Total Surface Area of Hemisphere
Total Surface Area of Hemisphere
A sphere is a three-dimensional object where every point on its outer surface is at an equal distance from its center. When a plane intersects the sphere exactly through its center, it divides the sphere into two congruent halves, each called a hemisphere. By joining these two hemispheres together, a complete sphere is formed.
Area of the Hemisphere
We can easily determine the surface area of a hemisphere due to the circular nature of the sphere's base. The surface area of a hemisphere can be divided into two main types: the total surface area and the curved surface area. These two categories differ in what they encompass.
Area of Curved Surface
The curved surface area is the area of the hemisphere's outer
surface.
Surface Area Total
The total surface area of a hemisphere includes both the area of its curved surface and the area of its circular base. Since a hemisphere is half of a sphere, we can start by finding the curved surface area (CSA) of the hemisphere.
The curved surface area of a sphere is given by the formula:
CSA of sphere = 4πr2
Since the hemisphere is half of the sphere, its curved surface area is:
CSA of hemisphere = (1/2) × 4πr2 = 2πr2
Thus, the curved surface area of a hemisphere is:
Curved surface area = 2πr2 square units
Next, we need to include the area of the circular base. The area of a circle is given by the formula:
Area of base = πr2
Therefore, the total surface area (TSA) of the hemisphere, which includes both the curved surface area and the base area, is:
TSA = 2πr2 + πr2 = 3πr2
In summary:
Total surface area of a hemisphere = 3πr2 square units
Here, π (pi) is a constant approximately equal to 3.14, and r represents the radius of the hemisphere.
Surface Area of a Hemisphere Example
To calculate the surface area of a hemisphere with a radius of 4 cm:
Curved Surface Area (CSA):
Formula: CSA = 2πr2
Substitute the radius (r = 4 cm):
CSA = 2 × 3.14 × 4 × 4
CSA = 3.14 × 32
CSA = 100.48 square centimeters
Total Surface Area (TSA):
Formula: TSA = 3πr2
Substitute the radius (r = 4 cm):
TSA = 3 × 3.14 × 4 × 4
TSA = 3.14 × 48
TSA = 150.72 square centimeters
Therefore, the curved surface area of the hemisphere is 100.48 square centimeters, and the total surface area is 150.72 square centimeters.
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Frequently Asked Questions on Total Surface Area of Hemisphere
The surface area of a hemisphere has two parts - the curved surface area (CSA) and the total surface area (TSA). The CSA is 2πr^2, while the TSA is 3πr^2, where r is the radius of the hemisphere. The volume of a hemisphere is (2/3)πr^3.
The curved surface area (CSA) of a hemisphere is the area of the curved outer surface, and it is calculated as 2πr^2. The total surface area (TSA) includes the CSA plus the area of the circular base, which is calculated as 3πr^2.
The formula for the total surface area (TSA) of a hemisphere is 3πr^2, where r is the radius of the hemisphere. This includes the curved surface area (2πr^2) plus the area of the circular base (πr^2).
The total surface area (TSA) of a hemisphere is 3πr^2 because it includes both the curved surface area (2πr^2) and the area of the circular base (πr^2). The TSA represents the complete outer surface area of the hemisphere.
The total surface area (TSA) of a cylinder is the sum of the curved surface area (2πrh) and the area of the two circular bases (2πr^2).
The TSA of a hemisphere is 3πr^2, where r is the radius. This includes the curved surface area (2πr^2) and the area of the circular base (πr^2).