Grade 9 Surface Area And Volume In 3v6y6x

ID : in-9-Surface-Area-and-Volume [1]

Class 9 Surface Area and Volume For more such worksheets visit www.edugain.com

Answer the questions (1)

A sphere is just enclosed inside a cube of volume 120 cm 3. Find the volume of the sphere.

(2)

A sphere is just enclosed inside a right circular cylinder. If surface area of sphere is 30 cm 2, find total surface area of cylinder.

(3)

Find the volume the biggest sphere which can fit in a cube of side 6b.

(4)

If the radius of a hemisphere is 4r, find its total surface area.

(5)

If a hemisphere and cone stands on equal bases, and have the same height. Find the ratio of their volumes.

(6)

If radius of two spheres are in ratio 5:3, find the ratio of their volumes.

Choose correct answer(s) from given choice (7)

Find the volume of the biggest cone that can fit inside a cube of side 4 cm. a.

c. (8)

(9)

16 π

cm 3

3 4π 3

cm 3

b.

d.

64 π

cm 3

3 32 π

cm 3

3

An sphere is expanded to a bigger sphere such that its volume increases by a factor of 8, find the change in its radius. a. 2 times

b. 8 times

c. 4 times

d. None of these

A cone made completely of metal (i.e. it is not hollow) has a base radius of 4 cm, and height of 2 cm. If we melt it and recast it into a sphere, what will be the radius of sphere? a. 3 cm

b. 1 cm

c. 4 cm

d. 2 cm

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ID : in-9-Surface-Area-and-Volume [2]

(10) If radius of a hemisphere is 4a, find its volume. a. 16/3 π a3

b. 18 π a3

c. 2/3 π a3

d. 128/3 π a3

(11) If radius of a sphere is 2a, find its surface area. a. 64 π a2

b. 36 π a2

c. 16 π a2

d. 4 π a2

(12) Find the surface area of the biggest sphere, which can fit in a cube of side 6a. a. 64 π a2

b. 4 π a2

c. 16 π a2

d. 36 π a2

(13) A sphere and a cone have the same radii. If the volume of the sphere is four times of the volume of the cone, find the ratio of the cone's height and radius. a. 1:3

b. 1:1

c. 3:1 (14) The radius of a cylinder is halved and the height is doubled. What is the area of the curved surface now compared to the previous surface area? a. same

b. half

c. double

d. four times

(15) A sphere and a right circular cylinder have the same radius. If the volume of the sphere is four times the volume of the cylinder, then what is the ratio of cylinder's height and radius? a. 4:3

b. 3:1

c. 1:3

d. 3:4

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ID : in-9-Surface-Area-and-Volume [3]

Answers (1)

20 π cm 3 Step 1 The volume of a sphere is of radius r =

4 3

π r3

Step 2 If it fits exactly within a cube, this means the length/width/height of the cube is the same as the diameter of the sphere i.e. 2r Step 3 The volume of a cube of side 2r = (2r)3 = 8r3 Step 4 8r3 = 120 cm 3 This means r = 15 Step 5 Putting this in the formula for the volume of the sphere, we get the volume = 20 π cm 3

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ID : in-9-Surface-Area-and-Volume [4]

(2)

45 cm 2 Step 1 There are three equations we need to know this type of question - the total area of a cylinder, the curved area of a cylinder, and the surface area of a sphere Step 2 The curved surface area of a cylinder of radius 'r' and height 'h' is 2πrh. Here we know the sphere will fit in exactly in the cylinder, so h=2r, and the formula now becomes 4πr2 Step 3 The total surface area of the same cylinder will be the sum of the curved area and the surface area of the two circles at top and bottom. So 4πr2 + 2πr2 = 6πr2 Step 4 And of course, the sphere will have the radius r too, so it's surface area is 4πr2 Step 5 From these equations, we see that for this case, the surface area of the sphere is the same as the curved surface are of the cylinder, and 2/3 of the total surface area of the cylinder Step 6 Here we know that surface area of sphere is 30 cm 2, and need to find total surface area of cylinder Step 7 Substituting from the equation above, we get total surface area of cylinder = 45 cm 2

(3)

36 π b3 Step 1 The biggest sphere that can fit inside a cube of side 6b will have a diameter of 6b (anything larger will not fit in, as opposite sides are separated by a distance of 6b. Step 2 This means that the radius of this sphere is (1/2)6b Step 3 The volume of a sphere of radius x is (4/3)πx3 Step 4 Therefore the volume of this sphere is (4/3)π((1/2)6b)3 Step 5 Solving for this gives us 36 π b3

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ID : in-9-Surface-Area-and-Volume [5]

(4)

48 π r2 Step 1 The surface area of a sphere of radius x is given by 4πx2 Step 2 The curved surface area of a hemisphere is half of that i.e. 2πx2 Step 3 To get the total surface area, we need to add to this the area of the circle at the base i.e. πx2br> Step 4 Adding them, we get area of hemisphere = 3πx2 Step 5 Here the radius is specified as 4r. Substituting this into the formula, we get the answer is (3π) x (4r)2 Step 6 This gives us the answer 48 π r2

(5)

2:1 Step 1 The volume of a hemishpere of radius 'r' is

2 3

πr3.

Step 2 The volume of a cone of radius 'r' and height 'h' is

1 3

π r2h

Step 3 From these equations we can cancel out the equal ( the heights are also equal) to find the ratio as 2:1

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ID : in-9-Surface-Area-and-Volume [6]

(6)

125:27 Step 1 The volume of a spheres of radius x is given by (4/3)πx3 Step 2 We see that the volume is proportional to the 3rd power of the radius To see this more clearly, assume the radii of these two spheres are 5x and 3x (note that this allows us to get the ratio of 5:3, which is the only thing we know about these radii) The volume of the first one then is 4/3π5x3, and the volume of the second one is 4/3π3x3 The ratio of the the volumes is therefore 4/3π(5x)3:4/3π(3x)3 This simplifies to (5x)3:(3x)3, and further to 53:33 Step 3 Therefore the answer is 125:27

(7)

16 π

a.

3

cm 3

Step 1 The volume of a cone is of radius r and height h =

1 3

π r2h

Step 2 Since we have to fit it inside a cube of side 4 cm, we see that the diameter of the cone will be 4 cm, and the height will be 4 cm (a cone larger than this in the diameter or the height will not fit inside the cube Step 3 So the radius of this cone is

4 2

=2

Step 4 Putting these values into the equation of the volume, we get the volume of the cone =

x

1 3

xπ

4 2 x4 2

Step 5 Solving we get the volume of the cone =

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16 π 3

cm 3

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ID : in-9-Surface-Area-and-Volume [7]

(8)

a. 2 times Step 1 The volume of a sphere of radius x =

4 3

π x3

The surface area of a sphere of radius x = 4π x2 Step 2 This means that the surface area will increase as a square of the increase in radius And the volume will increase as a cube of the increase in radius Step 3 Here we know that the volume increased by a factor of 8 Step 4 This means that the radius grew by the cube root of this value i.e. 8

1 3

Step 5 Solving this, we get 2 times (9)

d. 2 cm Step 1 The volume of a cone is of radius r and height h =

1 3

π r2h

Step 2 The volume of a sphere of radius x =

4 3

π x3

Step 3 We know that the cone of base radius 4 cm and height 2 was melted down Step 4 The volume of metal resulting from this =

1 3

π 42 2

Step 5 When we recast it into a sphere, we get a sphere of this volume Step 6 That is to say

4 3

π x3 =

1 3

π 42 2

Step 7 Solving for x, we get x = 2 cm

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ID : in-9-Surface-Area-and-Volume [8]

(10) d. 128/3 π a3 Step 1 The volume of a hemisphere of radius x is given by (4/3)πx3 Step 2 The volume of a hemisphere is half that i.e. (2/3)πx3 Step 3 Here the radius is specified as 4a. Substituting this into the formula, we get the answer is (2/3π) x (4a)3 Step 4 This gives us the answer 128/3 π a3 (11) c. 16 π a2 Step 1 The surface area of a sphere of radius x is given by 4πx2 Step 2 Here the radius is specified as 2a. Substituting this into the formula, we get the answer is (4π) x (2a)2 Step 3 This gives us the answer 16 π a2 (12) d. 36 π a2 Step 1 The biggest sphere that can fit inside a cube of side 6a will have a diameter of 6a (anything larger will not fit in, as opposite sides are separated by a distance of 6a. Step 2 This means that the radius of this sphere is

1 2

6a

Step 3 The surface area of a sphere of radius x is 4πx2 Step 4 Therefore the surface area of this sphere is 4π(

1 2

6a)2

Step 5 The answer is 36 π a2

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ID : in-9-Surface-Area-and-Volume [9]

(13) b. 1:1 Step 1 The volume of a cone is of radius r and height h = 1/3 π r2h Step 2 The volume of a sphere is of radius r = 4/3 π r3 Step 3 We know the volume of the sphere is four times of the volume of the cone 4/3 π r3 = 4 x (1/3 π r2h) h/r = 1:1 (14) a. same Step 1 The curved surface area of a cylinder is 2πrh Step 2 Here we halved the radius and doubled the height Step 3 Putting this into the formula, we see that the curved surface area becomes same (15) c. 1:3 Step 1 The volume of a sphere is of radius r =

4 3

π r3

Step 2 The volume of a cylinder of radius 'r' and height 'h' is πr2h. Step 3 Here, we are told the the volume of the sphere is four times times the volume of the cylinder So

4 3

π r3 = 4 x (πr2h)

Step 4 Cancelling out the , we get 3 x h = 1 x r Step 5 Therefore, the ratio of the cylinder's height to radius is 3:1

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