A sphere is a three-dimensional shape that is perfectly round. Every point on its surface is at the same distance from the center. The surface area of a sphere is the total area that covers its outer surface.
If the radius of the sphere is r, the surface area is calculated using the formula:
Surface Area of a Sphere = 4πr²
This formula shows that the surface area depends on the square of the radius. If the radius is doubled, the surface area becomes four times larger.
Area Formula using diameter:
Surface area of a sphere = πd²
Where r is the radius and d is the diameter 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 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 = the 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 is 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 is 4πr² square units.
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How to Find the Surface Area of a 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.
Solved Examples
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?