Radius of a cylinder is equal to its height. If the radius is taken as ‘ r ’, the volume of the cylinder is π r 2 × r = π r 3 . Like this find the volumes of the solids, with the following measures. Solids Measures Volume Cone radius = height = r Hemisphere radius = r Sphere radius = r What is the ratio of the volumes of the cone, hemisphere, cylinder and the sphere?

Given, Radius of a cylinder is equal to its height. If the radius is taken as ‘ r ’, the volume of the cylinder is π r 2 × r = π r 3 . Like this we need to find the volumes of the solids, with the following measures. Solids Measures Volume Cone radius = height = r 1 3 πr 2 h ⇒ 1 3 πr 2 × r ⇒ 1 3 πr 3 Hemisphere radius = r 2 3 πr 3 Sphere radius = r 4 3 πr 3 V c : V h : V c y : V s = 1 3 πr 3 : 2 3 πr 3 : πr 3 : 4 3 πr 3 = 1 3 : 2 3 : 1 : 4 3 = 1 : 2 : 3 : 4

Perimeter of the rectangle in the figure is 36 c e n t i m e t r e s . A C = 164 c e n t i m e t r e s . What is A B + B C ?

Perimeter of the rectangle in the figure is 36 c e n t i m e t r e s . A C = 164 c e n t i m e t r e s . We have to find the length of A B + B C . A B C D rectangle. Perimeter = 36 c m Perimeter of a rectangle = 2 × l + b Perimeter = 2 × A B + B C ⇒ 2 × A B + B C = 36 A B + B C = 36 2 = 18 Hence, the required answer is 18 c m

Perimeter of the rectangle in the figure is 36 c e n t i m e t r e s . A C = 164 c e n t i m e t r e s . Find length of A B .

A B C D is a rectangle. Perimeter = 36 c m Diagonal A C = 164 = 41 × 4 = 2 41 c m . A B + B C = 36 / 2 = 18 ( A B + B C ) 2 = 18 2 A B 2 + B C 2 + 2 A B × B C = ( 20 - 2 ) 2 = 400 – 80 + 4 = 324 A B 2 + B C 2 = A C 2 = ( 164 ) 2 164 × 164 + 2 A B × B C = 324 2 A B × B C = 324 – 164 = 160 A B × B C = 80 ( A B - B C ) 2 = A B 2 + B C 2 – 2 A B × B C = 164 – 2 × 80 = 4 A B - B C = 2 A B + B C = 18 2 A B = 20 A B = 10 Hence, the required answer is 10 c m .

The picture shows the shape of a boiler. Total height of the boiler is 12 metres and the diameter is 6 meters, height of the cylindrical part is 6 meters. How many litres can the boiler hold ? ( 1 m 3 = 1000 litre)

Total height of boiler = 12 m Diameter of cylindrical part = diameter of hemispherical part = 6 m Height of cone = Total Height - Height of cylinder - radius of hemisphere = 12 - 6 - 3 = 3 m Volume of boiler = Volume of cylinder + Volume of hemisphere + Volume of cone = 2 3 π r 3 + π r 2 h + 1 3 π r 2 H = 1 3 π r 2 2 r + 3 h + H = 1 3 π 3 2 6 + 18 + 3 = π × 9 × 27 3 = 81 × 22 7 = 254 . 571 m 3 = 254571 litres ( 1 m 3 = 1000 litre)

The picture shows the shape of a boiler. Total height of the boiler is 12 metres and the diameter is 6 meters, height of the cylindrical part is 6 meters. What is the height of the cone?

Total height of boiler = 12 m Diameter of cylindrical part = diameter of hemispherical part = 6 m Height of cone = Total Height - Height of cylinder - radius of hemisphere = 12 - 6 - 3 = 3 m Therefore, the height of the cone = 3 m .

The perimeter of the base of a square pyramid is 96 cm and its height is 16 cm . Find the lateral surface area.

Given, the perimeter of the base of a square pyramid is 96 cm and its height is 16 cm . The base of a square pyramid is a square. Perimeter of a square = 4 a , where a is the side length of the square. ∴ 4 a = 96 ⇒ a = 96 4 ⇒ a = 24 Slant height, l = a 2 2 + h 2 , where a is the length of the base edge of the square pyramid and h is its height Slant height, l = 24 2 2 + 16 2 = 144 + 256 = 400 = 20 Lateral surface area of a square pyramid = 2 a l , where a is the length of the base edge of the square pyramid and l is the slant height Lateral surface area of the square pyramid = 2 × 24 cm × 20 cm = 960 cm 2 Therefore, the lateral surface area of the square pyramid is 960 cm 2 .

The perimeter of the base of a square pyramid is 96 cm and its height is 16 cm . What is the slant height?

Given, the perimeter of the base of a square pyramid is 96 cm and its height is 16 cm . The base of a square pyramid is a square. Perimeter of a square = 4 a , where a is the side length of the square. ∴ 4 a = 96 ⇒ a = 96 4 ⇒ a = 24 Slant height, l = a 2 2 + h 2 , where a is the length of the base edge of the square pyramid and h is its height Slant height, l = 24 2 2 + 16 2 = 144 + 256 = 400 = 20 Therefore, the slant height of the square pyramid is 20 cm .

The perimeter of the base of a square pyramid is 96 cm and its height is 16 cm . What is the length of a base edge?

Given, the perimeter of the base of a square pyramid is 96 cm and its height is 16 cm . The base of a square pyramid is a square. Perimeter of a square = 4 a , where a is the side length of the square. ∴ 4 a = 96 ⇒ a = 96 4 ⇒ a = 24 Therefore, the length of a base edge of the square pyramid is 24 cm .

Radius of a cylinder is equal to its height. If the radius is taken as ‘ r ’, the volume of the cylinder is π r 2 × r = π r 3 . Like this find the volumes of the solids, with the following measures. Solids Measures Volume Cone radius = height = r Hemisphere radius = r Sphere radius = r What is the ratio of the volumes of the cone, hemisphere, cylinder and the sphere?

Given, Radius of a cylinder is equal to its height. If the radius is taken as ‘ r ’, the volume of the cylinder is π r 2 × r = π r 3 . Like this we need to find the volumes of the solids, with the following measures. Solids Measures Volume Cone radius = height = r 1 3 πr 2 h ⇒ 1 3 πr 2 × r ⇒ 1 3 πr 3 Hemisphere radius = r 2 3 πr 3 Sphere radius = r 4 3 πr 3 V c : V h : V c y : V s = 1 3 πr 3 : 2 3 πr 3 : πr 3 : 4 3 πr 3 = 1 3 : 2 3 : 1 : 4 3 = 1 : 2 : 3 : 4

A sector of radius 12 c e n t i m e t r e s and central angle 120 ° is rolled up into a cone. What is the central angle of the sector to be used to make a cone of base radius 2 c e n t i m e t r e s and height 4 c e n t i m e t r e s ?

Radius of cone = 2 c m Height of the cone = 4 c m l 2 = 4 2 + 2 2 l 2 = 16 + 2 = 18 l = 18 = 9 × 2 = 3 2 c m Radius of sector = Slant height of cone Length of arc of sector = Circumference of cone θ 360 ° × 2 π × l = 2 πr θ 360 ° = r l θ = 360 ° × r l θ = 360 ° × 2 3 2 θ = 360 ° 3 = 120 ° Hence, central angle of sector is 120 ° .

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The given figure is the lateral face of a square pyramid. $A B = A C = 25$ centimeters and $B D = D C = 15$ centimeters. What is the length of its base edge? [Take $\sqrt{2} = 1.414$ ]

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Important Questions on Solids

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EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Solids>Kerala Board-2019>Q 8

The given figure is the lateral face of a square pyramid. $A B = A C = 25$ centimeters and $B D = D C = 15$ centimeters

Find the lateral surface area of the pyramid.

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EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Solids>Kerala Board-2019>Q 9

Radius of a cylinder is equal to its height. If the radius is taken as ‘ $r$ ’, the volume of the cylinder is $π r^{2} \times r = π r^{3}$ . Like this find the volumes of the solids, with the following measures.

Solids	Measures	Volume
Cone	radius $=$ height $= r$
Hemisphere	radius $= r$
Sphere	radius $= r$

What is the ratio of the volumes of the cone, hemisphere, cylinder and the sphere?

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EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Solids>Kerala Board-2019>Q 10

Solids	Measures	Volume
Cone	radius $=$ height $= r$
Hemisphere	radius $= r$
Sphere	radius $= r$

A solid metal sphere of radius $6 cm$ is melted and recast into solid cones of radius $6 cm$ and height $6 cm$ . Find the number of cones.

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EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Solids>Kerala Board-2020>Q 11

The figure of a square sheet paper is shown below. Length of one side of the paper sheet is $36 cm$ and $A B = 10 cm$ . The shaded portion is cut out and folded into a square pyramid. If the length of the base of the pyramid is $k cm$ , find $k$ .

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10th Kerala Board

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EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Solids>Kerala Board-2020>Q 12

The figure of a square sheet paper is shown below. Length of one side of the paper sheet is $36 cm$ and $A B = 10 cm$ . The shaded portion is cut out and folded into a square pyramid. If the slant height of the pyramid is $k cm$ , find $k$ .

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10th Kerala Board

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EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Solids>Kerala Board-2020>Q 13

The figure of a square sheet paper is shown below. Length of one side of the paper sheet is $36 cm$ and $A B = 10 cm$ . The shaded portion is cut out and folded into a square pyramid. If the lateral surface area of the pyramid is $k {cm}^{2}$ , find $k$ .

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10th Kerala Board

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EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Solids>Kerala Board-2020>Q 14

Perimeter of the rectangle in the figure is $36 c e n t i m e t r e s$ . $A C = \sqrt{164} c e n t i m e t r e s .$ What is $A B + B C$ ?

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EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 8 - Solids>Kerala Board-2020>Q 15

Perimeter of the rectangle in the figure is $36 c e n t i m e t r e s$ . $A C = \sqrt{164} c e n t i m e t r e s .$ Find length of $A B$ .

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The given figure is the lateral face of a square pyramid. AB=AC=25 centimeters and BD=DC=15 centimeters. What is the length of its base edge? [Take 2=1.414]

Important Questions on Solids

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The given figure is the lateral face of a square pyramid. $A B = A C = 25$ centimeters and $B D = D C = 15$ centimeters. What is the length of its base edge? [Take $\sqrt{2} = 1.414$ ]