Volume of a Sphere
The volume of a sphere is one of the greatest inventions of Archimedes (287 BC -212 BC). He discovered that the volume of a purely solid sphere is two-thirds of the volume of the smallest cylinder that surrounds the sphere. Using this he also worked out that the volume of a cone, sphere, and a cylinder on the same base are in the ratio 1:2:3.
Archimedes subdivided the whole volume into small slices of known cross-sectional area and added them. The sum of the areas provides the total volume of the sphere. Later this technique was formulated as integral calculus, which is the backbone of modern science and mathematics.
What is Volume?
Volume is a property of three-dimensional objects and is defined as the space that an object encloses within it. If we take a fully filled glass of water and pour a solid spherical metal ball in it, then the amount of water that comes outside the glass gives the volume of the metal ball. It is measured in cubic units such as, cubic meter, cubic feet, cubic inch, etc.
What is a Sphere?
A sphere is a three dimensional shape. Objects like basketballs, soccer balls, and globes are examples of spheres. Every point on the surface of a sphere is equidistant from a fixed point. The fixed point is called the center of the sphere and the distance from the center to any point on the sphere is called the radius of the sphere. If we rotate a circle and observe the change in shape, then we can obtain a three dimensional sphere. Therefore, the rotation of a two-dimensional circle gives a three-dimensional sphere.
What is the Volume of a Sphere?
Archimedes conducted a neat experiment to find the volume of a sphere. He found the volume of a sphere by placing a solid spherical metal ball into a fully filled water container. On placing the metal ball into the container some water spills out. So, the amount of water that was removed is equal to the volume of the sphere that was placed inside it.
We can observe that a circle and a sphere are both round. The difference between the two shapes is that a sphere is a three-dimensional shape and a circle is a two-dimensional shape. This is the reason we can measure both the volume and the area of a sphere.
The volume of the sphere can be obtained theoretically using the volume of the sphere formula.
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Volume of Sphere formula
The formula used to find the volume of a sphere is:
Volume of the sphere \( =\frac{4}{3}\times \pi \times r^3 \) cubic unit.
Check out Volume of a sphere calculator
Volume of Hollow Sphere
A sphere that is made of walls of equal thickness is hollowed in such a way that it creates a sphere inside which has a smaller radius than the outer wall.
Volume of Hollow Sphere formula
The volume of the hollow sphere is calculated as,
Volume of the Hollow sphere \( =\frac{4}{3}\times \pi \times (R-r)^3 \) cubic unit.
Where ‘R’ is the outer radius of the sphere and ‘r’ is the internal radius of the sphere.
Volume of Hemisphere
A hemisphere is one half of a sphere. If we cut the sphere exactly in two equal halves using a plane then the hemisphere is obtained. The volume of the sphere is half of the volume of the sphere of the same radius.
Volume of Hemisphere formula
The volume of the hemi-sphere is calculated as,
Volume of the hemisphere \( =\frac{1}{2}\times \) volume of the full sphere.
\( =\frac{2}{3} \pi r^3 \) cubic units.
Solved Examples on Volume of a Sphere
Example 1: Find the volume of a sphere of radius 8 inches.
Solution:
The given radius of the sphere is r=8 inches.
\( V=\frac{4}{3} \pi r^3 \) [write the formula for volume]
\( =\frac{4}{3} \pi (8)^3 \) [substitute for r]
\( =\frac{2048}{3} \pi \) [simplify]
\( =2145.52 in.^3 \) [use a calculator]
The volume of sphere is about \( 2145.52 in.^3. \).
Example 2: Find the volume of a sphere of radius 5 feet. Round your answer to the nearest tenth.
Solution:
The given radius of the sphere is r=10 ft.
\( V=\frac{4}{3} \pi r^3 \) [write the formula for volume]
\( =\frac{4}{3} \pi (5)^3 \) [substitute for r]
\( =\frac{500}{3} \pi \) [simplify]
\( \approx 523.6 \) [use a calculator]
The sphere volume is about 523.6 cubic feet.
Example 3: Find the volume of the hemisphere of radius 6 inches.
Solution:
The given radius of the hemisphere is r=8 inches.
\( V=\frac{2}{3} \pi r^3 \) [write the formula for volume]
\( =\frac{2}{3} \pi (6)^3 \) [substitute for r]
\( =\frac{432}{3} \pi \) [simplify]
\( =452.57 in.^3 \) [use a calculator]
The volume of hemisphere is about \( =452.57 in.^3 \).
Example 4: Find the volume of a hollow sphere. The outer radius of the sphere is 5 feet and the thickness is 1 foot.
Solution:
The given outer radius of the hollow sphere is R=5 ft.
The inner radius of the sphere = Outer radius – thickness
\( =5-1=4 \)
\( V=\frac{4}{3} \pi (R-r)^3 \) [write the formula for volume]
\( =\frac{4}{3} \pi (5-4)^3 \) [substitute for R and r]
\( =\frac{4}{3} \pi \) [simplify]
\( \approx 4.19 \) [use a calculator]
The volume of the hollow sphere is about 523.6 cubic feet.
No, a sphere has no base. It has only a curved outer surface.
The height of a sphere in terms of radius of sphere is calculated as,
Height of the sphere = 2 × radius of the sphere
The volume of the sphere will be (2/3)rd of the cylinder whose height and radius of the base will be equal to the radius of the sphere.
No, if the volume of two solid spheres are equal then the radius will also be equal. The spheres will be identical.