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Surface Area of Sphere | CSA, TSA, Formula and Derivation

Last Updated : 18 Feb, 2024
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Surface Area of Sphere is the area occupied by the curved surface of the sphere. We know that, a sphere is a three-dimensional solid figure that is symmetrical in shape, with every point on its surface equidistant from its center.

Surface Area of Sphere

Let’s learn the formulas of Surface Area of Sphere and how to use them with the help of solved examples.

What is Surface Area of Sphere?

The surface area of a sphere is the region covered by the outer surface in the 3-dimensional space. It can be said that a sphere is the 3-dimensional form of a circle. The surface area of a sphere formula is given in terms of pi (π) and radius.

Formula Related to Surface Area of Sphere
Surface Area of a Sphere 4πr2
Surface Area with Diameter πd2
Curved Surface Area of a Sphere 4πr2
Total Surface Area of a Sphere 4πr2 (Same as curved surface area)

Surface Area of Sphere Formula

Formula for surface area of a sphere :

Surface Area of Sphere = 4πr2

Surface Area of Sphere Formula using diameter:

Surface Area of Sphere = πd2

where, r is the radius of the sphere.

Derivation of Surface Area of 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 a sphere and the radius of a cylinder to be the same, so the sphere can perfectly fit into the cylinder.

Therefore, the height of the sphere = height of a cylinder = the diameter of a sphere.

Surface Area of Sphere Derivation

Surface area of a sphere = 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

How to Find Surface Area of Sphere?

The surface area of a sphere is simply the area occupied by its surface. Let’s consider an example to see how to use its formula.

Example: Find the surface area of a sphere of radius 7 cm.

Step 1: Note the radius of the given sphere. Here, the radius of the sphere is 47 cm.

Step 2: We know that the surface area of a sphere = 4πr2. So, substitute the value of the given radius in the equation = 4 × (3.14) × (7)2 = 616 cm2.

Step 3: Hence, the surface area of the sphere is 616 square cm.

Curved Surface Area (CSA) of Sphere

The sphere has only one curved surface. Therefore, the curved surface area of the sphere is equal to the total surface area of the sphere, which is equal to the surface area of the sphere in general.

Therefore,

CSA of a Sphere = 4πr2

Total Surface Area (TSA) of Sphere

As the complete surface of the sphere is curved thus total Surface Area and Curved Surface Area are the same for the Sphere.

TSA of Sphere = CSA of Sphere

Related :

Surface Area of Sphere Examples

Let’s solve questions on the Surface Area of Sphere.

Example 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.

Example 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.

Example 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/-

Example 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.

Sphere Area Formula FAQs

What is Surface Area of Sphere?

The surface area of a sphere is given by the formula, 

Surface area = 4πr2

Where r is the radius of the sphere.

What is Surface Area of Hemisphere?

The surface area of a hemisphere is given by the sum of half of the sphere’s surface area and the base area (which is circular), that is,

S.A. = 2πr2 + πr2= 3πr2

Thus, the surface area of a hemisphere = 3πr2

What is Lateral Surface Area of Sphere?

The lateral surface area is equal to the surface area of the sphere, which is equal to the curved surface area of the sphere.

Why is Surface Area of Sphere 4 Times the Area of Circle?

Surface area of a sphere is four times the area of a circle because a sphere with radius, r can be thought of as composed of circles with the same radius. The formula for the surface area of a sphere, 4πr2 is derived from the fact that its surface area is equivalent to the lateral surface area of a cylinder with the same radius and height equal to the diameter of the sphere. The diameter is twice the radius, thus leading to the factor of four.

How Does Surface Area of Sphere Change When Radius is Halved?

When the radius of a sphere is halved, its surface area becomes one-fourth of the original surface area. This is because the surface area formula, 4πr2 , involves the square of the radius, so halving the radius squares one-fourth.

How does Surface Area of Sphere Change When Radius is Tripled?

When the radius of a sphere is tripled, its surface area increases by a factor of nine. This is because the surface area is proportional to the square of the radius (4πr2 ) , so tripling the radius (3r) leads to 4π(3r)2 = 9×4πr2 , which is nine times the original surface area.



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