Surface Areas and Volume Class 9 Worksheet Maths

Sol: The area of a circle of radius r is πr² 
Thus if the hemisphere is meant to include the base then the surface area is 2πr² + πr² = 3πr² 

Sol: Height (h) of conical tent = 10 m
Radius (r) of conical tent = 24 m
Let the slant height of the tent be l
l² = h² + r²
l² = (10)² + (24)²
l² = 100 + 576 
l² = 676 
l = √676

l = √26²
l = 26 m
Therefore, the slant height of the tent is 26 m.

Sol: Diameter of the base of the cone is 10.5 cm and slant height is 10 cm.
Curved surface area of a right circular cone of base radius, ['r']and slant height, l is πr.
Diameter, d = 10.5 cm
Radius, r = 10.5 / 2 cm= 5.25 cm
Slant height, l = 10 cm
Curved surface area = πrl
= 3.14 × 5.25 × 10 = 165 cm
Thus, curved surface area of the cone = 165 cm².

Sol: Given radius of sphere = 5.6 cm
Surface area of sphere = 4πr² 
= 4 × 3.14 × (5.6)² 
Surface area of sphere = 393.88 cm² 

Sol: Volume of the cone = 1/3 πr²h
Given
Slant height = l= 28 cm
Height of cone = h= 21 cm
Let radius of cone = r cm
l² = h² + r²
28² = 21² + r² 
28² - 21² = r² 
r² = 28² - 21² 
r² = (28 - 21)(28 + 21)
r² =(7)(49)
r = √7(49)
r = √7(7)² 
r = 7√7 cm
Volume of the cone = 1/3 πr² h

Sol: Given Diameter of sphere =14cm radius =7cm
surface area of sphere = 4πr² = 4π(7)² 
= 4 × 3.14 × 49
surface area of sphere = 616cm²  

Sol: The formula to calculate the volume of a hollow cylinder is given as, 
Volume of hollow cylinder =π(R²−r²)h cubic units,
where, 'R′ is the outer radius, ' r ' is the inner radius, and, ' h ' is the height of the hollow cylinder.

Sol: Given surface area of sphere =154cm²
Let radius of the sphere = r cm
4πr² = 1544 × 227 × r²
=154r²
Volume of sphere =4/3πr³ 
=179.67cm³

Sol: The volume of water the bowl contain =2/3πr³ 
Radius of hemisphere =r=3.5cm
The volume of water the bowl can contain =2/3πr³ 
= 2/3 × 22/7 × 3.5 × 3.5 × 3.5cm³
= 89.8cm³

Sol: The formula for the volume of a cone is:  13 π r² h

Sol: Let the radius of the sphere = r.
According to the question,
height and diameter of cylinder = diameter of sphere.
So, the radius of the cylinder = r
And, the height of the cylinder = 2r
We know that,
Volume of sphere = 2/3 volume of cylinder
Hence, the given statement “the volume of a sphere is equal to two-third of the volume of a cylinder whose height and diameter are equal to the diameter of the sphere” is true.

Sol: Let the original radius of the cone = r
Let height of the cone = h.
The volume of cone = 1/3 πr²h
Now, when radius of a height circular cone is halved and height is doubled, then
We can observe that the new volume = half of the original volume.
Hence, the given statement “if the radius of a right circular cone is halved and height is doubled, the volume will remain unchanged” is false.

Sol: Let radius of the cylinder = r
Height of the cylinder = h
Then, curved surface area of the cylinder, CSA = 2πrh
According to the question,
Radius is doubled and curved surface area is not changed.
New radius of the cylinder, R = 2r
New curved surface area of the cylinder, CSA’ = 2πrh …(i)
Alternate case:
When R = 2r and CSA’ = 2πrh
But curved surface area of cylinder in this case, CSA’= 2πRh = 2π(2r)h = 4πrh …(ii)
Comparing equations (i) and (ii),
We get,
Since, 2πrh ≠ 4πrh
equation (i) ≠ equation (ii)
Thus, if h = h/2 (height is halved)
Then,
CSA’ = 2π(2r)(h/2) = 2πrh
Hence, the given statement “If the radius of a cylinder is doubled and its curved surface area is not changed, the height must be halved” is true.

Sol: Formula for the volume of a sphere:

V = 43 π r³

If the radius is doubled (i.e., r becomes 2r), then the new volume V' is:

V' = 43 π (2r)³ = 43 π 8r³ = 8 × 43 π r³

Thus, doubling the radius increases the volume by a factor of 8, not 2.

Sol: The total surface area of a cone is the sum of its lateral surface area and the area of its circular base:

Lateral Surface Area = π r l

Area of Circular Base = π r²

Total Surface Area = π r l + π r²

For a cone with radius r and slant height l, the total surface area A is given by:

A = π r l + π r2