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Volume of a Hemisphere refers to the capacity of a hemisphere. A hemisphere is the precise half of a sphere geometrically. As a result, a hemisphere's volume is half that of a sphere. The volume of a hemisphere is given by the formula (2/3)πr3 cubic units, where ‘r’ refers to the radius of the hemisphere and π is a constant whose value is equal to 22/7 or 3.14 approximately. The northern and southern halves of the Earth, the left and right halves of our brain, a bowl, and an igloo are all examples of hemispheres in real life. The volume of a hemisphere, like its area, is an important mathematical calculation that aids in the solution of many mensuration problems.
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Key Terms: Hemisphere, Sphere, Three-Dimensional Shapes, Coordinates, Volume, Area, Three-Dimensional Geometry, Radius
Definition of Hemisphere
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A hemisphere is formed when a plane cuts across the sphere at the centre or in equal portions. "Hemi" means half, and a sphere is defined as a set of three-dimensional points, with all points on the surface being equidistant from the centre.
In geometry, a hemisphere is a three-dimensional shape formed by splitting a sphere into two equal half, each with one flat side. In real life, we come across a variety of hemisphere-shaped objects. For example, cutting around a watermelon in half yields a hemisphere-shaped watermelon, and cutting a cherry in half yields a hemisphere.
Hemisphere
Read More: Introduction to Three-Dimensional Geometry
Examples of Hemisphere
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- An igloo is a well-known example of a hemispheric object in everyday life. The curved face of the hemisphere is formed by the top of the igloo, while the flat face is formed by the base or the ground.
- The majority of the bowls are hemispheric in shape.
- A curved surface and a flat base can be seen on the mushroom cap if you look attentively. As a result, a mushroom is a classic example of a daily hemisphere-shaped item.
- A hat's shape is similar to that of a hemisphere since it has a curved top and a flat base.
- The earcups of a headphone are an excellent illustration of hemispheric items seen in our environment.
Volume of Hemisphere
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The total capacity of a hemisphere is the number of unit cubes covered inside that area, and the volume of a hemisphere is the number of unit cubes covered inside that space. The volume of a hemisphere is stated in cubic units, such as m3, cm3, in3, and so on.
The volume of a hemisphere equals the half the volume of a sphere, thus, it is expressed as,
Volume of Hemisphere = (2/3)πr3
Where
- ‘r’ refers to the radius of the hemisphere
- ‘π’ is a constant whose value is equal to 22/7 or 3.14 approximately.
Example: Calculate the volume of a hemisphere whose radius is 6 cm.
Solution: Radius (r) = 6 cm
Volume of Hemisphere = (2/3)πr3
= 2/3 × 3.14 × 6 × 6 × 6
= 452.16 cm3
Volume of Hemisphere
Read More: Surface Area of A Hemisphere
Equation of Hemisphere
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Let the distance from the centre i.e. the radius as R
x2 + y2 + z2 = R2
The equation of a hemisphere with the radius “R” at the point (x0, y0, z0) is written as
(x-x0)2 + (y-y0)2 + (z-z0)2 = R2
Then the spherical coordinates of the hemisphere will be
- x = r cos θ sin ∅
- y = r sin θ cos ∅
- z = r cos ∅
Read More: Coordinate Geometry
Things to Remember
- A hemisphere is a three-dimensional shape formed by splitting a sphere into two equal halves.
- The volume of a hemisphere is half the volume of a sphere or we can say two hemispheres are combined to form a sphere.
- The volume of the hemisphere is given as (2/3)πr3 cubic units, where ‘r’ denotes the radius of the hemisphere and π is equal to 22/7 or 3.14 approximately.
- The spherical coordinates of the hemisphere are x = r cos θ sin ∅, y = r sin θ cos ∅ and z = r cos ∅.
Sample Questions
Ques. Find the volume of a hemisphere having a radius of 5 cm? (3 Marks)
Ans. The radius of the hemisphere = 5 cm
Volume of a hemisphere = V = (2/3)πr3
= 2 × 3.14 × 5 × 5 × 5 / 3
= 261.67 cubic cm
Hence the volume of given hemisphere is 261.67 cubic cm.
Ques. The circumference of a hemispherical bowl's edge is 132 cm. Determine the capacity of the bowl. (3 Marks)
Ans. Circumference of the bowl = 132 cm
Circumference = 2πr
2πr = 132 cm
r = 132/2π
r = 21 cm
Now, Volume of Hemispherical bowl = V = (2/3)πr3
V = 2/3 × 22/7 × (21)3
= 19404 cm3
Hence the capacity of the bowl is 19404 cm3
Ques. What is the relation between the volume of a sphere and the volume of a hemisphere? (3 Marks)
Ans. A three-dimensional solid's volume is the amount of space it takes up. Cubic units are used to measure volume i.e. cm3, m3, etc. Before computing the volume, it should be made sure that all of the measurements are in the same unit.
Volume of Sphere = 4/3πr3
A hemisphere's volume is half of the volume of the associated sphere.
Volume of Hemisphere = 2/3πr3
Ques. Define Hemisphere in three-dimensional geometry. (2 Marks)
Ans. In general, a hemisphere refers to half of the world, such as the northern or southern hemisphere. In geometry, however, a hemisphere is a three-dimensional shape formed by splitting a sphere into two equal half, each with one flat side.
Ques. What are the curved surface area and total surface area of a hemisphere? (3 Marks)
Ans. The curved surface area refers to the area of the hemisphere's outer surface. The overall surface area includes the area of the curved surface as well as the area of the circular (base).
- The curved surface area of a hemisphere = 2πr² square units
- The total surface area of a hemisphere = 3πr² square units
Ques. A hemisphere has a volume of 2500 cm3. Calculate the radius of the hemisphere? (3 Marks)
Ans. The volume of a given hemisphere = 2500 cubic cm
Volume of a hemisphere = V = (2/3)πr3
2500 = (2/3)πr3
2500 × 3 / 2π = r3
r3 = 7500/2π
r3 = 1194.267
r = 3√(1194.267)
r = 10.61 cm
Hence, the radius of given hemisphere is 10.61 cm.
Ques. A sphere with a radius of 4 cm is divided into two equal half. Calculate the volume of each produced hemisphere. (3 Marks)
Ans. Radius of Sphere = 4cm
Then, radius of Hemisphere = 4cm
Volume of each Hemisphere = V = (2/3)πr3
V = (2/3)πr3
V = (2/3)π 43
V = 133.9 cubic cm
Hence the volume of each Hemisphere is 133.9 cubic centimetres.
Ques. For a hemisphere whose centre is at the origin, what is the cartesian equation? (3 Marks)
Ans. x2 + y2 + z2 = r2 is the cartesian equation for a hemispheric shape with its centre at the origin.
And, the spherical coordinates of the hemisphere will be
- x = r cos θ sin ∅
- y = r sin θ cos ∅
- z = r cos ∅
Ques. What is the volume of a sphere having a radius of 3 cm? (3 Marks)
Ans. Radius of sphere r = 3cm
Volume of a sphere = V = 4πr3/3 cubic units
V = 4/3 × 3.14 × 33
V = 4/3 × 3.14 × 3 × 3 × 3
V = 113.04 cm3
Hence the volume of given sphere is 113.04 cubic cm.
Ques. What is the volume of a sphere having a diameter of 10 cm? (3 Marks)
Ans. The diameter of given sphere = 10 cm
Then the radius = diameter/2 = 10/2 = 5 cm
Now, for volume of sphere,
Volume = V= 4πr3/3 cubic units
V = 4/3 π 53
V = 4/3 × 22/7 × 5 × 5 × 5
V = 4/3 × 22/7 × 125
V = 523.8 cm3
Hence the volume of given sphere is 523.8 cubic cm.
Ques. A plane passes through the centre of a sphere, forming a 12-foot-radius circle. What is the sphere's surface area? (2 Marks)
Ans. The radius of the given sphere r = 12 feet
Total surface area of the sphere (V) = 4 π r2
= 4 × 3.14 × 12 × 12
= 1808.64
Hence area of the sphere is 1808.64 square feet.
Ques. A spherical object has a surface area of 379.94 m2. What is the object's diameter? (3 Marks)
Ans. Surface area of given sphere = 379.94 m2
And surface area of a sphere = 4πr2
Now,
379.94 = 4 × 3.14 × r2
379.94 = 12.56 r2
r2 = 379.94/12.56
r2 = 30.25
r = √30.25
r = 5.5
Therefore, the radius of the spherical solid is 5.5 m.
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