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How to find the Surface Area of a Sphere?

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  • Last Updated : 12 May, 2022

In geometry, a sphere is a three-dimensional solid figure that is round in shape, with every point on its surface equidistant from its center. The distance between any point on the surface of a sphere and its center is called the “radius of a sphere.” A sphere is a three-dimensional figure that is defined in three dimensions, i.e., the x-axis, the y-axis, and the z-axis, whereas a circle is a two-dimensional figure that is defined in a plane. A globe, a football, etc., are some examples of a sphere that we see in our daily life.

Surface Area of a Sphere

The curved surface area covered by a three-dimensional spherical object in space is referred to as the total surface area of a sphere. The formula for the surface area of a sphere is given as,

The Surface Area of a Sphere = 4 π r2 square units

Where r = radius of the sphere.

Sphere

Derivation of Surface Area of a Sphere

The area occupied by the surface of a sphere in space is the surface area of a sphere. We know that a sphere is round in shape, so to calculate its surface area, we can connect it to a curved shape, such as a cylinder. A cylinder is a three-dimensional figure that has a curved surface with two flat surfaces on either side. Let’s consider that the radius of the sphere and the radius of a cylinder is the same. So the sphere can perfectly fit into a cylinder. Therefore, the height of the sphere is equal to the height of a sphere, i.e., the diameter of a sphere. This fact was proved by the mathematician, Archimedes, that the surface area of a sphere of radius “r” is equal to the lateral surface area of a cylinder of radius “r”.

Therefore,

The Surface area of a sphere = The Lateral surface area of a cylinder

We know that,

The lateral surface area of a cylinder = 2πrh, 

Where r is the radius of the cylinder and h is its height.

We have assumed that the sphere perfectly fits into the cylinder. So, the height of the cylinder is equal to the diameter of the sphere.

Height of the cylinder (h) = Diameter of the sphere (d) = 2r (where r is the radius)

Therefore,

The Surface area of a sphere = The Lateral surface area of a cylinder = 2πrh

Surface area of the sphere = 2πr × (2r) = 4πr2

Hence, 

The surface area of the sphere = 4πr2 square units

Where r = radius of the sphere

Determining the surface area of a sphere

Let’s consider an example to see how to determine the surface area of a sphere using its formula.

Example: Find the surface area of a sphere of radius 4 inches.

Step 1: Note the radius of the given sphere. Here, the radius of the sphere is 4 inches.

Step 2: We know that the surface area of a sphere = 4πr2. So, substitute the value of given radius in the equation = 4 × (3.14) × (4)2 = 200.96

Step 3: Hence, the surface area of the sphere = is 200.96 square inches.

Sample Problems

Problem 1: Calculate the total surface area of a sphere with a radius of 15 cm. (Take π = 3.14)

Solution:

Given, the radius of the sphere = 15 cm

We know that the total surface area of a sphere = 4 π r2 square units

= 4 × (3.14) × (15)2

= 2826 cm2

Hence, the total surface area of the sphere is 2826 cm2.

Problem 2: Calculate the diameter of a sphere whose surface area is 616 square inches. (Take π = 22/7)

Solution:

Given, the curved surface area of the sphere = 616 sq. in

We know,

The total surface area of a sphere = 4 π r2 square units

⇒ 4 π r2 = 616

⇒ 4 × (22/7) × r2 = 616

⇒ r2 = (616 × 7)/(4 × 22) = 49

⇒ r = √49 = 7 in

We know, diameter = 2 × radius = 2 × 7 = 14 inches

Hence, the diameter of the sphere is 14 inches.

Problem 3: Find the cost required to paint a ball that is in the shape of a sphere with a radius of 10 cm. The painting cost of the ball is ₨ 4 per square cm. (Take π = 3.14)

Solution:

Given, the radius of the ball = 10 cm

We know that,

The surface area of a sphere = 4 π r2 square units

= 4 × (3.14) × (10)2

= 1256 square cm

Hence, the total cost to paint the ball = 4 × 1256 = ₨ 5024/-

Problem 4: Find the surface area of a sphere whose diameter is 21 cm. (Take π = 22/7)

Solution: 

Given, the diameter of a sphere is 21 cm

We know,

diameter =  2 × radius

⇒ 21 = 2 × r ⇒ r = 10.5 cm

Now, the surface area of a sphere = 4 π r2 square units

= 4 × (22/7) × (10.5) 

= 1386 sq. cm

Hence, the total surface area of the sphere = 1386 sq. cm

Problem 5: Find the ratio between the surface areas of two spheres whose radii are in the ratio of 4:3. (Take π = 22/7)

Solution:

Given, the ratio between the radii of two spheres = 4:3

We know that,

The surface area of a sphere = 4 π r2

From the equation, we can say that the surface area of a sphere is directly proportional to the square of its radius.

Area ∝ (radius)2

⇒ A1/A2 = (r1)2/(r2)2

⇒ A1/A2 = (4)2/(3)2 = 16/9

Therefore, the ratio between the total surface areas of the given two spheres is 16:9.

Problem 6: Find the ratio between the radii of two spheres when their surface areas are in the ratio of 25:121. (Take π = 22/7)

Solution:

Given, the ratio between the total surface areas of two spheres = 25:121

We know that,

The total surface area of a sphere = 4 π r2

From the equation, we can say that the surface area of a sphere is directly proportional to the square of its radius.

Area ∝ (radius)2 ⇒ radius ∝ √Area

⇒ r1/r2 = √A1/√A2

⇒ r1/r2 = √25/√121 = 5/11

Therefore, the ratio between the radii of the given two spheres is 5:11.

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