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• Surface Area of a Sphere

What is a sphere? Is it the same as a circle?

Can you draw a circle on a paper? Yes, you can, because a circle is a plane closed figure whose every point lies at a constant distance (called radius) from a fixed point, which is called the centre of the circle.

Now if you paste a string along a diameter of a circular disc and rotate it as you had rotated the triangle in the previous section, you see a new solid (see Fig 13.18). What does it resemble? A ball? Yes. It is called a sphere.

Can you guess what happens to the centre of the circle, when it forms a sphere on rotation?

Of course, it becomes the centre of the sphere. So, a sphere is a three dimensional figure (solid figure), which is made up of all points in the space, which lie at a constant distance called the radius, from a fixed point called the centre of the sphere.

`color{green}("Note :")`

`"A sphere is like the surface of a ball. The word solid "`

`"sphere is used for the solid whose surface is a sphere."`

`color{red}("Activity :")` Have you ever played with a top or have you at least watched someone play with one? You must be aware of how a string is wound around it.

Now, let us take a rubber ball and drive a nail into it. Taking support of the nail, let us wind a string around the ball.

When you have reached the ‘fullest’ part of the ball, use pins to keep the string in place, and continue to wind the string around the remaining part of the ball, till you have completely covered the ball [see Fig. 13.19(a)]. Mark the starting and finishing points on the string, and slowly unwind the string from the surface of the ball.

Now, ask your teacher to help you in measuring the diameter of the ball, from which you easily get its radius. Then on a sheet of paper, draw four circles with radius equal to the radius of the ball.

Start filling the circles one by one, with the string you had wound around the ball [see Fig. 13.19(b)].

What have you achieved in all this?

The string, which had completely covered the surface area of the sphere, has been used to completely fill the regions of four circles, all of the same radius as of the sphere.

So, what does that mean? This suggests that the surface area of a sphere of radius `r = 4` times the area of a circle of radius `r = 4 × (π r^2)`

So,

where r is the radius of the sphere.

How many faces do you see in the surface of a sphere? There is only one, which is curved.

Now, let us take a solid sphere, and slice it exactly ‘through the middle’ with a plane that passes through its centre.

What happens to the sphere?

Yes, it gets divided into two equal parts (see Fig. 13.20)! What will each half be called? It is called a hemisphere. (Because ‘hemi’ also means ‘half’)

And what about the surface of a hemisphere? How many faces does it have?

Two! There is a curved face and a flat face (base).

The curved surface area of a hemisphere is half the surface area of the sphere, which is `1/2` of `4pir^2`

`color{green}("Therefore,")`

where r is the radius of the sphere of which the hemisphere is a part.

Now taking the two faces of a hemisphere, its surface area `2πr^2 + πr^2`

So,

Can you draw a circle on a paper? Yes, you can, because a circle is a plane closed figure whose every point lies at a constant distance (called radius) from a fixed point, which is called the centre of the circle.

Now if you paste a string along a diameter of a circular disc and rotate it as you had rotated the triangle in the previous section, you see a new solid (see Fig 13.18). What does it resemble? A ball? Yes. It is called a sphere.

Can you guess what happens to the centre of the circle, when it forms a sphere on rotation?

Of course, it becomes the centre of the sphere. So, a sphere is a three dimensional figure (solid figure), which is made up of all points in the space, which lie at a constant distance called the radius, from a fixed point called the centre of the sphere.

`color{green}("Note :")`

`"A sphere is like the surface of a ball. The word solid "`

`"sphere is used for the solid whose surface is a sphere."`

`color{red}("Activity :")` Have you ever played with a top or have you at least watched someone play with one? You must be aware of how a string is wound around it.

Now, let us take a rubber ball and drive a nail into it. Taking support of the nail, let us wind a string around the ball.

When you have reached the ‘fullest’ part of the ball, use pins to keep the string in place, and continue to wind the string around the remaining part of the ball, till you have completely covered the ball [see Fig. 13.19(a)]. Mark the starting and finishing points on the string, and slowly unwind the string from the surface of the ball.

Now, ask your teacher to help you in measuring the diameter of the ball, from which you easily get its radius. Then on a sheet of paper, draw four circles with radius equal to the radius of the ball.

Start filling the circles one by one, with the string you had wound around the ball [see Fig. 13.19(b)].

What have you achieved in all this?

The string, which had completely covered the surface area of the sphere, has been used to completely fill the regions of four circles, all of the same radius as of the sphere.

So, what does that mean? This suggests that the surface area of a sphere of radius `r = 4` times the area of a circle of radius `r = 4 × (π r^2)`

So,

`color{green}("Surface Area of a Sphere" = 4 π r^2)`

where r is the radius of the sphere.

How many faces do you see in the surface of a sphere? There is only one, which is curved.

Now, let us take a solid sphere, and slice it exactly ‘through the middle’ with a plane that passes through its centre.

What happens to the sphere?

Yes, it gets divided into two equal parts (see Fig. 13.20)! What will each half be called? It is called a hemisphere. (Because ‘hemi’ also means ‘half’)

And what about the surface of a hemisphere? How many faces does it have?

Two! There is a curved face and a flat face (base).

The curved surface area of a hemisphere is half the surface area of the sphere, which is `1/2` of `4pir^2`

`color{green}("Therefore,")`

`color{orange}("Curved Surface Area of a Hemisphere" = 2πr^2)`

where r is the radius of the sphere of which the hemisphere is a part.

Now taking the two faces of a hemisphere, its surface area `2πr^2 + πr^2`

So,

`color{navy}("Total Surface Area of a Hemisphere" = 3πr^2)`

Q 3210878719

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

Class 9 Chapter 13 Example 7

Class 9 Chapter 13 Example 7

The surface area of a sphere of radius 7 cm would be

`4pir^2= 4 xx(22)/7 xx7xx7 cm^2= 616cm^2`

Q 3210078810

Find (i) the curved surface area and (ii) the total surface area of a hemisphere of radius 21 cm.

Class Chapter 13 Example 8

Class Chapter 13 Example 8

The curved surface area of a hemisphere of radius 21 cm would be

`= 2πr2 = 2 × 22/7 × 21 × 21 cm^2 = 2772 cm^2`

(ii) the total surface area of the hemisphere would be

`3πr^2 = 3 xx22/7 21 × 21 cm^2 = 4158 cm^2`

Q 3230078812

The hollow sphere, in which the circus motorcyclist performs his stunts, has a diameter of 7 m. Find the area available to the motorcyclist for riding.

Class 9 Chapter 13 Example 9

Class 9 Chapter 13 Example 9

Diameter of the sphere = 7 m. Therefore, radius is 3.5 m. So, the riding space available for the motorcyclist is the surface area of the ‘sphere’ which is given by

`4πr^2 = 4 xx 22/7 × 3.5 × 3.5 m^2`

`= 154 m^2`

Q 3240078813

A hemispherical dome of a building needs to be painted (see Fig. 13.21). If the circumference of the base of the dome is 17.6 m, find the cost of painting it, given the cost of painting is Rs.5 per 100 `cm^2.`

Class 9 Chapter 13 Example 10

Class 9 Chapter 13 Example 10

Since only the rounded surface of the dome is to be painted, we would need to find the curved surface area of the hemisphere to know the extent of painting that needs to be done. Now, circumference of the dome = 17.6 m. Therefore, `17.6 = 2pir`

So, the radius of the dome `= 17.6 xx7/(2xx22) m = 2.8m`

The curved surface area of the dome `= 2πr^2`

`2xx(22)/7 xx2.8 x2.5 m^2`

`= 49.28 m^2`

Now, cost of painting `100 cm^2` is `Rs. 5.`

So, cost of painting `1 m^2 = Rs 500`

Therefore, cost of painting the whole dome.

`= Rs.500 × 49.28`

`= Rs 24640`