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## Mathematics Surface Areas And Volumes Worksheets for Class 9

We have provided **chapter-wise worksheets for class 9 Mathematics Surface Areas And Volumes** which the students can download in Pdf format for free. This is the best collection of Mathematics Surface Areas And Volumes standard 9th worksheets with important questions and answers for each grade 9th Mathematics Surface Areas And Volumes chapter so that the students are able to properly practice and gain more marks in Class 9 Mathematics Surface Areas And Volumes class tests and exams.

### Chapter-wise Class 9 Mathematics Surface Areas And Volumes Worksheets Pdf Download

1. Surface Area of a Cuboid and a Cube

2. Surface Area of a Right Circular Cylinder

3. Surface Area of a Right Circular Cone

4. Surface Area of a Sphere

5. Volume of a Cuboid

6. Volume of a Cylinder

7. Volume of a Right Circular Cone

8. Volume of a Sphere

**• Polyhedrons Shapes:(i) Cube:**

Cube whose edge = a

Diagonal of Cube = 3a

Lateral Surface Area of Cube = 4a

^{2}

Total Surface Area of Cube = 6a

^{2}

Volume of Cube = a

^{3}

**(ii) Cuboid:**

Cuboid whose length = l, breadth = b and height = h

Diagonal of Cuboid =√l2 + b^{2} + h^{2}

Lateral Surface Area of Cuboid = 2(l + b) h

Total Surface Area of Cuboid = 2 (lb + bh + hl)

Volume of Cuboid = lbh

**• Non-polyhedrons:(i) Cylinder:**

Cylinder whose radius = r, height = h

Curved Surface Area of Cylinder = 2πrh

Total Surface Area of Cylinder = 2π rh (r + h)

Volume of Cylinder = πr2h

**(ii) Cone:**

Cone having height = h, radius = r and slant height = l

Slant height of Cone (l) = r^{2} + h^{2}

Curved Surface Area of Cone = πrl

Total Surface Area of Cone = πr (r + l)

Volume of Cone = 1/3 πr2h

**(iii) Sphere:**

Sphere whose radius = r

Surface Area of a Sphere = 4πr^{2}

Volume of Sphere = 4/3πr^{2}

**(iv) Hemisphere:**

Hemisphere whose radius = r

Curved Surface Area of Hemisphere = 2πr^{2}

Total Surface Area of Hemisphere = 3πr^{2}

Volume of Hemisphere = 2/3 πr^{3}

**Question.****The total surface area of a cube is 96 cm ^{2}. The volume of the cube is:**(a) 8cm

^{3}

(b) 512cm

^{3}

(c) 64cm

^{3}

(d) 27cm

^{3}

**Answer**

C

** Question. The length of the longest pole that can be put in a room of dimensions (10 m × 10 m × 5m) is**(A) 15 m

(B) 16 m

(C) 10 m

(D) 12 m

**Answer**

A

**Question.****A cone is 8.4 cm high and the radius of its base is 2.1 cm. It is melted and recast into a sphere. The radius of the sphere is:**(A) 4.2 cm

(B) 2.1 cm

(C) 2.4 cm

(D) 1.6 cm

**Answer**

B

**Question.** The radius of a sphere is 2r, then its volume will be

**Answer**

D

** Question. In a cylinder, radius is doubled and height is halved, curved surface area will be**(A) halved

(B) doubled

(C) same

(D) four time

**Answer**

C

** Question. The radii of two cylinders are in the ratio of 2:3 and their heights are in the ratio of 5:3. The ratio of their volumes is:**(A) 10 : 17

(B) 20 : 27

(C) 17 : 27

(D) 20 : 37

**Answer**

B

** Question. The number of planks of dimensions (4 m × 50 cm × 20 cm) that can be stored in a pit which is 16 m long, 12m wide and 4 m deep is**(A) 1900

(B) 1920

(C) 1800

(D) 1840

**Answer**

B

**Question.****The lateral surface area of a cube is 256 m ^{2}. The volume of the cube is**

(A) 512 m

^{3}

(B) 64 m

^{3}

(C) 216 m

^{3}

(D) 256 m

^{3}

**Answer**

A

**Question.** The total surface area of a cone whose radius is (2/r) and slant height 2l is

**Answer**

B

**Question.****The radius of a hemispherical balloon increases from 6 cm to 12 cm as air is being pumped into it. The ratios of the surface areas of the balloon in the two cases is**(A) 1 : 4

(B) 1 : 3

(C) 2 : 3

(D) 2 : 1

**Answer**

A

** Question. The volume of the largest right circular cone that can be fitted in a cube whose edge is 2r equals to the volume of a hemisphere of radius r.Answer.** The height of the largest cone is 2r that can be fitted in a cube whose edge is 2r.

Its volume = 1/3 πr

^{2}(2r) 2/3 πr

^{3}

But 2/3 πr

^{3}is the volume of a hemisphere of radius r.

Hence, the given statement is true.

** Question. If the radius of a right circular cone is halved and height is doubled, the volume will remain unchanged.Answer. **Let the original radius of the cone be r and height be h.

The volume of cone = 1/3 πr

^{2}h

Now, when radius of a height circular cone is halved and height is doubled, then

We see that the volume becomes half of the original volume.

Hence, the given statement is false.

**Question.** If the radius of a cylinder is doubled and its curved surface area is not changed, the height must be halved.** Answer.** Let r be the radius and h the height of the cylinder.

Then the curved surface area = 2πrh

When the radius is double and the curved surface area is not changed, the height must be halved.

Then the curved surface area 2π (2r) h/2 which is equal to = 2πrh.

Hence, the given statement is true.

**Question.****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. Answer.** Let the radius of the sphere be r.

As it is given that height and diameter of cylinder are equal to the diameter of sphere.

So, the radius of the cylinder is r and the height of the cylinder = 2r

Now, volume of sphere = 2/3 volume of cylinder

⇒ 4/3 πr

^{3}= (πr

^{2}× 2r = 4/3 πr

^{3}

Hence, the given statement is true.

**Question.****In a right circular cone, height, radius and slant height do not always be sides of a right triangle.**** Answer.** In a right circular cone, height (h), radius(r) and slant height (l) are always the sides of right triangle i.e., l

^{2}+ r

^{2}= h

^{2}

Hence, the given statement is false.

**Question.****If the length of the diagonal of a cube is 6√3cm, then the length of the edge of the cube is 3 cm. Answer.** Let the length of the edge of the cube be a

Then the diagonal of the cube is 3a

Also, the length of the diagonal of a cube is 6√3.

∴ √3a = 6√3 ⇒ a = 6So, the length of the edge of the cube is 6 cm.

Hence, the given statement is false.

**Question.****A cylinder and a right circular cone are having the same base and same height. The volume of the cylinder is three times the volume of the cone.**** Answer.** Let r the radius and h be height of the cylinder and a right circular cone.

Now, volume of cylinder = πr

^{2}h

And Volume of cone = 1/3 πr

^{2}h

Clearly, volume of cylinder = 3(Volume of cone)

Hence, the given statement is true.

**Question.** A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their volumes is 1 : 2 : 3.** Answer.** A cone, a hemisphere and a cylinder stand on equal bases and same height.

Let radius of each of a cone, a hemisphere and cylinder be r (equal bases)

Height of hemisphere = r,

So, the height of the cone = r,

And height of cylinder = r.

i.e., the ratio of their volume is 1 : 2 : 3.

Hence, the given statement is true.

**Question.****If a sphere is inscribed in a cube, then the ratio of the volume of the cube to the volume of the sphere will be 6 : π. Answer. **Let a be the edge of the cube.

As the sphere is inscribe in a cube, the radius of the sphere is a/2

V

_{1}= Volume of cube = (edge)3 = a

^{3}

∴ V

_{1}: V

_{2}= a

^{3}: (1/6) πa

^{3}= 1 : π/6 = 6 : π

So, the ratio of the volume of the cube to the volume of the sphere will be 6 :π .

Hence, the given statement is true.

**Question.** If the radius of a cylinder is doubled and height is halved, the volume will be doubled.** Answer.** Let r be the radius and h the height of the cylinder.

∴ Original volume 2

(V

_{1}) = π r

^{2}h.

When radius of cylinder is doubled and height is halved (V

_{2})

Hence, the given statement is true.

Volume of a sphere = 4/3 πr^{3}

= 4/3 x 3.14 x (2)3 = 100.48 / 3 = 33.49cm^{3}

Volume of 16 such spheres = (33.49 × 16) cm^{3} = 535.84 cm^{3}

When 16 spheres are packed, the box is filled with preservative liquid.

Volume of the preservative liquid = 10^{24} – 535.84 cm^{3} = 488.16 cm^{3}

** Question. Find the amount of water displaced by a solid spherical ball of diameter 4.2 cm, when it is completely submerged in water.Answer.** Amount of water displaced by a solid spherical ball = Volume of solid spherical ball.

Hence, the amount of water displaced by solid spherical ball when it completely immersed in water = 38.808 cm^{3}

**Question.** A storage tank is in the form of a cube. When it is full of water, the volume of water is 15.625 m^{3}. If the present depth of water is 1.3 m, find the volume of water already used from the tank.** Answer.** Let the edge of the cube be x.

Then, its volume = x

^{3},

When the cube (storage tank) is full of water, the volume of water is 15.625 m

^{3}

So, the volume of cube (storage tank) = 15.625 m

^{3}

∴ Edge of the cube = 2.5 m

Present depth of water in the tank = 1.3 m

Remaining depth = 2.5m – 1.3m = 1.2m

∴ Volume of water already used in the tank = 2.5 × 2.5 × 1.2 = 7.5 m^{3}

**Question.** How many square metres of canvas is required for a conical tent whose height is 3.5 m and the radius of the base is 12 m?** Answer.** Area of canvas required = πrl

**Question.****A cylindrical roller 2.5 m in length, 1.75 m in radius when rolled on a road was found to cover the area of 5500 m ^{2}. How many revolutions did it made?**

**Length (height) of cylinder roller is 2.5 m and radius of the roller is 1.75 m.**

**Answer.**In one revolution area covered = lateral surface area of the cylinder

2πrh = 2 x 22/7 x 1.75 x 2.5m

^{2}

44 × 0.25 × 2.5 = 27.5m

Total area on the road covered by cylinder roller = 5500 m

^{2}.

Hence, number of revolution made by the roller

**Question.****Two solid spheres made of the same metal have weights 5920 g and 740 g, respectively. Determine the radius of the larger sphere, if the diameter of the smaller one is 5 cm.**** Answer.** Two sides spheres made of the same metal have weights 5920 g and 740 g, respectively. Mass per unit volume is called the density.

Hence, the radius of the larger sphere = 5 cm.

**Question.****A right triangle with sides 6 cm, 8 cm and 10 cm is revolved about the side 8 cm. Find the volume and the curved surface of the solid so formed. Answer. **When a right triangle is revolved about me of its sides (containing the right angles), a cone is formed. Now, we have,

Radius (r) of the base of the cone = 6 cm

Height (h) of the cone = 8 cm

And the slant height (l) = 10 cm

Now, volume of the cone = 1/2 πr^{2}h = 1/3 x 22/7 x (6)^{2} x 8

= 22 12 8 / 7 = 301.71cm^{3}

Curved surface area of cone = πrl = 22/7 x 6 x 10 = 1320

= 188.57cm^{2}

**Question.** A small village, having a population of 5000, requires 75 litres of water per head per day. The village has got an overhead tank of measurement 40 m × 25 m × 15 m. For how many days will the water of this tank last?** Answer.** Water contained in overhead tank = (40 × 25 × 15) m

^{3}

= (40 × 25 × 15 × 1,000) litres

Water needed for 5,000 village for one day

= (5,000 × 75) litres = 375000 litres

∴ Number of days the water of the tank last

= 40 x 25 x 15 x 1000 / 375000 = 15000 / 375 = 40 days

** Question. A shopkeeper has one spherical laddoo of radius 5cm. With the same amount of material, how many laddoos of radius 2.5 cm can be made?Answer. **Numbers of ladoos = Volume of spherical ladoo of radius 5cm / Volume of one spherical ladoo of radius 2.5cm

**Question.** A school provides milk to the students daily in a cylindrical glasses of diameter 7 cm. If the glass is filled with milk upto an height of 12 cm, find how many litres of milk is needed to serve 1600 students.** Answer.** Radius of cylinder glass = 7 ÷ 2 = 3.5 cm.

Glass is filled with milk upto an height of 12 cm.

Quantity of milk needed for 1600 students = (462 × 1600) cm^{3}

= 739.2 litres.

**Question.** A cylindrical tube opened at both the ends is made of iron sheet which is 2 cm thick.**If the outer diameter is 16 cm and its length is 100 cm, find how many cubic** **centimeters of iron has been used in making the tube?**** Answer.** A cylinder tube is made iron sheet. Which is 2cm thick.

Its outer radius = 16 ÷ 2 = 8

Thickness of iron sheet = 2 cm

And its inner radius = 8 cm – 2 cm = 6 cm.

Height of cylinder = 100 cm

Quantity of iron used in making the tube = π (R

^{2}− r

^{2})h

** Question. A cloth having an area of 165 m^{2} is shaped into the form of a conical tent of radius 5m** (i) Number of students who can sit in the tent if a student, on an average, occupies 5/7)m

(i) How many students can sit in the tent if a student, on an average, occupies (5/7)m^{2} on the ground?

(ii) Find the volume of the cone.

Answer.

^{2}on the ground.

** Question. A sphere and a right circular cylinder of the same radius have equal volumes. By what percentage does the diameter of the cylinder exceed its height?Answer. **Volume of sphere

= 4 / 3 πr

^{3}

And volume of right circular cylinder 2 = πr h

∴ Volume of sphere = volume of right circular cylinder [Given]

⇒ 4 / 3 πr

^{3}h ⇒ = (4/3) r

Diameter of the cylinder exceeds its height by

⇒ 2r – (4/3) r = 2r/2

Percentage by which diameter of the cylinder exceeds its hieght

**Question.****The water for a factory is stored in a hemispherical tank whose internal diameter is 14 m. The tank contains 50 kilolitres of water. Water is pumped into the tank to fill to its capacity. Calculate the volume of water pumped into the tank.**** Answer.** Internal diameter of hemispherical tank = 14 m

∴ Internal radius of hemispherical tank = 14 m ÷ 2 = 7 m

Volume of hemispherical tank = 2/3 πr

^{3}= 2/3 x 22/7 x (7)

^{3}

= 44 x 49 / 3 = 718.66m

^{3}

The tank contains 50 Kilolitres of water = 50,000 litres = (50,000 / 1,000) m

^{3}= 50m

^{3}Volume of water pumped into the tank = 718.66 m

^{3}– 50 m

^{3}= 668.66 m

^{3}

**Question.** The volumes of the two spheres are in the ratio 64 : 27. Find the ratio of their surface areas.** Answer.** Let V

_{1}and V

_{2}be the volumes of two spheres.

Now, Let SA_{1} and SA_{2} be the surface areas of two spheres.

So, the ratio of the surface area of the two spheres is 16:9.

**Question.****A cube of side 4 cm contains a sphere touching its sides. Find the volume of the gap in between.**** Answer.** Side of cube = 4 cm.

Volume of cube 3 3 = (4) = 4 × 4 × 4 = 64cm

As cube contains a sphere touching its sides, so the diameter of the sphere = 4 cm.

Radius of sphere = 4 cm ÷ 2 = 2 cm

Volume of sphere = 4/3 πr

^{2}= 4 / 3 x 22 / 7 x (2)

^{3}

= 88 x 8 / 21 = 704 / 21 = 33.52cm

^{2}

Volume of gap in between them = 64 cm

^{3 }– 33.52 cm

^{3}= 30.48 cm

^{3}.

**Question.** A semi-circular sheet of metal of diameter 28 cm is bent to form an open conical cup. Find the capacity of the cup.** Answer.** Diameter of semicircular sheet is 28 cm. It is bent to from an open conical cup. The radius of sheet becomes the slant height of the cup. The circumference of the sheet becomes the circumference of the base of the cone.

∴ l = Slant height of conical cup = 14 cm.

Let r cm be the radius and h cm be the height of the conical cup circumference of conical

cup of the semicircular sheet

∴ 2πr = π ×14 ⇒ r = 7cm

**Question.****30 circular plates, each of radius 14 cm and thickness 3cm are placed one above the another to form a cylindrical solid. Find:(i) the total surface area(ii) volume of the cylinder so formed.**Radius of one circular plate = 14 cm

**Answer.**

Thickness of one circular plate = 3 cm.

As the plates are placed one above the other, so the height of the cylinder formed by placing 30 plates = 30 × 3 = 90 cm

(i) Total surface area of the cylinder

## Mathematics Surface Areas And Volumes Worksheets for Class 9 as per CBSE NCERT pattern

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