Surface Area & Volume of Sphere | Mathematics (Maths) Class 9

Surface Area of a Sphere

The surface area of a sphere is defined as the region covered by its outer surface in three-dimensional space.  A Sphere is a three-dimensional solid having a round shape, just like a circle. The formula of total surface area of a sphere in terms of pi (π) is given by:

Surface area =  4 πr² square units

The difference between a sphere and a circle is that a circle is a two-dimensional figure or a flat shape, whereas, a sphere is a three-dimensional shape. Therefore, the area of circle is different from  area of sphere.
Area of circle = πr²

Definition
From a visual perspective, a sphere has a three-dimensional structure that forms by rotating a disc that is circular with one of the diagonals.
Let us consider an instance where spherical ball faces are painted. To paint the whole surface, the paint quantity required has to be known beforehand. Hence, the area of every face has to be known to calculate the paint quantity for painting the same. We define this term as the total surface area.
The surface area of a sphere is equal to the areas of the entire face surrounding it.

Formula
The surface area of a sphere formula is given  by,
A = 4 πr² square units
For any three-dimensional shapes, the area of the object can be categorised into three types. They are:

• Curved Surface Area
• Lateral Surface Area
• Total Surface Area

1. Curved Surface Area: The curved surface area is the area of all the curved regions of the solid.
2. Lateral Surface Area: The lateral surface area is the area of all the regions except bases (i.e., top and bottom).
3. Total Surface Area: The total surface area is the area of all the sides, top and bottom the solid object.

In case of a Sphere, it has no flat surface.
Therefore, the Total surface area of a sphere = Curved surface area of a sphere.

Solved Examples
Q.1. Calculate the cost required to paint a football which is in the shape of a sphere having a radius of 7 cm. If the painting cost of football is INR 2.5/square cm. (Take π = 22 / 7)
Solution: We know,
The total surface area of a sphere = 4 πr² square units
= 4 × (22 / 7) × 7 × 7
= 616 cm²
Therefore, total cost of painting the container = 2.5 × 616 = Rs. 1540.

Q.2. Calculate the curved surface area of a sphere having radius equals to 3.5 cm. (Take π = 22 / 7)
Solution: We know,
Curved surface area = Total surface area = 4 πr² square units
= 4 × (22 / 7) × 3.5 × 3.5
Therefore, the curved surface area of a sphere = 154 cm².

Volume Of Sphere

The volume of sphere is the capacity it has. The shape of the sphere is round and three -dimensional. It has three axes such as x-axis, y-axis and z-axis which defines its shape. All the things like football and basketball are examples of the sphere which have volume.
The volume here depends on the diameter of radius of the sphere since if we take the cross-section of the sphere, it is a circle. The surface area of sphere is the area or region of its outer surface. To calculate the sphere volume, whose radius is ‘r’ we have the below formula:

Volume of a sphere = 4 / 3 πr³

Now let us learn here to derive this formula and also solve some questions with us to master the concept.
If you consider a circle and a sphere, both are round. The difference between the two shapes is that a circle is a two-dimensional shape and sphere is a three-dimensional shape which is the reason that we can measure Volume and area of a Sphere.

Sphere Volume
The sphere is defined as the three-dimensional round solid figure in which every point on its surface is equidistant from its centre. The fixed distance is called the radius of the sphere and the fixed point is called the centre of the sphere. When the circle is rotated, we will observe the change of shape. Thus, the three-dimensional shape sphere is obtained from the rotation of the two-dimensional object called a circle.
Archimedes’ principle helps us find the volume of a spherical object. It states that when a solid object is engaged in a container filled with water, the volume of the solid object can be obtained. Because the volume of water that flows from the container is equal to the volume of the spherical object.

Solved Examples
Q.1. Find the volume of a sphere whose radius is 3 cm?
Solution: Given: Radius, r = cm
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.

Q.2. Find the volume of sphere whose diameter is 10 cm.
Solution: 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.