Class 10 maths Areas Related to Circles Hots Questions

Q 1 :

The radius of the largest right circular cone that can be cut out from a cube of edge 4.2 cm is

2.1 cm
4.2 cm
3.1 cm
2.2 cm

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(1)

The diameter of the largest right circular cone that can be cut out from a cube of edge 4.2 cm.

$∴$ $2 r = 4.2 c m \Rightarrow r = 2.1 c m$

⇒ Radius of the largest right circular cone = 2.1 cm.

Q 2 :

Volume and surface area of a solid hemisphere are numerically equal. What is the diameter of hemisphere?

9 units
6 units
4.5 units
18 units

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(1)

Volume of hemisphere = Surface area of hemisphere

$\Rightarrow \frac{2}{3} π r^{3} = 3 π r^{2} \Rightarrow r = \frac{9}{2} u n i t s$

$∴$ d = 9 units

Q 3 :

Two identical solid hemispheres of equal base radius are stuck along their bases. The total surface area of the combination is

$π r^{2}$
$2 π r^{2}$
$3 π r^{2}$
$4 π r^{2}$

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(4)

The resultant solid will be a sphere of radius r whose total surface area is $4 π r^{2}$

Q 4 :

The sum of the length, breadth and height of a cuboid is $6 \sqrt{3}$ cm and the length of its diagonal is $2 \sqrt{3}$ cm. The total surface area of the cuboid is

$48 {cm}^{2}$
$72 {cm}^{2}$
$96 {cm}^{2}$
$108 {cm}^{2}$

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(3)

$Given: l + b + h = 6 \sqrt{3} cm$ ...(i)

$and the length of its diagonal = 2 \sqrt{3} cm \Rightarrow \sqrt{l^{2} + b^{2} + h^{2}} = 2 \sqrt{3}$

Squaring both sides, we get $l^{2} + b^{2} + h^{2} = 12$ ...(ii)

From eq. (i), ${(l + b + h)}^{2} = {(6 \sqrt{3})}^{2}$

$\Rightarrow l^{2} + b^{2} + h^{2} + 2 (l b + b h + h l) = 108$

$\Rightarrow 12 + 2 (l b + b h + h l) = 108$ $[From eq. (iii)] \Rightarrow 2 (l b + b h + h l) = 96$

Hence, total surface area of the cuboid is $96 {cm}^{2}$

Q 5 :

The volume of the largest right circular cone that can be carved out from a solid cube of edge 2 cm is :

$\frac{4 π}{3}$ cu cm
$\frac{5 π}{3}$ cu cm
$\frac{8 π}{3}$ cu cm
$\frac{2 π}{3}$ cu cm

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(4)

Radius of cone = 1 cm

Height of cone = 2 cm

Volume of cone $= \frac{1}{3} π r^{2} h = \frac{1}{3} π {(1)}^{2} \times 2 = \frac{2 π}{3} cu cm$

Q 6 :

A solid sphere is cut into two hemispheres. The ratio of the surface areas of sphere to that of two hemispheres taken together, is:

1 : 1
1 : 4
2 : 3
3 : 2

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(3)

The total surface area of a sphere = $4 π r^{2}$

The surface area of one hemi-sphere = $3 π r^{2}$

$∴$ The total surface area of two hemi-spheres = $6 π r^{2}$

Required Ratio = $\frac{4 π r^{2}}{6 π r^{2}} = \frac{4}{6} = \frac{2}{3}$

Q 7 :

A cap is cylindrical in shape, surmounted by a conical top. If the volume of the cylindrical part is equal to that of the conical part, then the ratio of the height of the cylindrical part to the height of the conical part is :

1 : 2
1 : 3
2 : 1
3 : 1

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(2) 1 : 3

Q 8 :

A solid sphere is cut into two hemisphere. The ratio of the surface areas of sphere to that of two hemisphere taken together, is:

1 : 1
1 : 4
2 : 3
3 : 2

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(3)

Let r be the radius of sphere.

$S u r f a c e a r e a o f s p h e r e = 4 π r^{2} T o t a l s u r f a c e a r e a o f e a c h h e m i s p h e r e = C u r v e d s u r f a c e a r e a = 2 π r^{2}$

$B a s e a r e a = π r^{2} Total = 2 π r^{2} + π r^{2} = 3 π r^{2} S o f o r t w o h e m i s p h e r e s :$

$3 π r^{2} + 3 π r^{2} = 6 π r^{2} R e q u i r e d r a t i o : = \frac{4 π r^{2}}{3 π r^{2} + 3 π r^{2}} = \frac{4 π r^{2}}{6 π r^{2}} = \frac{4}{6} = \frac{2}{3}$

Q 9 :

The shape of a gilli, in the gilli-danda game (see figure), is a combination of:

two cylinders
a cone and a cylinder
two cones and a cylinder
two cylinders and a cone

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(3)

The shape of a gilli, in the gilli-danda game is a combination of two cones (ABC and DEF) and a cylinder (BDEC).

Q 10 :

If the surface area of a sphere is 616 cm², its diameter (in cm) is:

7
14
21
56

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(2)

$4 π r^{2} = 616 \Rightarrow r^{2} = 616 \times \frac{1}{4} \times \frac{7}{22} = 49 \Rightarrow r = 7 cm Diameter = 2 r = 2 \times 7 = 14 cm$

Q 11 :

The total surface area of a hemisphere of radius 7 cm is:

$588 π c m ²$
$392 π c m ²$
$147 π c m ²$
$98 π c m ²$

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(3)

Here, R = 7cm

Total surface area of hemisphere:

$= 3 π R^{2} = (3 π \times 7 \times 7) {cm}^{2} = 147 π {cm}^{2}$

Q 12 :

For a cuboid with length (l), breadth (b) and height (h), which of the following statements are true?

(i) Total surface area of the cuboid $= 2 (l b + b h + l h) square units .$

(ii) Diagonal of the cuboid has a length of $\sqrt{l^{2} + b^{2} + h^{2}} units .$

(iii) If the cuboid was a room, the area of its walls would be $(2 l + 2 b + 2 h) square units .$

(iv) All cuboids are cube.

Choose the correct option from the following:

(i) and (ii)
(ii) and (iii)
(i) and (iv)
(ii) and (iv)

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(1)

(i) Total surface area $= S u m o f a r e a s o f a l l f a c e s = l b + b h + l h + b h + l h + l b = 2 (l b + b h + l h)$

(ii) Diagonal length $= \sqrt{l^{2} + b^{2} + h^{2}}$

The diagonal joins two opposite vertices of the cuboid.

(iii) Sum of the areas of the walls $= l h + b h + l h + b h = 2 h (l + b)$

(iv) All cubes are cuboid but all cuboids are not cube.

Q 13 :

The appearance of an ice-cream cone is a combination of:

(i) Cube
(ii) Hemisphere
(iii) Cone
(iv) Cylinder

Choose the correct option from the following:

(i) and (iii)
(i) and (iv)
(ii) and (iii)
(ii) and (iv)

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(3)

The shape of ice-cream is a combination of a hemisphere and a cone.

Q 14 :

We convert a solid sphere into a cylinder, then which of the following statements hold true?

(i) Surface area might change but volume stays the same.
(ii) Volume changes but surface area remains the same.
(iii) Volume of the sphere > volume of the cylinder
(iv) Nothing changes

Choose the correct option from the following:

(ii) and (iii)
Only (i)
(iii) and (iv)
(i) and (ii)

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(2)

When an entire object is converted from one shape to another, their volume remains the same.

∴ Volume of sphere = Volume of cylinder

Surface area may or may not change.

Therefore,

Option (b) is correct.

Q 15 :

A cylinder has a radius of 3 cm and a height of 5 cm. Which of the following represents its total surface area and curved surface area?

(i) Total surface area = $48 π c m ²$
(ii) Curved surface area = $62 π c m ²$
(iii) Curved surface area = $30 π c m ²$
(iv) Total surface area = $3 π c m ²$

Choose the correct option from the following:

Only (i)
(ii) and (iv)
(iii) and (iv)
(i) and (iii)

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(4)

Total surface area of cylinder $= curved surface area + 2 \times area of base$

$= 2 π r h + 2 π r^{2} = 2 π (3) (5) + 2 π (3)^{2} = 2 π (3) (3 + 5) = 48 π {cm}^{2} C u r v e d s u r f a c e a r e a o f c y l i n d e r = 2 π r h = 2 π (3) (5) = 30 π {cm}^{2} S o, (i) a n d (i i i) a r e c o r r e c t .$

Q 16 :

Two cones have their heights in the ratio 1 : 3 and radii in the ratio 3 : 1. Then ratio of their volume is:

1 : 3
2 : 3
3 : 1
1 : 1

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(3)

Let height of the cones are h and 3h respectively and their radii are 3r and r respectively.

∴ Ratio of volumes = $\frac{volume of first cone}{volume of second cone} = \frac{\frac{1}{3} π (3 r)^{2} \times h}{\frac{1}{3} π r^{2} \times 3 h} = \frac{9}{3} = \frac{3}{1} ∴ R e q u i r e d r a t i o i s 3 : 1 .$

Q 17 :

12 spheres of the same size are made from melting a solid cylinder of 16 cm diameter and 2 cm height. The diameter of each sphere is:

$\sqrt{3} cm$
2 cm
3 cm
4 cm

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(4)

$R a d i u s o f c y l i n d e r = \frac{16}{2} = 8 cm$

Let r cm be the radius of each sphere.

hen one solid shape is converted into another, the volume remains the same

$∴ Volume of the solid cylinder = 12 \times volume of one sphere$

$π \times (8)^{2} \times 2 = 12 \times \frac{4}{3} π r^{3}$

$128 = 16 r^{3} \Rightarrow r^{3} = 8 \Rightarrow r = 2 cm$

$Diameter of sphere = 2 r = 2 \times 2 = 4 cm$

$∴ Option (d) is correct.$

Q 18 :

The volume of the largest right circular cone that can be carved out from a solid cube of edge 2 cm is:

$\frac{4 π}{3} c u c m$
$\frac{5 π}{3} c u c m$
$\frac{8 π}{3} c u c m$
$\frac{2 π}{3} c u c m$

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(4)

The largest cone that can be carved out from solid cube of edge 2 cm will be the cone whose diameter of base and height both are equal to the edge of the cube.

From the figure, radius of cone = r = 1 cm and height of cone = h = 2 cm

∴ Volume of cone formed

$= \frac{1}{3} π r^{2} h$

$= \frac{1}{3} \times π \times 1^{2} \times 2$

$= \frac{2 π}{3} {cm}^{3}$

Q 19 :

A right circular hollow cylinder has an outer radius of R and inner radius of r. Which of the following statements hold true if h is the height of the cylinder?

(i) Curved surface area of the cylinder $= 2 π (R + r) h$

(ii) Total surface area of the cylinder $= 2 π (R + r) (R + h - r)$

(iii) Total volume of the cylinder $= π (R^{2} - r^{2}) h$

(iv) Total volume of the cylinder $= π (R^{2} + r^{2}) h$

Choose the correct option from the following:

Only (iv)
(i) and (iii)
(i), (ii) and (iii)
(i), (ii) and (iv)

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(3)

Curved surface area = External curved surface area + Internal curved surface area

$= 2 π R h + 2 π r h = 2 π (R + r) h$

$(ii) Total surface area = 2 π R h + 2 π r h + 2 (π R^{2} - π r^{2})$

$= 2 π (R + r) h + 2 π (R + r) (R - r)$

$= 2 π (R + r) (R + h - r)$

$(iii) Total volume = π R^{2} h - π r^{2} h = π h (R^{2} - r^{2})$

$∴ Option (c) is correct.$

Q 20 :

There is a cylinder circumscribing the hemisphere such that their bases are common and their height is same. The respective ratio (hemisphere : cylinder) of volume and curved surface area are:

(i) Ratio of volume = 1 : 3 (ii) Ratio of CSA = 1 : 2

(iii) Ratio of volume = 2 : 3 (iv) Ratio of CSA = 1 : 1

Choose the correct option from the following:

(i) and (iv)
(ii) and (iii)
(i) and (iii)
(iii) and (iv)

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(4)

Height of cylinder = r
Radius of sphere = r

$V o l u m e o f c y l i n d e r = V_{1} = π (r^{2}) (r) = π r^{3}$

$V o l u m e o f h e m i s p h e r e = V_{2} = \frac{2}{3} π r^{3}$

$∴ \frac{V_{1}}{V_{2}} = \frac{π r^{3}}{\frac{2}{3} π r^{3}} = \frac{3}{2} ∴ V_{2} : V_{1} = 2 : 3$

$R a t i o o f C S A = \frac{CSA of hemisphere}{CSA of cylinder} = \frac{2 π r^{2}}{2 π r (r)} = 1 : 1$

Topic Question Set

Q 1 :

The radius of the largest right circular cone that can be cut out from a cube of edge 4.2 cm is

Q 2 :

Volume and surface area of a solid hemisphere are numerically equal. What is the diameter of hemisphere?

Q 3 :

Two identical solid hemispheres of equal base radius are stuck along their bases. The total surface area of the combination is

Q 4 :

The sum of the length, breadth and height of a cuboid is $6 \sqrt{3}$ cm and the length of its diagonal is $2 \sqrt{3}$ cm. The total surface area of the cuboid is

Q 5 :

The volume of the largest right circular cone that can be carved out from a solid cube of edge 2 cm is :

Q 6 :

A solid sphere is cut into two hemispheres. The ratio of the surface areas of sphere to that of two hemispheres taken together, is:

Q 7 :

A cap is cylindrical in shape, surmounted by a conical top. If the volume of the cylindrical part is equal to that of the conical part, then the ratio of the height of the cylindrical part to the height of the conical part is :

Q 8 :

A solid sphere is cut into two hemisphere. The ratio of the surface areas of sphere to that of two hemisphere taken together, is:

Q 9 :

The shape of a gilli, in the gilli-danda game (see figure), is a combination of:

Q 10 :

If the surface area of a sphere is 616 cm², its diameter (in cm) is:

Q 11 :

The total surface area of a hemisphere of radius 7 cm is:

Q 13 :

The appearance of an ice-cream cone is a combination of:

(i) Cube
(ii) Hemisphere
(iii) Cone
(iv) Cylinder

Choose the correct option from the following:

Q 16 :

Two cones have their heights in the ratio 1 : 3 and radii in the ratio 3 : 1. Then ratio of their volume is:

Q 17 :

12 spheres of the same size are made from melting a solid cylinder of 16 cm diameter and 2 cm height. The diameter of each sphere is:

Q 18 :

The volume of the largest right circular cone that can be carved out from a solid cube of edge 2 cm is:

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Topic Question Set

Q 1 : The radius of the largest right circular cone that can be cut out from a cube of edge 4.2 cm is

Q 2 : Volume and surface area of a solid hemisphere are numerically equal. What is the diameter of hemisphere?

Q 3 : Two identical solid hemispheres of equal base radius are stuck along their bases. The total surface area of the combination is

Q 4 : The sum of the length, breadth and height of a cuboid is 63cm and the length of its diagonal is 23cm. The total surface area of the cuboid is

Q 5 : The volume of the largest right circular cone that can be carved out from a solid cube of edge 2 cm is :

Q 6 : A solid sphere is cut into two hemispheres. The ratio of the surface areas of sphere to that of two hemispheres taken together, is:

Q 7 : A cap is cylindrical in shape, surmounted by a conical top. If the volume of the cylindrical part is equal to that of the conical part, then the ratio of the height of the cylindrical part to the height of the conical part is :

Q 8 : A solid sphere is cut into two hemisphere. The ratio of the surface areas of sphere to that of two hemisphere taken together, is:

Q 9 : The shape of a gilli, in the gilli-danda game (see figure), is a combination of:

Q 10 : If the surface area of a sphere is 616 cm², its diameter (in cm) is:

Q 11 : The total surface area of a hemisphere of radius 7 cm is:

Q 13 : The appearance of an ice-cream cone is a combination of: (i) Cube (ii) Hemisphere (iii) Cone (iv) Cylinder Choose the correct option from the following:

Q 16 : Two cones have their heights in the ratio 1 : 3 and radii in the ratio 3 : 1. Then ratio of their volume is:

Q 17 : 12 spheres of the same size are made from melting a solid cylinder of 16 cm diameter and 2 cm height. The diameter of each sphere is:

Q 18 : The volume of the largest right circular cone that can be carved out from a solid cube of edge 2 cm is:

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Q 1 :

The radius of the largest right circular cone that can be cut out from a cube of edge 4.2 cm is

Q 2 :

Volume and surface area of a solid hemisphere are numerically equal. What is the diameter of hemisphere?

Q 3 :

Two identical solid hemispheres of equal base radius are stuck along their bases. The total surface area of the combination is

Q 4 :

The sum of the length, breadth and height of a cuboid is $6 \sqrt{3}$ cm and the length of its diagonal is $2 \sqrt{3}$ cm. The total surface area of the cuboid is

Q 5 :

The volume of the largest right circular cone that can be carved out from a solid cube of edge 2 cm is :

Q 6 :

A solid sphere is cut into two hemispheres. The ratio of the surface areas of sphere to that of two hemispheres taken together, is:

Q 7 :

A cap is cylindrical in shape, surmounted by a conical top. If the volume of the cylindrical part is equal to that of the conical part, then the ratio of the height of the cylindrical part to the height of the conical part is :

Q 8 :

A solid sphere is cut into two hemisphere. The ratio of the surface areas of sphere to that of two hemisphere taken together, is:

Q 9 :

The shape of a gilli, in the gilli-danda game (see figure), is a combination of:

Q 10 :

If the surface area of a sphere is 616 cm², its diameter (in cm) is:

Q 11 :

The total surface area of a hemisphere of radius 7 cm is:

Q 13 :

The appearance of an ice-cream cone is a combination of:

(i) Cube
(ii) Hemisphere
(iii) Cone
(iv) Cylinder

Choose the correct option from the following:

Q 16 :

Two cones have their heights in the ratio 1 : 3 and radii in the ratio 3 : 1. Then ratio of their volume is:

Q 17 :

12 spheres of the same size are made from melting a solid cylinder of 16 cm diameter and 2 cm height. The diameter of each sphere is:

Q 18 :

The volume of the largest right circular cone that can be carved out from a solid cube of edge 2 cm is: