/2 × (Curved Surface Area of Sphere)
CSA = 1/2 (4πr2)
CSA = 2πr2
Base Area of Hemisphere
Base of hemisphere is in a circular shape, and therefore, the formula for the base area of the hemisphere is equal to the area of a circle.
Base Area of Hemisphere = πr2
Total surface area of a hemisphere is defined as the total space covered by the surface of the hemisphere. The total surface area is given by the sum of its curved surface area and base area. The formula for total surface area equals three times the product of the pi (π) and the square of the radius of the hemisphere.
Total Surface Area of Hemisphere
Total Surface Area of Hemisphere = 3πr2
where,
- π is constant with value of 3.14
- r is radius of hemisphere
Derivation of the Formula of Total Surface Area of a Hemisphere
Total surface area for a hemisphere is the sum of the curved surface area of the hemisphere and the area of its circular base since a hemisphere is just half of a sphere with a circular base. Therefore, the total surface area of a hemisphere can be expressed as:
Total Surface Area of a Hemisphere = Curved Surface Area of Hemisphere + Base Area of Hemisphere
⇒ TSA = 2πr2 + πr2
TSA = 3πr2
Surface area of a hollow hemisphere can be understood by considering its components. A hollow hemisphere possesses two diameters, as the presence of the inner hollow hemisphere introduces a smaller diameter. Observing closely, the surface area of a hollow hemisphere comprises three main parts:
- Curved surface area of Outer Hemisphere
- Curved surface area of Inner Hemisphere
- Area of Remaining Base
Derivation of the Formula Surface Area of a Hollow Hemisphere
Let’s break down the areas in order to obtain the surface area of a hollow hemisphere:
- Crved Surface area of Outer Hemisphere = 2πR2
- Curved Surface area of Inner Hemisphere = 2πr2
- Base Area of Hollow Hemisphere = π(R2 – r2)
Therefore, total surface area of hemisphere = 2πR2 + 2πr2 + π(R2 – r2)
TSA = 2πR2 + 2πr2 + πR2 – πr2
TSA = 3πR2 + πr2
Surface Area of a Hollow Hemisphere
Total Surface Area of a Hollow Hemisphere with Closed Base
Total surface area of a hollow hemisphere with a closed base consists of the following components:
- Curved Surface Area of Larger Hemisphere: It is calculated using the formula for the surface area of a sphere: 2πR 2, where R is the radius of the larger hemisphere.
- Curved Surface Area of the Smaller Hemisphere: Similarly, the curved surface area of the smaller hemisphere is also 2πr2, where r is the radius of the smaller hemisphere.
- Area of Remaining Base: Base area of the hollow hemisphere is the difference between the areas of the bases of the larger and smaller hemispheres. The area of the larger base is 2πR 2, and the area of the smaller base is 2πr2.
Total surface area of the hollow hemisphere is the sum of these three components:
Total Surface Area = 2πR 2 + 2πr2 + (πR 2 − πr2)
= 2πR 2 + 2πr2 + πR 2 − πr2
= 3πR 2 + πr2
This formula accounts for the surface area of both the curved portions and the closed base of the hollow hemisphere.
How to Find Surface Area of a Hemisphere?
The surface area of a hemisphere can be found by following easy steps based on what type of hemisphere is given. If a solid hemisphere is given, the formula of total surface area and curved surface area can be used based on the requirement, and if a hollow hemisphere is given, the formula for a hollow hemisphere must be used. Following are the steps that can be followed to obtain the surface areas based on the requirement.
How to Find Curved Surface Area of a Hemisphere
The formula for the curved surface area of a hemisphere when the given radius is “r” is 2πr2. Below are the steps provided to find the curved surface area of a hemisphere:
- Note down the radius of the hemisphere.
- Put the “r” value in the formula for the curved surface area of a sphere, that is, CSA = 2πr2.
- Present the final answer in square units.
Example: Calculate the curved surface area of a hemisphere radius of 5 m. (Use π = 3.14).
We have,
r = 5
Using the formula we get,
CSA = 2πr2
⇒ CSA = 2 (3.14) (5)2
⇒ CSA = 157 sq. m
How to Find Total Surface Area of a Hemisphere
The formula for the total surface area of a hemisphere when the given radius is “r” is 3πr2. Below are the steps provided to find the curved surface area of a hemisphere:
- Note down the radius of the hemisphere.
- Put the “r” value in the formula for the total surface area of a sphere, that is, TSA = 3πr2.
- Present the final answer in square units.
Example: Calculate the total surface area of a hemisphere diameter of 16cm.
Solution:
We have,
Using the formula we get,
TSA = 3πr2
⇒ TSA = 3 (3.14) (8)2
⇒ TSA = 602.88 sq. cm.
How to Find Surface Area of a Hollow Hemisphere
The formula for the surface area of a hollow hemisphere when the given radius is “r” is 3πR2 + πr2. Below are the steps provided to find the surface area of a hollow hemisphere:
- Note down the radius of the hemisphere.
- Put the “r” value in the formula for the total surface area of a sphere, that is, TSA = 3πR2 + πr2.
- Present the final answer in square units.
Summary of Surface Area of Hemisphere
A hemisphere is also called a semi-sphere or half a sphere. It is formed when a plane divides a sphere into two equal parts. When a sphere is sliced at the exact centre along its diameter, two equal hemispheres are generated. The surface area of a hemisphere, also known as the total surface area of a hemisphere, is the sum of all the areas of its faces, including the curved surface and the base.
The base area of a hemisphere is the area of the flat surface at the bottom of the hemisphere.
- Base Area of Hemisphere = πr2
- Total Surface Area (TSA) of Hemisphere = 3πr2
The surface area of a hollow hemisphere can be calculated using the following formula the sum of the inner surface, an outer surface, and the area of the base which is in the shape of a ring.
- Curved Surface Area of a Hollow Hemisphere = 2π(R2 + r2)
- Total Surface Area of a Hollow Hemisphere = 3πR2 + πr2
Where r is the radius of the hemisphere and in the case of the hollow hemisphere R and r are the outer and inner radius of the hollow hemisphere.
Solved Questions on Surface Area of Hemisphere
Question 1: Calculate the total surface area of a hemisphere radius of 4 m.
Solution:
We have,
r = 4
Using the formula we get,
TSA = 3πr2
⇒ TSA = 3 (3.14) (4)2
⇒ TSA = 150.72 sq. m
Question 2: Calculate the radius of a hemisphere if its total surface area is 200 sq. m.
Solution:
We have,
A = 200
Using the formula we get,
A = 3πr2
⇒ r2 = A/3π
⇒ r2 = 200/3 (3.14)
⇒ r = 4.60 m
Question 3: Calculate the radius of a hemisphere if its total surface area is 350 sq. m.
Solution:
We have,
A = 200
Using the formula we get,
A = 3πr2
⇒ r2 = A/3π
⇒ r2 = 350/3 (3.14)
⇒ r = 6.09 m
Question 4: Calculate the curved surface area of a hemisphere radius of 4 m.
Solution:
We have,
r = 4
Using the formula we get,
CSA = 2πr2
⇒ CSA = 2 (3.14) (4)2
⇒ CSA = 100.48 sq. m
Question 5: Calculate the radius of a hemisphere if its curved surface area is 790 sq. m.
Solution:
We have,
A = 790
Using the formula we get,
A = 2πr2
⇒ r2 = A/2π
⇒ r2 = 790/2 x (3.14)
⇒ r = √125.79 ⇒ 11.22m
Hemisphere Surface Area Practice Questions
High Order Thinking Skills Questions are very important to hone your understanding and conceptual clarity about the topic. Solve the following HOTS questions on Surface Area of a Hemisphere to build conceptual confidence:
Question 1: A dome-shaped building has a radius of 10 meters. The dome is painted on both the inside and the outside. What is the total area that gets painted? How would the paint required change if the radius of the dome was increased by 10%?
Question 2: A company manufactures hemispherical bowls with a radius of 5 cm. If the cost of the material used to make the bowl is proportional to its surface area, how would the cost change if the company decided to manufacture bowls that are 20% larger in radius?
Question 3: A planet is approximately a sphere. If we consider the Northern Hemisphere, what would be the change in its surface area if the radius of the planet increased by 1%?
Question 4: A hemispherical tank with a radius of 2 meters is used to store water. If the tank is expanded by increasing its radius by 50%, how much more water can it store? How does this relate to the change in the surface area of the tank?
Question 5: An igloo is built in the shape of a hemisphere. If the radius of the igloo is doubled, how does this affect the surface area of the igloo? How would this change affect the amount of ice needed to build the igloo?
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