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Volume of a Sphere is referred to as the measurement of the space that it occupies. The volume of a sphere is measured in cubic units, such as m3, cm3, in3, etc. The shape of the sphere is round and three-dimensional that has no edges or vertices. It has three axes which are, the x-axis, y-axis, and z-axis which define its shape. Here, we will be learning to find the volume of a sphere by deducing the formula of the volume of a sphere.
Read More: Difference between Area and Volume
Key Terms: Sphere, Volume of Sphere, Radius, Diameter, Volume, Three-dimensional Shape, Cone, Cylinder
What is Volume of Sphere?
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The sphere is a three-dimensional structure. Like a circle, the sphere also possesses a center and a radius. All the points at a distance of radius in space together form a sphere. Twice the distance of a radius is called its diameter.
Also Read: Edges, Faces, and Vertices
Volume of a Sphere Formula
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The formula for the volume of a sphere is calculated using the radius of the three-dimensional structure. Spheres can be of two types, solid and hollow. The formula which is used to calculate the volume of a solid sphere is given below:
| Volume of Sphere = 4/3 πr3 |
Where, r is the Radius of the Sphere.
In the case of a hollow sphere, there is a thin layer that has mass or volume, and it makes the hollow sphere. So, the volume is calculated as the
Volume of Hollow Sphere = (Volume of Outer Sphere) – (Volume of Inner Sphere)
| Volume of Hollow Sphere = 4/3 πR3 - 4/3 πr3 |
Where
- R = Radius of Outer Sphere
- r = Radius of the Inner Sphere
Volume of sphere needs the value of the radius or its diameter. In the case of diameter, the value has to be halved to get the radius of the sphere.
Also read: Sphere Formula
Derivation of the Volume of a Sphere
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The formula can be derived using two methods, the Arrhenius method, and by integration method. Both of the methods used to derive the formula are explained below:
Arrhenius Method
According to this method, the volume of a sphere is calculated using the volume of a cone and a cylinder that have an equal radius, cross-section area and also height equal to twice the radius. According to this method, the volume of such a cylinder is equal to the sum of the volume of a cone, and sphere, further these volumes are in the ratio 1:2:3 (cylinder: sphere: cone).
Thus,
Volume of a cylinder = Volume of sphere + Volume of a cone
Volume of sphere = Volume of cylinder – Volume of cone
Volume of cylinder = 3 x Volume of cone
Volume of a cylinder = πr2h …….. (1)
πr2h = 3 x Volume of cone
Volume of the cone = πr2h/3 …….. (2)
Volume of sphere = Volume of cylinder – Volume of cone
Volume of sphere = πr2h - πr2h/3
Volume of sphere = 2 πr2h/3
h = 2r
Volume of sphere = 2 πr2 2r/3
Volume of sphere = 4 πr3/3
The volume of a solid sphere = 4/3 π r3
Also Read: Quadrilateral
Integration Method
Integration Method
Integration method assumes that the sphere is made up of circular discs having thickness dy and radius x. They are placed at a different height from the center of the sphere, have radius r, and have varied radiuses. The disc selected for the derivation is placed at the height z from the central plane.
According to Pythagoras' theorem:
x2 + z2 = r2
x = r2 - z2
Area of the disk thus is π (r2 – z2)
Now we need to calculate the area of all the discs that would make up the sphere, thus we apply integration to the formula above from +r to -r.
V = r∫ -r (r2–z2)dy
V = π (zr2 – z2/3)
V = π [2/3r3 – (-2/3 r3)]
Thus, the volume of a solid sphere = 4/3 π r3
Also Read: Surface Areas and Volumes Revision Notes
How to Calculate Volume of a Sphere?
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The volume of a sphere is the space occupied within it. It can be calculated with the above formula, which we have already derived. In order to find the volume of a given sphere, follow the steps given below:
- Check the value of the radius of the given sphere. If the diameter of the sphere is known, then divide it by 2, to get the radius
- Then take the cube of the radius r3
- Now multiply it with (4/3)π
- Finally, add the units to the final answer.
Solved Examples on Volume of Sphere
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Example 1: Find the volume of a sphere having a radius of 5 cm.
Solution: Given, the volume of a sphere formula,
V = 4/3 πr3
= (4/3) x 3.14 x 53
= (4/3) x 3.14 x 5 x 5 x 5
= 523.3 cm3
Example 2: Find the radius of the sphere when the volume of a sphere is 523 cubic yards.
Solution: Given,
V = 523 cubic yards
Volume of a sphere, V = 4/3 πr3
523 = (4/3) x 3.14 x r3
523 = 4.19r3
Divide both sides by 4.19
r3 = 124.82
3√r3 = 3√124.82
r = 5
So, the radius of the sphere is 5 yards.
Example 3: Find the volume of a sphere with a radius of 24 mm.
Solution: As we know the radius is half of the diameter, then
r = 24/2 = 12 mm
The volume of a sphere = 4/3 πr3
By substitution, we get
V = (4/3) x 3.14 x 12 x 12 x 12 = 7734.6 mm3
Example 4: The radius of an inflated spherical balloon is 7 feet. If air is leaking from the balloon at a constant rate of 26 cubic feet per minute, then how long will it take for the balloon to be completely deflated?
Solution: Given that
Volume of the spherical balloon = 4/3 πr3
Volume = 4/3 x 3.14 x 7 x 7 x 7 = 1436.03 cubic feet
Now, divide the volume of the balloon by the rate of leakage
Time in minutes = 1436.03 cubic feet/26 cubic feet = 55 minutes
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Things to Remember
- A sphere is a three-dimensional structure.
- The properties of a sphere are similar to a circle.
- The volume of a solid sphere = 4/3 πr3
- The volume of hollow sphere = 4/3 πR3 - 4/3 πr3
- The surface area of a sphere is 4 πr2
- The formula to derive the formula to calculate the volume of a sphere can be of two ways: Arrhenius and integration method.
Read More: Perimeter of Parallelogram
Sample Questions
Ques. Find the volume of a ball having a radius of 6 m. (2 Marks)
Ans. Radius of the ball = 6 m
Volume = 4/3 πr3
V = 4 x 22 x 6x6x6/3 x 7
Volume = 905.14 m3
Ques. Find the volume of a sphere having a diameter of 10 m. (3 Marks)
Ans. Diameter = 10 m
Radius = Diameter/2 = 5 m
Volume = 4/3 πr3
V = 4 x 22 x 5x5x5/3 x 7
Volume = 523.8 m3
Ques. A spherical vessel has an inner diameter of 14 m, find its capacity. (3 Marks)
Ans. Capacity of the vessel = volume
Diameter = 14 m
Radius = 7 m
Volume = 4/3 πr3
V = 4 x 22 x 7x7x7/3 x 7
Volume = 1378.6 m3
Ques. A sphere is made up of a thick sheet having a thickness of 1 cm. The inner radius is 5 cm. Find the volume of metal used. (4 Marks)
Ans. Volume of metal used = volume of metal and hollow sphere.
The volume of hollow sphere = 4/3 πR3 - 4/3 πr3
R’ = outer radius = 5 + 1 = 6 cm
R” = inner radius = 5 cm
Volume of hollow sphere = 4/3 πR3 - 4/3 πr3
The volume of hollow sphere = 4/3 π (R3 - r3)
The volume of hollow sphere = 4/3 π (125 - 108)
The volume of hollow sphere = 71.2 cm3
Ques. A metal cube has a side of 7 cm. How many spheres can be made out of it having radius 1 cm. (2 Marks)
Ans. Side of cube = 7cm
Volume of cube = 441 cm3
Volume of sphere to be made = 4/3 πr3 = 4.18 cm3
Number of spheres that can be made = 441/4.18 cm3 = 105.5 spheres
Ques. A sphere has a volume of 1000 m3, what is its diameter? (2 Marks)
Ans. Volume = 1000 m3 = 4/3 πr3
r = 31000x3x7/4x22
Radius = 6.2 m
Ques. A cone and a sphere have equal volume and equal radius, find the ratio of the height of the cone and the radius of the sphere. (2 Marks)
Ans. Volume of sphere = volume of the cone
4/3 πr3 = 1/3 πr2h
r = h/4
h = 4r
Thus, the ratio of r: h is 4:1
Ques. A cylinder has a height equal to three times its radius. Its volume is equal to the volume of a sphere having a radius of 4 cm. Find the radius of the cylinder. (4 Marks)
Ans. Height of cylinder = 3 r’ (radius of cylinder)
Volume of cylinder = volume of sphere
πr’2h = 4/3 πr3
πr’2h = 4 x 22 x 4x4x4/3 x 7
πr’2h = 268.19
πr’2 3r’ = 268.19
3πr’3 = 268.19
r’ = 50.2 cm
Ques. Find the volume of a ball having a radius of 5 cm. (2 Marks)
Ans. Radius of the ball = 5 cm
Volume = 4/3 πr3
V = 4 x 22 x 5x5x5/3 x 7
Volume = 523.8 cm3
Ques. Find the volume of a sphere having a diameter of 20 cm. (2 Marks)
Ans. Diameter = 20 cm
Radius = Diameter/2 = 10 cm
Volume = 4/3 πr3
V = 4 x 22 x 10x10x10/3 x 7
Volume = 4190.4 cm3
Ques. Find the volume of a hemisphere ball having a radius of 7 cm. (2 Marks)
Ans. Radius of the ball = 7 m
Volume = 4/3 πr3
V = 4 x 22 x 7x7x7/3 x 7
Volume = 1378.6 m3
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