The surface area of a 3D shape is the total area of all of the faces. It involves the flat as well as the curved faces. A hemisphere is a three-dimensional form of a semicircle. The total surface area of hemisphere is the area occupied by the curved surface as well as the flat circular surface of the hemisphere. Circular shapes take the shape of a hemisphere when observed as three-dimensional structures, e.g., a bowl or a mushroom head.
Let’s learn how to find the surface area of hemisphere and its methods and formulas.
Surface Area of Hemisphere
The area occupied by the surface/boundary of a hemisphere is known as the surface area of hemisphere. It is always measured in square units. As it has a flat circular base, thus it has a total surface area as well as a curved surface area.
The surface area of hemisphere is measured as the “number of square units” ($cm^{2}$, $m^{2}$, $in^{2}$, $ft^{2}$, etc.). There are two types of surface areas of hemisphere
- Curved Surface Area
- Total Surface Area
Curved Surface Area of Hemisphere Or CSA Of Hemisphere
The curved surface area of hemisphere is the area covered by its curved face(surface). The CSA of a hemisphere is half of the surface area of a sphere which is $4 \pi r^{2}$.
Therefore, the curved surface area of a hemisphere with radius $r$ can be calculated using the following formula
$ \text{Curved surface area of hemisphere} = \frac {1}{2} \left( \text{curved surface area of a sphere} \right) = \frac {1}{2} \left(4 \pi r^{2} \right) = 2 \pi r^{2}$.
where $r$ is the radius of the hemisphere
Total Surface Area of Hemisphere Or TSA Of Hemisphere
The Total Surface Area of hemisphere is the total space occupied by the curved surface and the flat base surface(circular in shape) of the hemisphere. The TSA of a hemisphere is calculated by finding the sum of the areas of its curved surface and base surface.
The total surface area of hemisphere formula is
$\text{Surface area of a hemisphere} = \text{Curved Surface Area} + \text{Base Area} = 2 \pi r^{2} + \pi r^{2} = 3 \pi r^{2}$.
where $r$ is the radius of the hemisphere.
Surface Area of a Hollow Hemisphere
A hollow hemisphere can be considered as made up of two hemispheres, one is called the inner hemisphere and the other one is called the outer hemisphere, thus forming a hemisphere with three surfaces
- Outer Curved Surface Area
- Inner Curved Surface Area (Hollow Area)
- Flat Area (Rim – Two Concentric Circles)
There are two types of surface areas of a hollow hemisphere
- Curved Surface Area Of Hemisphere Or CSA Of Hemisphere
- Total Surface Area Of Hemisphere Or TSA Of Hemisphere
Curved Surface Area of Hollow Hemisphere Or CSA Of Hemisphere
Surface area of a sphere = $4 \pi r^{2}$.
Therefore, for a hollow hemisphere of outer radius $R$ and inner radius $r$, the formula for a curved surface area of a hemisphere is given by
- $\text{Curved Surface Area of Outer Surface} = 2\pi R^{2}$
- $\text{Curved Surface Area of Inner Surface} = 2\pi r^{2}$
Total Surface Area of Hollow Hemisphere Or TSA Of Hemisphere
The total surface area of a hollow hemisphere consists of three parts.
- Outer Curved Surface Area
- Inner Curved Surface Area (Hollow Area)
- Flat Area (Rim – Two Concentric Circles)
The total surface area of the hollow hemisphere = (Curved Surface Area of Outer Surface) + (Curved Surface Area of Inner Surface) + (Surface Area of Flat Ring)
These component areas can be calculated using the following formulas.
Curved Surface Area of Outer Surface = $\pi R^{2}$
Curved Surface Area of Inner Surface = $\pi r^{2}$
Area of the Flat Ring = $\pi \left(R^{2} – r^{2} \right)$
Therefore, total surface area of the hollow hemisphere = $2 \pi R^{2} + 2 \pi r^{2} + \pi \left(R^{2} – r^{2} \right) = 2 \pi R^{2} + 2 \pi r^{2} + \pi R^{2} – \pi r^{2} = 3 \pi R^{2} + \pi r^{2}$.
$\text{Total Surface Area of a Hollow Hemisphere} = 3 \pi R^{2} + \pi r^{2}$.
Examples
Ex 1: Find the curved surface area of a hemisphere whose radius is $3.5 cm$.
Radius of hemisphere $r = 3.5 cm$.
Curved surface area of a hemisphere = $2 \pi r^{2} = 2 \times \frac {22}{7} \times 3.5^{2} = 77 cm^{2}$.
Ex 2: Find the total surface area of a hemisphere whose radius is $10.5 in$.
Radius of hemisphere $r = 10.5 in$.
Total surface area of a hemisphere = $3 \pi r^{2} = 3 \times \frac {22}{7} \times 10.5^{2} = 1039.5 in^{2}$.
Ex 3: Find the surface area of a hollow hemisphere whose outer radius measures $5 cm$ and inner radius is $4.5 cm$.
Outer radius of the hemisphere $R = 5 cm$
Inner radius of the hemisphere $r = 4.5 cm$
Surface area of a hollow hemisphere = $3 \pi R^{2} + \pi r^{2} = 3 \times \frac {22}{7} \times 5^{2} + \frac {22}{7} \times 4.5^{2} = 299.36 cm^{3}$.
Hemisphere – A 3D Solid Shape
The word hemisphere comes from the two words Hemi(Greek hemisus meaning half) and sphere. So hemisphere is a 3D geometric shape that is half of a sphere with one side flat and the other side as a circular bowl. A hemisphere is formed when a sphere is cut at the exact centre along its diameter leaving behind two equal parts. The flat side of the hemisphere is known as the base or the face of the hemisphere.
The important elements of a hemisphere are as follows:
- Radius: The length of the line segment drawn between the centre of the hemisphere to any point on its surface. If $O$ is the centre of the hemisphere and $A$ is any point on its surface, then the distance $OA$ is its radius.
- Diameter: The length of the line segment from one point on the surface of the hemisphere to the other point which is exactly opposite to it, passing through the centre is called the diameter of the hemisphere. The length of the diameter is exactly double the length of the radius.
- Circumference: The length of the great semicircle of the hemisphere is called its circumference.
Properties of a Hemisphere
Since a hemisphere is the exact half of a sphere, a hemisphere and a sphere have quite similar properties. The properties of the hemisphere are
- A hemisphere has a curved surface area.
- Just like a sphere, there are no edges and no vertices in a hemisphere.
- It is not a polyhedron since polyhedrons are made up of polygons, but a hemisphere has one circular base and one curved surface.
- The diameter of a hemisphere is a line segment that passes through the centre and touches the two opposite points on the base of the hemisphere.
- The radius of a hemisphere is a line segment from the centre to a point on the curved surface of the hemisphere.
Practice Problems
- What is the total surface area of a hemisphere whose radius is $5.5$ metres?
- If the curved surface area of the heimsphere is $98.56 cm^{2}$, then find the radius of the hemisphere.
- If the curved surface area of the solid hemisphere is $2772$ sq. cm, then find its total surface area.
- Radii of two solid hemispheres are in the ratio of $2:3$. Find the ratio of their curved surface areas and the ratio of their total surface areas.
FAQs
What is the surface area of a hemisphere?
The surface area of a hemisphere is the sum of areas of the outer curved surface and flat circular surface. The area of an outer curved surface of a hemisphere is $2 \pi r^{2}$ and the area of the flat circular surface is $\pi r^{2}$. Therefore, the surface area of a hemisphere is $3 \pi r^{2}$.
What is the formula for the surface area of a hemisphere?
The formula for the surface area of the hemisphere is $3 \pi r^{2}$, where $r$ is the radius of the hemisphere.
What is the total surface area of a hollow hemisphere?
For a hollow hemisphere, of inner radius (r) and outer radius (R) the total surface area of a hollow hemisphere = outer curved surface area + inner curved surface area + base area.
This can be written as $2 \pi \left(R^{2} + r^{2} \right) + \pi \left(R^{2} – r^{2} \right)$, which on simplification becomes $3 \pi R^{2} + \pi r^{2}$.
Conclusion
The surface area of hemisphere is the area occupied by the curved surface of the hemisphere in a three-dimensional space. In the case of a hemisphere, there are two types of surface area, viz, curved surface area and total surface area.
Recommended Reading
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