Surface Area of Sphere | Formula, Derivation and Solved Examples
Last Updated :
15 Jul, 2024
A sphere is a three-dimensional object with all points on its surface equidistant from its center, giving it a perfectly round shape. The surface area of a sphere is the total area that covers its outer surface.
To calculate the surface area of a sphere with radius r, we use the formula:
Surface Area of Sphere = 4πr2
This formula shows that the surface area of a sphere is directly proportional to the square of its radius. For example, if the radius of a sphere is doubled, its surface area increases by a factor of four. Let’s learn more about the formula for the Surface Area of the Sphere with solved examples.
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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) |
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 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
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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
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Solved Examples on Surface Area of Sphere
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.
Practice Problems – Surface Area of Sphere
Problem 1: Find the surface area of a sphere with a radius of 5 cm. Use π=3.14.
Problem 2: A sphere has a diameter of 10 inches. Calculate its surface area.
Problem 3: Determine the surface area of a sphere whose radius is 7 meters.
Problem 4: The radius of a sphere is 15 cm. What is its surface area in square centimeters?
Problem 5: If a sphere with a radius of 8 cm is cut into two hemispheres, what is the surface area of each hemisphere?
Problem 6: A sphere has a surface area of 500 square meters. What is the radius of the sphere?
Problem 7: Calculate the surface area of a sphere with a radius of 12 cm.
Problem 8: The diameter of a sphere is 16 inches. Find its surface area.
Problem 9: A spherical balloon has a radius of 21 cm. Determine its surface area in square centimeters.
Problem 10: A sphere has a radius of 0.5 meters. Find its surface area in square meters.
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