Volume and Surface area of Sphere

Volume and surface area of a sphere are important concepts in geometry, used in many fields like science, engineering, and everyday situations. A sphere is a perfectly round shape in 3D, with all points on its surface equally distant from the center. The Surface area of a sphere refers to the total area that covers the outer surface of the sphere. It measures the extent of the sphere's outer layer where as the volume of a sphere refers to the amount of space enclosed within the sphere.

In this article, we will discuss the volume and surface area of sphere, their formula, calculation, uses and some solved examples of Volume and surface area of sphere

Surface Area of a Sphere

The area covered by the outer surface of the sphere is called the surface area of the sphere. A sphere do not have any flat surfaces therefore,

Curved Surface Area of a Sphere = Lateral Surface Area of a Sphere = Total Surface Area of a Sphere

Surface area of sphere is = 4πr²

Example: If the radius of a sphere is given as 14cm, then find its surface area. (you can use π = 3.14 for your convenience).

Solution:

It is given that the Radius of the sphere is 14cm.

Now, the Surface area of sphere = 4πr² = 4 × π × (14)² = 24 cm²

Volume of the Sphere

The volume of a sphere is the space occupied by the interior of the sphere. Draw a semicircle on a piece of paper and rotate it 360 degrees to make a sphere. There are two types of spheres which are Solid spheres and hollow spheres. The volumes of the two types of spheres are different. In the next section, we will learn about volumes.

Volume of Solid Sphere

Let, r = radius of the sphere then

Volume of sphere = (4/3)πr³

Volume of Hollow Sphere

Let R = radius of the Outer sphere, r = radius of the inner sphere then

 The volume of hollow sphere = Volume of Outer Sphere - Volume of Inner Sphere

Volume of hollow sphere  =  (4/3)πR³ - (4/3)πr³ = (4/3)π(R³ - r³)

Let's consider some examples for better understanding.

Example 1: Find the volume of the sphere which has a radius of 6 cm.

Solution:

It is given that the radius of the sphere is 6cm.

Now, Volume of sphere = (4/3)πr³ =  ((4/3) × π × (6cm)³) = 904.779 cm³ 

Example 2: Find the volume of a sphere whose inner radius is 5cm and the outer radius is 8 cm.

Solution:

Outer radius of sphere R = 8 cm.

and Inner radius of sphere r = 5 cm.

Now,  Volume of hollow sphere   = (4/3)π(R³ - r³) = (4/3)π((8cm)³ - (5cm)³) = 1621.062 cm³.

Sphere Formulas

There are three main formulas for a sphere, including formulas for the diameter of the sphere, the surface area of ​​the sphere, and the volume of the sphere. All of these formulas are listed in the table below.

Hemisphere

Hemisphere is half of the sphere. In another word, if a sphere is cut into two symmetrical pieces through the center then it is called a hemisphere. Because it is half of a sphere then the volume and surface area of hemisphere are half of the volume and surface area of a sphere.

Properties of Sphere

The following are the properties of a sphere:

• It has no vertex or edge.
• This is not a polyhedron.
• All of the points in the sphere have the same distance to the center.
• It only has a curved face, It does not have any flat face.
• It is perfectly symmetrical.

Comparison Between Circle and Sphere

Read More,

• Volume and Surface Area
• Surface Area formulas
• Volume and Surface area of Cylinder

Sample Questions on Volume and Surface Area of Sphere

Question 1: A baseball is 80mm in diameter. Find the baseball's volume. (π=3.14)

Solution:

We are given that Diameter is 80 mm i.e., D = 80mm.

As we know, D = 2r 

Thus, r = 40mm

Now,  Volume of sphere = (4/3)πr³ = (4/3)π(40mm)³ = 268082.573 mm³

Question 2:  Hollow spheres melt into the same small hollow sphere. The inner and outer radii of the larger sphere are 5 cm and 7 cm, respectively. If the inner and outer radii of the small spheres are 3 cm and 4 cm, respectively, how many small spheres can be formed? (π=3.14)

Solution:

We know that Volume of the sphere = (4/3)πr³

Now, Volume of the bigger sphere = volume of the sphere with outer radius - volume of the sphere with inner radius

⇒  Volume of the bigger sphere =  (4/3) × π × (7cm)³ -  (4/3)π(5cm)³

⇒  Volume of the bigger sphere =  (4/3) × π × (343-125) = (4/3) × π × (218) cm³

In the same way, Volume of smaller sphere = (4/3) × π × (4cm)³  - (4/3) × π × (3cm)³

⇒  Volume of the smaller sphere =   (4/3) × π × (64-27) = (4/3) × π × (37) cm³

hence, the number of spheres that can be formed = volume of the bigger sphere/ volume of the smaller sphere

therefore, the number of spheres that can be formed = (4/3) × π × (218) cm³/ (4/3) × π × (37) cm³

⇒ the number of spheres that can be formed = 5.92 ≈ 6 spheres.

Question 3: When you change the shape of an object from a sphere to a cylinder, then the volume of the cylinder increases, decreases, or remains unchanged. (π=3.14)

Solution:

Volume is a scalar quantity that describes the volume of 3D space surrounded by nearby surfaces.

When transforming a body into another body, the amount of material remains the same, so the volume of the body does not change.

Hence, the volume remain Unchanged.

Question 4: The surface of the sphere is 500 cm². If you change the radius to reduce the area by 50%, then find the radius. (π=3.14)

Solution:

Since the area get reduced by 50%, we can say that

New surface area = 50% of the original area

⇒ 4πr² = 1/2 × 500 

⇒ r² = ( 1/2 × 500 ) / 4π

⇒ r²=250/12.56

⇒ r² = 19.8945 

⇒ r = 4.46 cm

Practice Problems on Volume and Surface Area of Sphere

Problem 1: Calculate the surface area of a sphere with a radius of 5 cm.

Problem 2: Find the volume of a sphere with a diameter of 10 meters.

Problem 3: Determine the radius of a sphere if its surface area is 314 square meters.

Problem 4: Calculate the radius of a sphere that has a volume of 500 cubic centimeters.

Problem 5: Find the volume of a hemisphere with a radius of 4 cm.