VOLUME OF SPHERE AND HEMI-SPHERE

Volume of Sphere

A Volume of a sphere is a measure of the space it can occupy. A sphere is a 3-D shape that has no edges or vertices. Here, in this short lesson, we will discuss how to find the volume of a sphere, derive the formula for a sphere, and learn how to apply formulas. Once you understand this topic, you will learn how to solve problems with the volume of a sphere.

What is the Volume of the Sphere?

A volume of a sphere is the amount of space occupied inside the sphere. A sphere is defined as a 3-D round solid figure in which every point on its surface is equidistant from its center. The fixed distance is called the sphere's radius, and the fixed point is called the sphere's center. As we rotate the circle, we will observe a change in shape. Thus, the three-dimensional shape of a sphere is obtained by rotating a 2-D object known as a circle.

Archimedes' principle helps us for finding the volume of a spherical object. The principle states that when a solid object is plugged into a container that is filled with water, then the volume of the solid object can be obtained because the volume of water that flows out of the container equals the volume of the spherical object.

Derivation of Volume of Sphere Formula

Let's take that the volume of the sphere is made up of many thin circular disks that are arranged one over the other, as shown in the figure above. The circular discs have continuously varying diameters, which are positioned collinearly with their centers. Now select any of the disks. A thin disk with a radius "r" and a thickness "dy" is located at a distance y from the x-axis. The volume can therefore be written as the product of the area of ​​the circle and its thickness dy.

The radius of a circular disk 'r' can also be expressed in terms of the vertical dimension (y) using the Pythagorean theorem.

Now, the vol. of the disc element dV can be expressed as:

dV =(πr²)dy

dV =π (R²-y²) dy

Substitute the limits:

V = π[(R³ - R³/3) - (-R³ + R³/3)]

simplify the expression we get,

V = π[2R³ - 2R³/3]

V = π/3[6R³ - 2R³]

V = π/3(4R³)

Therefore, the volume of sphere formula is:

V = 4/3 π R³

How to Calculate Volume of Sphere?

A volume of a sphere is the space occupied by it. It can be calculated using the above formula, which we have already derived. To calculate the volume of a sphere, do the following:

• Check out the radius of the given sphere. If the diameter of the sphere is given, divide it by 2 to get the radius
• Find a cube of radius r³
• Then multiply that by (4/3)π
• The answer will be the volume of the sphere

Volume of Solid Sphere

If the "r" is the radius of the sphere and the v is the volume of the sphere. Therefore, the volume of the sphere is V = (4/3)πr³

Volume of Hollow Sphere

If the R is the radius of the outer sphere and r is the inner radius of the sphere and the V is the volume of the sphere. Then, the volume of the sphere is:

Volume of Sphere, V = Volume of Outer Sphere - Volume of Inner Sphere = (4/3)πR³ - (4/3)πr³ = (4/3)π(R³ - r³)

Solved Examples of Sphere

Q1. Calculate the volume of the sphere whose radius is 3 cm.

Ans. Given: Radius, r = cm

The volume of a sphere = 4/3 πr3 cubic units

V = 4/3 x 3.14 x 33

V = 4/3 x 3.14 x 3 x 3 x 3

V = 113.04 cm3

Q2. Find the volume of the sphere whose diameter is 10 cm.

Ans. Given, diameter = 10 cm

So, radius = diameter/2 = 10/2 = 5 cm

As per the formula of sphere volume, we know;

Volume = 4/3 πr3 cubic units

V = 4/3 π 53

V = 4/3 x 22/7 x 5 x 5 x 5

V = 4/3 x 22/7 x 125

V = 523.8 cu.cm.

Volume of Hemisphere

In geometry, we have studied various types of 3-D shapes. In 3-D shapes, solids have three dimensions: length, height, and breadth. We know that 3-D shapes do not lie on a piece of paper. Usually, most three-dimensional objects are obtained by rotating two-dimensional objects. One of the best examples of a 3D shape is a sphere, which is obtained by rotating a 2D shape called a circle. Our Earth is a perfect example of a sphere that is spherical.

What is the Volume of a Hemisphere?

A sphere is known as a set of 3-D points, and all points lying on the surface are equidistant from the center. When a plane intersects a sphere at the center or in equal parts then, it forms a hemisphere. Thus, we can say that a hemisphere is exactly half of a sphere. In general, a sphere forms exactly two hemispheres. One such good example of a hemisphere is our Earth. Our Earth consists of two hemispheres, the Southern Hemisphere and the Northern Hemisphere.

Volume of Hemisphere Formula

As from the above statement, the volume of a hemisphere is half the volume of a sphere, therefore, it is expressed as:

Volume of a hemisphere = 1/2 of 4πr³/3 = 1/2 × 4πr³/3 = 2πr³/3, where r is the radius of the hemisphere

How to Find the Volume of a Hemisphere?

• Step 1: Check the radius of a given hemisphere.
• Step 2: Place the given value of the radius in the formula.
• Step 3: After substituting the value of the radius, you will get the final answer

Solved Example of Hemisphere

Q1. Find the surface area of a sphere of radius 7 cm.

Ans. The surface area of a sphere of radius 7 cm would be

4πr² = 4 x 22/7 x 7 x 7 cm² = 616 cm²

Q2. Find (i) the curved surface area and (ii) the total surface area of a hemisphere of radius 21 cm.

Ans. (i) Curved surface area of a hemisphere of radius 21 cm would be

2πr² = 2 x 22/7 x 21 x 21cm² = 2772 cm²

(ii) Total surface area of the hemisphere would be

3πr² = 3 x 22/7 x 21 x 21cm² = 4158 cm²

Q3. A hemispherical bowl has a radius of 3.5 cm. What would be the volume of water it would contain?

Ans. The volume of water the bowl can contain

=2/3πr³

2/3 x 22/7 x 3.5 x 3.5 x 3.5cm³ = 89.8 cm³

Q4. A hemispherical dome of a building needs to be painted (figure). If the circumference of the base of the dome is 17.6 m, find the cost of painting it, given the cost of painting is Rs 5 per 100 cm².

Ans. Circumference of the dome = 17.6 m. Therefore, 17.6 = 2πr

So, the radius of the dome = 17.6 x 7/2 x 22 m = 2.8 m

The curved surface area of the dome = 2πr²

= 2 x 22/7 x 2.8 x 2.8m²

= 49.28 m²

Now, cost of painting 100 cm² is Rs. 5.

So, cost of painting 1 m² = Rs. 500

Therefore, cost of painting the whole dome

= 500 × 49.28

= Rs. 24640