A cone has a slant height of 15 cm and a base radius of 9 cm. What is the volume of the cone if its height is 12 cm? (Use π = 3.14)

Correct Option: A. Solution: Volume = \frac{1}{3}πr²h = \frac{1}{3} × 3.14 × 9² × 12 = 1017.36 cm³.

Surface Areas and Volumes

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CBSE Learning Objectives – Key Concepts & Skills You Must Know

Understand the concepts of surface areas and volumes of various geometric shapes.
Calculate the volume of a sphere using the formula: Volume of a Sphere = $\frac{4}{3} \pi r^3$ .
Determine the volume of a hemisphere using the formula: Volume of a Hemisphere = $\frac{2}{3} \pi r^3$ .
Compute the surface area of a sphere using the formula: Surface Area of a Sphere = $4 \pi r^2$ .
Calculate the curved surface area of a cone using the formula: Curved Surface Area of a Cone = $\pi r l$ , where $l$ is the slant height.
Find the total surface area of a cone using the formula: Total Surface Area of a Cone = $\pi r l + \pi r^2$ .
Apply the principles of volume displacement to find the volume of irregular objects.

CBSE Revision Notes & Quick Summary for Last-Minute Study

Chapter 11: Surface Areas and Volumes

11.1 Surface Area of a Right Circular Cone

Curved Surface Area:
- Formula: πrl
- Where:
  - r = base radius
  - l = slant height
Total Surface Area:
- Formula: πrl + πr² = πr(l + r)

11.2 Surface Area of a Sphere

Surface Area:
- Formula: 4πr²
- Where: r = radius of the sphere

11.3 Volume of a Right Circular Cone

Volume:
- Formula: (1/3)πr²h
- Where:
  - r = base radius
  - h = height

11.4 Volume of a Sphere

Volume:
- Formula: (4/3)πr³
- Where: r = radius of the sphere
Volume of a Hemisphere:
- Formula: (2/3)πr³

Summary of Key Formulas

Curved Surface Area of a Cone: πrl
Total Surface Area of a Cone: πr(l + r)
Surface Area of a Sphere: 4πr²
Volume of a Cone: (1/3)πr²h
Volume of a Sphere: (4/3)πr³
Volume of a Hemisphere: (2/3)πr³

Examples

Example 1: Find the curved surface area of a right circular cone with slant height 10 cm and base radius 7 cm.
- Solution: Curved Surface Area = πrl = 22/7 * 7 * 10 cm² = 220 cm²
Example 2: Find the volume of a sphere of radius 11.2 cm.
- Solution: Volume = (4/3)π(11.2)³ cm³ = 5887.32 cm³ (approx)

Exercises

Find the volume of a sphere whose radius is:
- (i) 7 cm
- (ii) 0.63 m
Find the amount of water displaced by a solid spherical ball of diameter:
- (i) 28 cm
- (ii) 0.21 m
The diameter of a metallic ball is 4.2 cm. What is the mass of the ball, if the density of the metal is 8.9 g/cm³?
The diameter of the moon is approximately one-fourth of the diameter of the earth. What fraction of the volume of the earth is the volume of the moon?
How many litres of milk can a hemispherical bowl of diameter 10.5 cm hold?
A hemispherical tank is made up of an iron sheet 1 cm thick. If the inner radius is 1 m, then find the volume of the iron used to make the tank.
Find the volume of a sphere whose surface area is 154 cm².
A dome of a building is in the form of a hemisphere. From inside, it was white-washed at the cost of ₹ 4989.60. If the cost of white-washing is ₹ 20 per square metre, find:
- (i) inside surface area of the dome
- (ii) volume of the air inside the dome.
Twenty-seven solid iron spheres, each of radius r and surface area S, are melted to form a sphere with surface area S'. Find:
- (i) radius r' of the new sphere
- (ii) ratio of S and S'.
A capsule of medicine is in the shape of a sphere of diameter 3.5 mm. How much medicine (in mm³) is needed to fill this capsule?

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

Common Mistakes and Exam Tips

Common Pitfalls

Misunderstanding Volume Formulas: Students often confuse the volume formulas for spheres, cones, and hemispheres. Ensure you memorize the correct formulas:
- Volume of a Sphere: $V = \frac{4}{3} \pi r^3$
- Volume of a Cone: $V = \frac{1}{3} \pi r^2 h$
- Volume of a Hemisphere: $V = \frac{2}{3} \pi r^3$
Incorrect Unit Conversions: Be careful with unit conversions, especially when dealing with measurements in cm and m. Always convert to the same unit before performing calculations.
Neglecting to Use π Correctly: Some students forget to use the value of π correctly. In exercises, it is often given as $\pi = \frac{22}{7}$ or 3.14. Make sure to apply the correct value as specified.
Forgetting to Include All Parts of Surface Area: When calculating the total surface area of cones or hemispheres, remember to include both the curved surface area and the base area if applicable.

Exam Tips

Practice with Different Shapes: Work on problems involving various shapes (spheres, cones, hemispheres) to become familiar with their properties and formulas.
Draw Diagrams: Whenever possible, draw diagrams to visualize the problem. This can help in understanding the relationships between different dimensions.
Check Your Work: After solving a problem, take a moment to review your calculations and ensure that you have used the correct formulas and units.
Time Management: Allocate your time wisely during exams. If you find a problem too challenging, move on and return to it later if time permits.

CBSE Quiz & Practice Test – MCQs, True/False Questions with Solutions

Multiple Choice Questions

96π cm²

108π cm²

120π cm²

132π cm²

Correct Answer: B

Solution:

The total surface area of a cone is given by

\pi r l + \pi r^2

. Here,

r = 6

cm and

l = 10

cm. Thus, the total surface area is

\pi \times 6 \times 10 + \pi \times 6^2 = 60\pi + 36\pi = 96\pi

cm².

1 cm

2 cm

3 cm

4 cm

Correct Answer: B

Solution:

The volume of the cone is

V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi \times 3^2 \times 4 = 12\pi \text{ cm}^3

. For the cylinder with the same base radius,

V_{cylinder} = \pi r^2 H

. Setting the volumes equal,

12\pi = \pi \times 3^2 \times H

. Solving for

H

gives

H = 2 \text{ cm}

6 cm

9 cm

12 cm

15 cm

Correct Answer: C

Solution:

The height of the frustum is the difference between the height of the original cone and the smaller cone. Thus, the height of the frustum is

18 - 6 = 12 \text{ cm}

10 cm

12 cm

14 cm

16 cm

Correct Answer: A

Solution:

The slant height

l

of a cone can be found using the Pythagorean theorem:

l = \sqrt{r^2 + h^2}

. Substituting the given values,

l = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \text{ cm}

50 cm³

75 cm³

100 cm³

125 cm³

Correct Answer: B

Solution:

The volume of a cone is one-third the volume of a cylinder with the same base and height. Therefore, the volume of the cone is

\frac{1}{3} \times 150 = 50

cm³.

8 cm

12 cm

16 cm

24 cm

Correct Answer: A

Solution:

The volume of the cone is

V_{\text{cone}} = \frac{1}{3} \pi r^2 h

. For the cylinder with the same base radius, the height of water

h_{\text{water}}

will be such that the volume

V_{\text{cylinder}} = \pi r^2 h_{\text{water}}

equals

V_{\text{cone}}

. Therefore,

h_{\text{water}} = \frac{1}{3} \times 24 = 8

cm.

754.76 cm²

615.44 cm²

550 cm²

300 cm²

Correct Answer: B

Solution:

First, find the slant height

l

using

l = \sqrt{r^2 + h^2}

l = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25

cm. Total surface area is

\pi r l + \pi r^2 = 3.14 \times 7 \times 25 + 3.14 \times 7^2 = 549.5 + 153.86 = 703.36

cm².

718.24 cm³

719.24 cm³

720.24 cm³

721.24 cm³

Correct Answer: A

Solution:

The volume of the hemisphere is

V_h = \frac{2}{3} \pi r^3 = \frac{2}{3} \times 3.14 \times 7^3 = 718.24

cm³. The volume of the cone is

V_c = \frac{1}{3} \pi r^2 h = \frac{1}{3} \times 3.14 \times 7^2 \times 7 = 359.12

cm³. The volume of the space not occupied by the cone is

V_h - V_c = 718.24 - 359.12 = 359.12

cm³.

13 cm

14 cm

15 cm

16 cm

Correct Answer: A

Solution:

Using the Pythagorean theorem, the slant height

l

is calculated as

l = \sqrt{r^2 + h^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13

cm.

7 cm

6 cm

8 cm

9 cm

Correct Answer: A

Solution:

The curved surface area of a cone is given by

\pi r l

. Here,

\pi = \frac{22}{7}

l = 14

cm, and the curved surface area is 308 cm². Thus,

\frac{22}{7} \times r \times 14 = 308

. Solving for

r

, we get

r = 7

cm.

84.78 cm³

84.6 cm³

84.5 cm³

85 cm³

Correct Answer: A

Solution:

The volume of a cone is given by

\frac{1}{3} \pi r^2 h

. Substituting the given values:

\frac{1}{3} \times 3.14 \times 3^2 \times 9 = 84.78

cm³.

26 m

25 m

24 m

23 m

Correct Answer: A

Solution:

Using the Pythagorean theorem,

l^2 = r^2 + h^2

. Substituting the given values,

l^2 = 10^2 + 24^2 = 100 + 576 = 676

. Thus,

l = 26

424.26 cm²

508.68 cm²

678.24 cm²

754.32 cm²

Correct Answer: B

Solution:

The total surface area of a cone is given by

\pi r l + \pi r^2

. Substituting the given values,

\pi = 3.14

r = 9

cm, and

l = 15

cm, we get:

\text{Total Surface Area} = 3.14 \times 9 \times 15 + 3.14 \times 9^2 = 424.26 + 254.34 = 508.68 \text{ cm}^2

15 cm

16 cm

14 cm

13 cm

Correct Answer: A

Solution:

Using the Pythagorean theorem for the cone,

l^2 = r^2 + h^2

. Substituting the given values:

17^2 = 8^2 + h^2

289 = 64 + h^2

h^2 = 225

h = 15

cm.

1:2

1:4

1:8

1:16

Correct Answer: B

Solution:

The surface area of a sphere is

4\pi r^2

. If the radius doubles, the new radius is

2r

, and the new surface area is

4\pi (2r)^2 = 16\pi r^2

. The ratio of the new surface area to the original is

16\pi r^2 : 4\pi r^2 = 4:1

188.4 cm²

180 cm²

200 cm²

220 cm²

Correct Answer: A

Solution:

The curved surface area of a cone is given by the formula

\pi rl

, where

r

is the base radius and

l

is the slant height. Thus, the curved surface area is

3.14 \times 6 \times 10 = 188.4 \text{ cm}^2

1017.36 cm³

1018 cm³

1020 cm³

1021 cm³

Correct Answer: A

Solution:

Volume = \frac{1}{3}πr²h = \frac{1}{3} × 3.14 × 9² × 12 = 1017.36 cm³.

5 cm

4 cm

6 cm

7 cm

Correct Answer: C

Solution:

Using the Pythagorean theorem, the slant height l is given by l² = r² + h². So, l = √(3² + 5²) = √(9 + 25) = √34 = 6 cm.

15 cm

16 cm

17 cm

18 cm

Correct Answer: A

Solution:

The slant height

l

of a cone can be found using the Pythagorean theorem:

l = \sqrt{h^2 + r^2}

. Here,

h = 9

cm and

r = 12

cm. So,

l = \sqrt{9^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15

cm.

6 cm

7 cm

8 cm

9 cm

Correct Answer: A

Solution:

The volume of the cone is given by

V_{cone} = \frac{1}{3} \pi r^2 h

. First, find the height of the cone using the Pythagorean theorem:

l^2 = r^2 + h^2 \Rightarrow 15^2 = 9^2 + h^2 \Rightarrow h = 12 cm

. Now, calculate the volume of the cone:

V_{cone} = \frac{1}{3} \times 3.14 \times 9^2 \times 12 = 1017.36 \text{ cm}^3

. The volume of the sphere is

V_{sphere} = \frac{4}{3} \pi R^3

. Equating the volumes,

\frac{4}{3} \pi R^3 = 1017.36

. Solving for

R

, we get

R = 6 \text{ cm}

100 cm³

150 cm³

200 cm³

250 cm³

Correct Answer: B

Solution:

The volume of a cone is

\frac{1}{3}

of the volume of a cylinder with the same base and height. Therefore, the volume of the cone is

\frac{1}{3} \times 300 = 100

cm³.

550 m²

600 m²

700 m²

750 m²

Correct Answer: A

Solution:

The curved surface area of a cone is given by

\pi rl

, where

r

is the radius. Here, the radius

r = \frac{14}{2} = 7 \text{ m}

. Thus, the curved surface area is

3.14 \times 7 \times 25 = 550 \text{ m}^2

15 cm

14 cm

16 cm

13 cm

Correct Answer: B

Solution:

Total surface area of a cone is

\pi r l + \pi r^2

. Given

\pi r l + \pi r^2 = 616

, solve for

l

3.14 \times 7 \times l + 3.14 \times 7^2 = 616

. Simplifying gives

21.98l + 153.86 = 616

, so

21.98l = 462.14

, thus

l \approx 14

cm.

2:1

4:1

8:1

16:1

Correct Answer: C

Solution:

The volume of a sphere is given by

V = \frac{4}{3} \pi r^3

. The original volume is

V_1 = \frac{4}{3} \pi \times 5^3

and the new volume is

V_2 = \frac{4}{3} \pi \times 10^3

. The ratio

\frac{V_2}{V_1} = \frac{10^3}{5^3} = 8:1

21 cm

22 cm

23 cm

24 cm

Correct Answer: A

Solution:

The curved surface area of a cone is given by

\pi r l

. Therefore,

3.14 \times 5 \times l = 330

. Solving for

l

, we get

l = \frac{330}{3.14 \times 5} \approx 21 \text{ cm}

452.16 cm³

339.12 cm³

226.08 cm³

113.04 cm³

Correct Answer: B

Solution:

The volume

V

of a cone is given by

V = \frac{1}{3} \pi r^2 h

. The radius

r

is half of the diameter, so

r = 6

cm. The height of the water is 12 cm. Thus, the volume of the water is

V = \frac{1}{3} \times 3.14 \times 6^2 \times 12 = 339.12

cm³.

330.86 cm²

331.86 cm²

332.86 cm²

333.86 cm²

Correct Answer: B

Solution:

Total surface area

= \pi r l + \pi r^2 = 3.14 \times 7 \times 10 + 3.14 \times 7^2 = 219.8 + 153.86 = 331.86 \text{ cm}^2

10 cm

9 cm

11 cm

12 cm

Correct Answer: A

Solution:

Using the Pythagorean theorem,

l^2 = r^2 + h^2

. Substituting the given values:

l^2 = 6^2 + 8^2 \Rightarrow l^2 = 36 + 64 \Rightarrow l^2 = 100 \Rightarrow l = 10

cm.

254.34 cm²

282.6 cm²

204.1 cm²

314 cm²

Correct Answer: B

Solution:

Total surface area of a cone is given by πrl + πr². Here, r = 5 cm and l = 13 cm. So, the total surface area = 3.14 × 5 × 13 + 3.14 × 5² = 204.1 + 78.5 = 282.6 cm².

It doubles

It triples

It quadruples

It remains the same

Correct Answer: C

Solution:

The surface area of a sphere is

4\pi r^2

. If the radius is doubled, the new surface area is

4\pi (2r)^2 = 16\pi r^2

, which is four times the original surface area.

314 cm³

314.2 cm³

314.5 cm³

315 cm³

Correct Answer: A

Solution:

The volume of a cone is given by

\frac{1}{3} \pi r^2 h

. Substituting the given values,

\frac{1}{3} \times 3.14 \times 5^2 \times 12 = 314 \text{ cm}^3

12 cm

10 cm

8 cm

15 cm

Correct Answer: A

Solution:

Using the Pythagorean theorem for the right triangle formed by the height, radius, and slant height of the cone, we have

l^2 = r^2 + h^2

. Substituting the given values:

13^2 = 5^2 + h^2

. This simplifies to

169 = 25 + h^2

, giving

h^2 = 144

. Therefore,

h = 12

cm.

314 cm³

314.67 cm³

315 cm³

315.67 cm³

Correct Answer: B

Solution:

Volume

= \frac{1}{3} \pi r^2 h = \frac{1}{3} \times 3.14 \times 5^2 \times 12 = 314.67 \text{ cm}^3

6 cm

7 cm

8 cm

9 cm

Correct Answer: B

Solution:

The curved surface area

A = \pi r l

. Given

A = 308

cm² and

l = 14

cm, we have

308 = 3.14 \times r \times 14

. Solving for

r

r = \frac{308}{3.14 \times 14} \approx 7

cm.

18 minutes

20 minutes

22 minutes

24 minutes

Correct Answer: B

Solution:

The volume of the cone is given by

V = \frac{1}{3} \pi r^2 h

. Substituting

r = 3

h = 10

m, and

 ext{π} = 3.14

, we get

V = \frac{1}{3} \times 3.14 \times 3^2 \times 10 = 94.2 \text{ m}^3

. At a rate of 5 m³/min, it will take

\frac{94.2}{5} = 18.84

minutes, approximately 20 minutes to fill the tank.

513.6 cm³

102.67 cm³

153.86 cm³

615.44 cm³

Correct Answer: A

Solution:

Volume of a cone is given by

\frac{1}{3} \pi r^2 h

. Here, r = 7 cm and h = 10 cm. So, the volume =

\frac{1}{3} \times 3.14 \times 7^2 \times 10 = 513.6

cm³.

1130.4 m²

942 m²

1256 m²

1570 m²

Correct Answer: A

Solution:

The total surface area of a cone is given by

\pi r l + \pi r^2

. Here,

r = 10

m and

l = 26

m. So, the total surface area is

3.14 \times 10 \times 26 + 3.14 \times 10^2 = 3.14 \times 260 + 3.14 \times 100 = 816.4 + 314 = 1130.4

m².

₹3,080

₹3,080.40

₹3,080.80

₹3,080.20

Correct Answer: A

Solution:

First, calculate the slant height:

l = \sqrt{r^2 + h^2} = \sqrt{7^2 + 24^2} = 25

m. Curved surface area =

\pi r l = 3.14 \times 7 \times 25 = 549.5

m². Total cost =

549.5 \times 70 = ₹3,080

37.68 cm³

37.68 cm²

37.68 m³

37.68 m²

Correct Answer: A

Solution:

The volume of a cone is given by

\frac{1}{3} \pi r^2 h

. Substituting the given values:

\pi = 3.14

r = 3

cm,

h = 4

cm, we get:

\frac{1}{3} \times 3.14 \times 3^2 \times 4 = 37.68

cm³.

1:1

1:2

1:3

1:4

Correct Answer: C

Solution:

The volume of a cone is

\frac{1}{3} \pi r^2 h

and the volume of a cylinder is

\pi r^2 h

. Therefore, the ratio of their volumes is

\frac{1}{3}

35 cm

36 cm

37 cm

38 cm

Correct Answer: B

Solution:

The volume of the sphere is

V = \frac{4}{3} \pi r^3 = \frac{4}{3} \times 3.14 \times 5^3 = 523.33

cm³. This volume is equal to the volume of the cone, which is

\frac{1}{3} \pi R^2 h

, where

R = 3

cm is the base radius of the cone. Solving for

h

, we have

523.33 = \frac{1}{3} \times 3.14 \times 3^2 \times h \Rightarrow h = 36

cm.

4 cm

6 cm

8 cm

10 cm

Correct Answer: A

Solution:

The volume of the cone is

V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi \times 5^2 \times 12 = 100\pi \text{ cm}^3

. The volume of the cylinder with the same base radius is

V_{cylinder} = \pi r^2 H

, where

H

is the height of the water. Setting the volumes equal,

100\pi = \pi \times 5^2 \times H

. Solving for

H

gives

H = 4 \text{ cm}

21.2 minutes

18.8 minutes

14.1 minutes

11.3 minutes

Correct Answer: A

Solution:

The volume of the cone is

V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \times 3.14 \times 3^2 \times 9 = 84.78

m³. At a rate of 2 m³/min, the time taken to fill the tank is

\frac{84.78}{2} = 42.39

minutes. Rounding to one decimal place gives 21.2 minutes.

204.1 cm²

204.2 cm²

204.3 cm²

204.4 cm²

Correct Answer: B

Solution:

The curved surface area of a cone is given by the formula

\pi r l

where

r

is the base radius and

l

is the slant height. Substituting the given values:

\pi = 3.14

r = 5

cm,

l = 13

cm, we get:

3.14 \times 5 \times 13 = 204.2

cm².

₹1884

₹1570

₹1880

₹1575

Correct Answer: A

Solution:

The radius of the base

r = \frac{10}{2} = 5

m. The slant height

l

of the cone is found using the Pythagorean theorem:

l = \sqrt{h^2 + r^2} = \sqrt{12^2 + 5^2} = 13

m. The curved surface area of the cone is

\pi r l = 3.14 \times 5 \times 13 = 204.1

m². The total cost is then

204.1 \times 50 = ₹10205

13 m

10 m

15 m

17 m

Correct Answer: A

Solution:

Using the Pythagorean theorem, the slant height l is given by l² = r² + h². So, l = √(5² + 12²) = √(25 + 144) = √169 = 13 m.

10 cm

12 cm

15 cm

20 cm

Correct Answer: A

Solution:

The curved surface area of a cone is given by

\pi r l

. Substituting the given values, we have

3.14 \times 5 \times l = 314

. Solving for

l

, we get

l = \frac{314}{15.7} = 20

cm.

12 cm

9 cm

10 cm

13 cm

Correct Answer: A

Solution:

To find the height of the cone, use the Pythagorean theorem:

l^2 = r^2 + h^2

where

l = 15

cm and

r = 9

cm. Thus,

15^2 = 9^2 + h^2

which simplifies to

225 = 81 + h^2

. Solving for

h

, we get

h^2 = 144

, so

h = 12

cm.

10.9 cm

11.0 cm

11.1 cm

11.2 cm

Correct Answer: C

Solution:

The volume of the cone is

V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \times 3.14 \times 3^2 \times 9 = 84.78

cm³. The base area of the cylinder is

\pi r^2 = 3.14 \times 3^2 = 28.26

cm². The additional height of water due to the submerged cone is

\frac{84.78}{28.26} = 3

cm. Therefore, the new height of the water is

10 + 3 = 11.1

cm.

47.1 cm²

75.4 cm²

94.2 cm²

113.1 cm²

Correct Answer: A

Solution:

The total surface area

A = \pi r l + \pi r^2

. Given

r = 3

cm and

l = 5

cm,

A = 3.14 \times 3 \times 5 + 3.14 \times 3^2 = 47.1

cm².

423.9 cm²

450 cm²

500 cm²

520 cm²

Correct Answer: A

Solution:

First, find the slant height

l

using

l = \sqrt{r^2 + h^2} = \sqrt{9^2 + 15^2} = \sqrt{81 + 225} = \sqrt{306} \approx 17.5 \text{ cm}

. The total surface area is

\pi rl + \pi r^2 = 3.14 \times 9 \times 17.5 + 3.14 \times 9^2 = 423.9 \text{ cm}^2

10 m

14 m

15 m

20 m

Correct Answer: C

Solution:

The slant height

l

of a cone is given by

l = \sqrt{r^2 + h^2}

. Here,

r = \frac{16}{2} = 8

m and

h = 12

m. Thus,

l = \sqrt{8^2 + 12^2} = \sqrt{64 + 144} = \sqrt{208} \approx 14.42

m, rounded to the nearest integer, it is 15 m.

1,272 cm³

1,272.6 cm³

1,273 cm³

1,273.5 cm³

Correct Answer: A

Solution:

Volume of a cone =

\frac{1}{3} \pi r^2 h = \frac{1}{3} \times 3.14 \times 9^2 \times 15 = 1,272

cm³.

100 cm³

200 cm³

300 cm³

400 cm³

Correct Answer: C

Solution:

The volume

V

of a cone is given by

V = \frac{1}{3}\pi r^2 h

. The radius

r

is half of the diameter, so

r = 5

cm. Thus,

V = \frac{1}{3} \times 3.14 \times 5^2 \times 12 \approx 300

cm³.

1:9

1:27

1:3

1:8

Correct Answer: B

Solution:

The volume of a cone is proportional to the cube of its height when the base radius is proportional to the height. Therefore, the ratio of the volumes is

\left(\frac{4}{12}\right)^3 = \left(\frac{1}{3}\right)^3 = \frac{1}{27}

2:1

3:2

1:1

4:3

Correct Answer: D

Solution:

The volume of the cone is

V_{cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \times 3.14 \times 7^2 \times 24

. The volume of the hemisphere is

V_{hemisphere} = \frac{2}{3} \pi r^3 = \frac{2}{3} \times 3.14 \times 7^3

. Calculating both:

V_{cone} = 1232.16

cm³ and

V_{hemisphere} = 1648.8

cm³. The ratio is

1232.16:1648.8 = 4:3

550 m²

600 m²

700 m²

750 m²

Correct Answer: A

Solution:

The slant height

l

of the cone is calculated using Pythagoras' theorem:

l = \sqrt{r^2 + h^2} = \sqrt{7^2 + 24^2} = 25

m. The curved surface area is

\pi r l = 3.14 \times 7 \times 25 = 550

m².

4 cm

5 cm

6 cm

7 cm

Correct Answer: B

Solution:

The volume of the cone is given by

V = \frac{1}{3} \pi r^2 h

. The slant height

l = 10

cm, and the base radius

r = 6

cm. Using the Pythagorean theorem, we find the height

h

of the cone:

l^2 = r^2 + h^2 \Rightarrow 10^2 = 6^2 + h^2 \Rightarrow h = 8

cm. The volume of the cone is then

V = \frac{1}{3} \times 3.14 \times 6^2 \times 8 = 301.44

cm³. This volume is equal to the volume of the sphere, which is

\frac{4}{3} \pi R^3

, where

R

is the radius of the sphere. Solving for

R

, we get

301.44 = \frac{4}{3} \times 3.14 \times R^3 \Rightarrow R^3 = 72 \Rightarrow R = 5

cm.

True or False

Correct Answer: True

Solution:

When a right-angled triangle is rotated around one of its perpendicular sides, it forms a right circular cone.

Correct Answer: True

Solution:

The total surface area of a right circular cone is the sum of its curved surface area and the area of its base, given by

\pi r l + \pi r^2

Correct Answer: True

Solution:

The volume of a hemisphere is half the volume of a sphere because a hemisphere is literally half of a sphere. The formula for the volume of a sphere is

\frac{4}{3}\pi r^3

, and for a hemisphere, it is

\frac{2}{3}\pi r^3

Correct Answer: True

Solution:

The formula for the curved surface area of a cone is

\pi r l

, where

r

is the radius of the base and

l

is the slant height.

Correct Answer: False

Solution:

A right circular cone is not a prism. Prisms are solids with two congruent bases connected by parallelogram faces. A cone has a circular base and a single vertex.

Correct Answer: True

Solution:

The formula for the curved surface area of a cone is

\pi r l

, where

r

is the base radius and

l

is the slant height.

Correct Answer: True

Solution:

The slant height

l

of a right circular cone can be calculated using the Pythagorean theorem:

l^2 = r^2 + h^2

, where

r

is the radius and

h

is the height.

Correct Answer: False

Solution:

A cone has one-third the volume of a cylinder with the same base radius and height.

Correct Answer: False

Solution:

The volume of a hemisphere is half the volume of a sphere with the same radius.

Correct Answer: False

Solution:

A hemisphere has half the volume of a sphere with the same radius. The volume of a sphere is

\frac{4}{3}\pi r^3

, while the volume of a hemisphere is

\frac{2}{3}\pi r^3

Correct Answer: False

Solution:

A right circular cone is not a pyramid. Pyramids have polygonal bases and triangular faces converging to a point, while a cone has a circular base and a curved surface.

Correct Answer: False

Solution:

The volume of a cone is one-third the volume of a cylinder with the same base radius and height.

Correct Answer: True

Solution:

A sphere is a three-dimensional object formed by rotating a circle around its diameter, creating a surface where every point is equidistant from the center.

Correct Answer: True

Solution:

The curved surface area of a cone is given by the formula

\pi r l

, where

r

is the base radius and

l

is the slant height, which is equivalent to the circumference of the base (

2\pi r

) times the slant height.

Correct Answer: True

Solution:

The total surface area of a cone includes both the curved surface area and the area of its circular base.

Correct Answer: False

Solution:

In a right circular cone, the slant height is the hypotenuse of the right triangle formed by the height and the radius, and is therefore longer than the height.

Correct Answer: True

Solution:

When a right-angled triangle is rotated around one of its perpendicular sides, it forms a right circular cone.

Correct Answer: True

Solution:

The slant height

l

of a right circular cone can be calculated using the Pythagorean theorem:

l^2 = r^2 + h^2

, where

r

is the radius and

h

is the height.

Correct Answer: True

Solution:

The volume of a right circular cone is given by

V = \frac{1}{3}\pi r^2 h

, which is one-third the volume of a cylinder with the same base and height.

Correct Answer: False

Solution:

A right circular cone must have a circular base. If the base is not circular, it cannot be called a right circular cone.

Correct Answer: True

Solution:

A right circular cone is considered a type of pyramid because it has a base that is connected to a single apex point, unlike a prism which has two congruent bases.

Correct Answer: True

Solution:

A sphere is defined as a three-dimensional object where every point on its surface is at a constant distance from its center.

Correct Answer: True

Solution:

In a right circular cone, the slant height is the hypotenuse of the right triangle formed by the height and the radius, making it longer than the height.

Correct Answer: True

Solution:

In a right circular cone, the line joining its vertex to the center of its base is perpendicular to the base.

Correct Answer: False

Solution:

The slant height of a right circular cone is not necessarily equal to the radius of its base. It is the hypotenuse of the right triangle formed by the height and the radius.

Surface Areas and Volumes

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CBSE Learning Objectives – Key Concepts & Skills You Must Know

CBSE Revision Notes & Quick Summary for Last-Minute Study

Chapter 11: Surface Areas and Volumes

11.1 Surface Area of a Right Circular Cone

11.2 Surface Area of a Sphere

11.3 Volume of a Right Circular Cone

11.4 Volume of a Sphere

Summary of Key Formulas

Examples

Exercises

CBSE Exam Tips, Important Questions & Common Mistakes to Avoid

Common Mistakes and Exam Tips

Common Pitfalls

Exam Tips

Multiple Choice Questions

Q1.A right circular cone has a base radius of 6 cm and a slant height of 10 cm. What is the total surface area of the cone?

Correct Answer: B

Q2.A solid cone with base radius 3 cm and height 4 cm is melted and recast into a cylinder with the same base radius. What will be the height of the cylinder?

Correct Answer: B

Q3.A right circular cone is cut by a plane parallel to its base, resulting in a smaller cone and a frustum. If the height of the original cone is 18 cm and the height of the smaller cone is 6 cm, what is the height of the frustum?

Correct Answer: C

Q4.If the height of a right circular cone is 8 cm and the base radius is 6 cm, what is the slant height?

Correct Answer: A

Q5.A cone and a cylinder have the same base radius and height. If the volume of the cylinder is 150 cm³, what is the volume of the cone?

Correct Answer: B

Q6.A cone has a base radius of 7 cm and a height of 24 cm. If the cone is filled with water and poured into a cylinder with the same base radius, what will be the height of the water in the cylinder?

Correct Answer: A

Q7.A cone has a base radius of 7 cm and a height of 24 cm. What is the total surface area of the cone? (Use π=3.14\pi = 3.14π=3.14)

Correct Answer: B

Q8.A hemisphere has a radius of 7 cm. If a cone with the same base radius and height is placed inside the hemisphere, what is the volume of the space not occupied by the cone? (Use π = 3.14)

Correct Answer: A

Q9.What is the slant height of a right circular cone with a base radius of 5 cm and a height of 12 cm?

Correct Answer: A

Q10.If the curved surface area of a cone is 308 cm² and the slant height is 14 cm, what is the radius of the base?

Correct Answer: A

Q11.Find the volume of a cone with a height of 9 cm and a base radius of 3 cm. (Use π=3.14\pi = 3.14π=3.14)

Correct Answer: A

Q12.A conical tent has a base radius of 10 m and a height of 24 m. What is the slant height of the tent?

Correct Answer: A

Q13.A right circular cone has a slant height of 15 cm and a base radius of 9 cm. Calculate the total surface area of the cone. Use π=3.14\pi = 3.14π=3.14.

Correct Answer: B

Q14.A cone has a slant height of 17 cm and a base radius of 8 cm. What is the height of the cone?

Correct Answer: A

Q15.A spherical balloon has a radius of 7 cm. If the balloon is inflated to double its original radius, what is the ratio of the new surface area to the original surface area?

Correct Answer: B

Q16.A right circular cone has a base radius of 6 cm and a slant height of 10 cm. What is the curved surface area of the cone? (Use π = 3.14)

Correct Answer: A

Q17.A cone has a slant height of 15 cm and a base radius of 9 cm. What is the volume of the cone if its height is 12 cm? (Use π = 3.14)

Correct Answer: A

Q18.If the height of a right circular cone is 5 cm and the base radius is 3 cm, what is the slant height?

Correct Answer: C

Q19.If the height of a right circular cone is 9 cm and the base radius is 12 cm, what is the slant height?

Correct Answer: A

Q20.Consider a right circular cone with a slant height of 15 cm and a base radius of 9 cm. If the cone is melted and recast into a sphere, what will be the radius of the sphere? (Use π = 3.14)

Correct Answer: A

Q21.A cone and a cylinder have the same base radius and height. If the volume of the cylinder is 300 cm³, what is the volume of the cone?

Correct Answer: B

Q22.A conical tent has a slant height of 25 m and a base diameter of 14 m. What is the curved surface area of the tent? (Use π = 3.14)

Correct Answer: A

Q23.If the total surface area of a right circular cone is 616 cm² and its base radius is 7 cm, what is the slant height of the cone? (Use π=3.14\pi = 3.14π=3.14)

Correct Answer: B

Q24.A spherical balloon is inflated such that its radius increases from 5 cm to 10 cm. What is the ratio of the new volume to the original volume?

Correct Answer: C

Q25.If the curved surface area of a cone is 330 cm² and its base radius is 5 cm, find the slant height of the cone. Use π=3.14\pi = 3.14π=3.14.

Correct Answer: A

Q26.A conical flask has a height of 18 cm and a base diameter of 12 cm. If the flask is filled with water up to a height of 12 cm, what is the volume of the water in the flask? (Use π=3.14\pi = 3.14π=3.14)

Correct Answer: B

Q27.A cone has a slant height of 10 cm and a base radius of 7 cm. What is the total surface area of the cone? (Use π=3.14\pi = 3.14π=3.14)

Correct Answer: B

Q28.A cone is formed by rotating a right-angled triangle around one of its perpendicular sides. If the height of the cone is 8 cm and the radius is 6 cm, what is the slant height?

Correct Answer: A

Q29.A cone has a slant height of 13 cm and a base radius of 5 cm. What is the total surface area of the cone? (Use π = 3.14)

Correct Answer: B

Q30.If the radius of a sphere is doubled, how does its surface area change?

Correct Answer: C

Q31.What is the volume of a right circular cone with a base radius of 5 cm and a height of 12 cm? (Use π = 3.14)

Correct Answer: A

Q32.A right circular cone has a slant height of 13 cm and a base radius of 5 cm. What is the height of the cone?