A hemisphere is a three-dimensional shape that is exactly half of a sphere. It is formed when a sphere is cut into two equal parts through its center. A hemisphere has one flat circular face and one curved surface.
Properties of a Hemisphere
Some important properties of the Hemisphere are:
- A hemisphere is exactly half of a sphere.
- The radius of a hemisphere is equal to the radius of the sphere.
- It has one flat circular base and one curved surface.
- A hemisphere has no edges and no vertices.
- It is not a polyhedron because it has a curved surface and a circular base.
- The diameter of a hemisphere is a line segment passing through the center and joining two opposite points on the base.
- The radius of a hemisphere is the line segment from the center to any point on its curved surface.
Volume of Hemisphere
The volume of a hemisphere refers to the amount of space it occupies. It is measured in cubic units such as cubic meters (m³), cubic centimeters (cm³), or cubic inches (in³).
Since a hemisphere is half of a sphere, its volume is also half the volume of a sphere. The formula for the volume of a sphere is 4πr³⁄3. Therefore, dividing this by 2 gives the formula for the volume of a hemisphere.
Therefore, the volume of a hemisphere is given by:
Volume of hemisphere = 2πr³⁄3
Where r is the radius of the hemisphere and the value of π is 22/7 or 3.142.
Surface Area of Hemisphere
The formula for the surface area of a hemisphere can be given for both a solid and a hollow hemisphere.
The surface area of a hemisphere is the total area covered by its surfaces. It can be divided into two parts:
Curved Surface Area of a Hemisphere
The curved surface area of a hemisphere is the area of its curved part only. Since a hemisphere is half of a sphere, its curved surface area is half the surface area of a sphere.
Curved surface area of a hemisphere = 2πr²
where r is the radius of the hemisphere and the value of π is 22/7 or 3.142.
Total Surface Area of a Hemisphere
The total surface area of a hemisphere includes both the curved surface and the flat circular base. The base of a hemisphere is a circle.
Total surface area of a hemisphere = Curved surface area + Area of base
= 2πr² + πr² = 3πr²
Where r is the radius of the hemisphere and the value of π is 22/7 or 3.142.
Hollow Hemisphere
A hollow hemisphere is a hemispherical shape that has thickness. It has two curved surfaces—an inner surface and an outer surface. Therefore, it has two radii: the inner radius (r) and the outer radius (R). The thickness of a hollow hemisphere is given by (R − r).
The study of a hollow hemisphere includes:
- Curved Surface Area
- Total Surface Area
- Volume
Curved Surface Area of Hollow Hemisphere
The curved surface area of a hollow hemisphere is the sum of the curved surfaces of the inner and outer hemispheres.
Curved surface area of outer hemisphere = 2πR²
Curved surface area of inner hemisphere = 2πr²
Curved surface area of hollow hemisphere = 2πR² + 2πr² = 2π(R² + r²)
Total Surface Area of Hollow Hemisphere
The total surface area of a hollow hemisphere includes:
- Curved surface area of the inner hemisphere
- Curved surface area of the outer hemisphere
- Area of the circular ring at the base
Area of circular ring = π(R² − r²)
Total Surface Area of Hollow Hemisphere = Curved Surface Area of Hollow + Area of Circular Ring
= 2π(R² + r²) + π(R² − r²) = 3πR² + πr²
Volume of Hollow Hemisphere
The volume of a hollow hemisphere is found by subtracting the volume of the inner hemisphere from the volume of the outer hemisphere.
Volume of hollow hemisphere = 2⁄3 π(R³ − r³)
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Solved Examples of Hemisphere Formula
Example 1: Determine the total surface area for a hemisphere having a radius of 25 cm.
Solution:
Total surface area of hemisphere = 3πr2
Given Radius = 25 cm
Total surface area of hemisphere = 3πr2
= 3 × 22/7 × 25 × 25
= 41250/7
= 5892.85 cm2Thus, the total surface area of hemisphere = 5892.85 cm2
Example 2: Determine the volume of a hemisphere having a radius of 22.6 units.
Solution:
Volume of hemisphere = 2/3πr3
Given Radius = 22.6 inch
Volume of hemisphere = 2/3πr3
= 2/3 × 3.14 × 22.6 × 22.6 × 22.6
= 72,491.14/3= 24,163.71 inch3
Thus, the volume of hemisphere is 24,163.71 inch3
Example 3: The radius of a hemisphere is 15.4 cm. Determine its curved surface area.
Solution:
Curved surface area of hemisphere = 2πr2
Given Radius = 15.4 cm
Curved surface area of hemisphere = 2πr2
= 2 × 22/7 × 15.4 ×15.4
= 10,435.04/7
= 1,490.72 cm2Thus, the Curved surface area of hemisphere is 1,490.72 cm2