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Surface Area of a Sphere(Definition, Formulas & Examples)

surface area of a sphere

The surface area of a 3D shape is the total area of all of the faces. It involves the flat as well as the curved faces. A sphere is a three-dimensional form of a circle. The surface area of a sphere is the area occupied by the curved surface of the sphere. Circular shapes take the shape of a sphere when observed as three-dimensional structures, e.g., a globe or a soccer ball.

Let’s learn how to find the surface area of a sphere and its methods and formulas.

 

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Maths can be really interesting for kids

What is the Surface Area of a Sphere?

The area covered by the outer surface of the sphere in three-dimensional space is known as the surface area of a sphere. The surface area of a sphere is expressed in square units such as $m^{2}$, $cm^{2}$, $ft^{2}$, $in^{2}$, etc.  

Since a sphere has only one surface/face, the surface area of a sphere is also known as the total surface area of a sphere.

To find the total surface area of a sphere we multiply $4$ by $\pi r^{2}$, i.e., the formula to calculate the total surface area of a sphere is $4 \pi r^{2}$, where $\pi$ is a mathematical constant and its approximate value taken during calculation is $\frac {22}{7}$ or $3.14$.

Note: In the case of a sphere there is no Curved Surface Area or Lateral Surface Area.

Derivation of Total Surface Area of Sphere Formula

A sphere has a curved surface, therefore, its surface area can be related to the surface area of a cylinder (a cylinder also has a curved surface along with flat surfaces).

Now, if the radius of a cylinder is the same as the radius of a sphere, it means that the sphere can fit into the cylinder perfectly. This means that the height of the cylinder is equal to the height of the sphere.

So, this height can also be called as the diameter of the sphere.

surface area of a sphere

The curved surface area of a cylinder of base radius $r$ and height $h$ is $2 \pi r h$. Replacing $h$ in the formula with the diameter of a sphere, i.e., $2r$, we get the surface area of a sphere as $2 \pi r \times 2r = 4 \pi r^{2}$.

Total Surface Area of Sphere Examples

Ex 1 Find the surface area of a sphere of radius $3.5 cm$.

Radius of a sphere $r = 3.5 cm$

Surface area of a sphere = $4 \pi r^{2} = 4 \times \frac {22}{7} \times 3.5^{2} = 154 cm^{2}$

Ex 2: The surface area of a sphere is $2464 in^{2}$. Find the radius of the sphere. 

The surface area of a sphere = $2464 in^{2}$.

Therefore, $4 \pi r^{2} = 2464 => 4 \times \frac {22}{7} \times r^{2} = 2464$

$ => r^{2} = \frac {2464 \times 7}{4 \times 22} => r^{2} = 196 => r = \sqrt{196} => r = 14 in$.

Ex 3: The radius of a sphere is doubled. How much the surface area of the sphere will change?

Let the radius of the sphere be $r$ units

Then the surface area of the sphere is  $4 \pi r^{2}$

When the radius is doubled, the new radius becomes $2r$

And the surface area of the sphere will become $4 \pi \left(2r \right)^{2} = 4 \pi \times 4r^{2} = 16 \pi r^{2}$.

Now, $\frac {16 \pi r^{2}}{4 \pi r^{2}} = 4$.

Therefore, the surface area of the sphere increases $4$ times when the radius of the circle is doubled.

Ex 4: The cost of leather is ₹$50$ per square metre. Find the cost of leather required to manufacture $1000$ footballs of radius $0.12 m$. (Take $\pi$ as $3.14$).

Radius of a football $r = 0.12 m$.

The amount of leather required to manufacture one ball is the same as the total surface area of the spherical ball.

Surface area of sphere = $4 \pi r^{2}$

Leather required for one football = $4 \times 3.14 \times 0.12^{2} = 0.180864 m^{2}$

Therefore, leather required for $1000$ footballs = $0.180864 \times 1000 = 180.864 m^{2}$.

Cost of leather for $1 m^{2}$ = ₹$50$

Therefore, total cost to manufacture $1000$ footballs = $180.864 \times 50 =$ ₹$9043.20$.

Maths in Real Life

Important Terms Associated With Sphere

The important elements of a sphere that you should before learning the different formulas are as follows:

  • Radius: The length of the line segment drawn between the center of the sphere to any point on its surface. If $O$ is the centre of the sphere and $A$ is any point on its surface, then the distance $OA$ is its radius. 
  • Diameter: The length of the line segment from one point on the surface of the sphere to the other point which is exactly opposite to it, passing through the centre is called the diameter of the sphere. The length of the diameter is exactly double the length of the radius.
  • Circumference: The length of the great circle of the sphere is called its circumference.
surface area of a sphere

Practice Problems

  1. Find the surface area of a spherical ball of radius $4.6 cm$.
  2. What is the surface area of a sphere of diameter $12 in$?
  3. How much the surface area of a sphere decreases when its radius becomes one-fourth?
  4. Find the radius of a sphere of surface area $5544 mm^{2}$.
  5. Find the surface area of the following (Take $\pi = 3.14$)
    • Earth, radius = $6,371 km$ 
    • Moon, radius = $1,737.4 km$
    • Sun, radius = $6,96,340 km$
    • Jupiter, radius = $69,911 km$

FAQs

What is the surface area of a sphere?

The surface area of a sphere is the total area that is covered by its outer surface. The formula for the surface area of a sphere depends on the radius or the diameter of the sphere and is mathematically expressed as $4 \pi r^{2}$, where $r$ is the radius of the sphere.

Why is the surface area of a sphere 4 times the area of a circle?

A string that completely covers the surface area of a sphere can completely cover the area surface of exactly four circles. Since the area of a circle is $\pi r^{2}$, therefore, the surface area of a sphere is $4 \pi r^{2}$.

What are the curved surface area and total surface area of a sphere?

A sphere has just one surface and that is curved. Since there is no flat surface in a sphere, the curved surface area of a sphere is equal to its total surface area of the sphere which is $4 \pi r^{2}$.

What is the surface area of a sphere formula in terms of its diameter?

The surface area of a sphere in terms of its radius is $4 \pi r^{2}$. Since, $r = \frac {d}{2}$, therefore, replacing $r$ with $\frac {d}{2}$, we get $4 \pi \left(\frac {d}{2} \right)^{2} = 4 \pi \times \frac {d^{2}}{4} = \pi d^{2}$.

Therefore, the surface area of a sphere formula in terms of its diameter is $\pi d^{2}$.

Conclusion

The surface area of a sphere is the area occupied by the curved surface of the sphere in a three-dimensional space and is calculated by using the formula $4 \pi r^{2}. In the case of a sphere, there is only one measurement for the surface area which is the total surface area and the lateral or curved surface area does not exist in this case.

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