Grade 9 Surface Area and Volume

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ID : ae-9-surface-area-and-volume [1] Grade 9 Surface Area and Volume For more such worksheets visit www.edugain.

(2) If the radius of a hemisphere is 2y, find its curved surface area. () A cone completely made of metal (i.e. it is not hollow) has a base radius of 8 cm, and height of 4 cm.

(5) A sphere is expanded to a bigger sphere such that its radius increases by a factor of, find the change in its surface area.

1 ID : ae-9-surface-area-and-volume [1] Grade 9 Surface Area and Volume For more such worksheets visit Answer the questions (1) The radius of a cylinder is halved and the height is tripled. What is the area of the curved surface when compared to the same area previously? (2) If the radius of a hemisphere is 2y, find its curved surface area. () A cone completely made of metal (i.e. it is not hollow) has a base radius of 8 cm, and height of 4 cm. If we melt it and recast it into a sphere, what will be the radius of the sphere? (4) Find the volume of the biggest cone that can fit inside a cube of side 5 cm. (5) A sphere is expanded to a bigger sphere such that its radius increases by a factor of, find the change in its surface area. (6) If a cone and cylinder stands on equal bases, and have the same height. Find the ratio of their volumes. (7) Find the surface area of the biggest sphere which can fit inside a cube of side 4a. (8) A sphere and a right circular cylinder have the same radius. If the volume of the sphere is five times of the volume of the cylinder, what is the ratio of the height of the cylinder to its radius? (9) The heights of two cylinders are in the ratio 2:7 and their radii are in the ratio 7:1. Find the ratio of their volumes. (10) If radius of a sphere is 4b, find its volume. (11) If surface area of a cube is 24 cm 2, find its volume. (12) If the radius of two hemispheres are in the ratio 1:5, find the ratio of their volumes. (1) A sphere and a cone have the same radius. If the volume of the sphere is one third of the volume of the cone, find the ratio of the height and radius of the cone. (14) If the radii of two hemispheres are in ratio :4, find the ratio of their surface area. (15) A sphere is just enclosed inside a right circular cylinder. If the volume of the cylinder is 240 cm, find the volume of the sphere Edugain ( All Rights Reserved Many more such worksheets can be generated at

2 Answers ID : ae-9-surface-area-and-volume [2] (1) 1.5 times The curved surface area of a cylinder is 2πrh. Here, we halved the radius and tripled the height. Putting this into the formula, we see that the curved surface area becomes 1.5 times. (2) 8πy 2 The surface area of a sphere of radius x is given by 4πx 2. The curved surface area of a hemisphere of same radius x, is half of that of a sphere, i.e. 2π x 2. Although, the above written expressions are valid when the radius is x. We will have to replace it by 2y as per the question. The curved surface area of the hemisphere = 2π (2y) 2 = 2π 4y 2 = 8πy 2. Hence, curved surface area of the hemisphere is 8πy 2.

3 () 4 cm ID : ae-9-surface-area-and-volume [] The volume of a cone is of radius r and height h = 1 πr 2 h The volume of a sphere of radius x = 4 πx We know that the cone of base radius 8 cm and height 4 cm was melted down. The volume of metal resulting from this = 1 π(8) 2 (4) As we melt the cone to recast it into a sphere. The volume of the sphere formed will be equal to the volume of the cone. Step 6 Hence, we have 4 πx = 1 π(8) 2 (4) Step 7 Solving for x, we get x = 4 cm. Step 8 Hence, radius of sphere = 4 cm

4 ID : ae-9-surface-area-and-volume [4] (4) 125π 12 cm The volume of a cone is of radius r and height h = 1 πr 2 h. Since we have to fit it inside a cube of side 5 cm, we see that the diameter of the cone will be 5 cm, and the height will be 5 cm (A cone larger than this in diameter or height will not fit inside the cube). So, the radius of this cone = 5 2 cm = 2.5 cm Putting these values into the equation of the volume, we get the volume of the cone = 1 π ( 5 2 ) 2 5 cm On solving, we get the volume of the cone = 125π 12 cm

5 (5) 9 times ID : ae-9-surface-area-and-volume [5] The volume of a sphere of radius x = 4 πx. The surface area of a sphere of radius x = 4πx 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. Here, we know that the radius is increased by a factor of. This means that the surface area would have increased by a square of this value i.e. by 2. Solving this, we get the answer as 9 times. (6) 1: The volume of a cylinder of radius 'r' and height 'h' is πr 2 h. The volume of a cone of radius 'r' and height 'h' is 1 π r 2 h From these equations we can cancel out the equal terms (remember the heights are also equal) to find the ratio as 1:

6 ID : ae-9-surface-area-and-volume [6] (7) 16πa 2 The biggest sphere that can fit inside a cube of side 4a will have a diameter of 4a (anything larger will not fit in, as opposite sides are separated by a distance of 4a). This means that the radius of this sphere is 1 2 4a = 2a. We know that the surface area of a sphere of radius x is 4πx 2. So, the surface area of the given sphere of radius 2a is 4π(2a) 2 = 4π 4a 2 = 16πa 2 (8) 4:15 The volume of a sphere of radius 'r' = 4 πr The volume of a cylinder of radius 'r' and height 'h' = πr 2 h Here, we are told the the volume of the sphere is five times of the volume of the cylinder. So, 4 πr = 5 (πr 2 h) Solving the above equation, we get 15h = 4r. Therefore, the ratio of the height of the cylinder to its radius is 4:15.

7 (9) 14:1 ID : ae-9-surface-area-and-volume [7] The volume of a cylinder is πr 2 h. To compare the ratio of the volumes we can ignore the constant multiplier π as it will be present in both the volumes. We are told that the ratio of the heights of two cylinders is 2:7. So let us represent the height of the first cylinder as 2h and that of the second cylinder as 7h. Similarly, their radii are in the ratio 7:1. So, let us represent the radius of the first cylinder as 7r and that of the second cylinder as r. So the ratio of the volumes is (7r) 2 2h : (r) 2 7h. This can be simplified to 49r 2 2h : r 2 7h. Step 6 Simplifying this further, we get the ratio of the volumes of the two cylinders is 14:1. (10) 256 πb The volume of a sphere of radius x is given by 4 πx. The above expression is valid when the radius is x. We will have to replace it by 4b as per the question: Volume of sphere = 4 π (4b) = 4 π 64b = 256 πb. This gives us the answer as 256 πb.

8 (11) 8 cm ID : ae-9-surface-area-and-volume [8] For a cube with length of side equal to a, we know that: Surface area = 6 a 2 Volume = a We are given that the surface area of the cube is 24 cm 2. This means 6 a 2 = 24 cm 2 or, a = 2 cm Now that we know the length of the side, we can find the volume of the cube as: a = 8 cm (12) 1:125 The volume of a sphere of radius x is given by 4 πx. The volume of a hemisphere is half of that i.e. 2 πx. We see that the volume is proportional to the rd power of the radius. To see this more clearly, assume the radii of these two hemispheres as x and 5x. Note that this allows us to get the ratio of 1:5, which is the only thing we know about these radius. The volume of the first hemisphere will become 2 πx and that of the second one will become 2 π(5x) = 2 π 125x = 250 πx. Hence, we can see that the ratio of the volumes of two given hemispheres is: 2 πx : 250 πx = 2 : 250 = 1:125

9 (1) 12:1 ID : ae-9-surface-area-and-volume [9] We know that the volume of a cone with radius r and height h = 1 πr 2 h. We also know that the volume of a sphere with radius r = 4 πr. We have been told that the volume of the sphere is one third the volume of the cone. Therefore, 4 πr = 1 ( 1 πr 2 h) or, h r = 12:1 Thus, the ratio of the height and radius of the cone is 12:1. (14) 9:16 The surface area of a sphere of radius x is given by 4πx 2. The surface area of a hemisphere is πx 2 (half of the surface area of the sphere, plus the area of the base, which is a circle of radius x. Assume the radii of these two hemispheres are x and 4x. Note that this allows us to get the ratio of :4, which is the only thing we know about these radii. The surface area of the first hemisphere will become π (x) 2 or, 27πx 2 and the surface area of the second hemisphere will become π (4x) 2 = 48πx 2. Hence, we can see that the ratio of the surface area of two given hemispheres is 27πx 2 : 48πx 2 = 27 : 48 = 9:16

10 ID : ae-9-surface-area-and-volume [10] (15) 160 cm There are three equations we need to know in this type of question - the volume of a cylinder, the volume a sphere, and the remaining volume of the gap between the sphere and the cylinder. The volume of a cylinder of radius 'r' and height 'h' is πr 2 h. Here, we know the sphere will fit in exactly in the cylinder, so h=2r, and the formula now becomes 2πr. The sphere will have the radius 'r' so its volume is 4 πr. The volume of the gap between the cylinder and the sphere is all the volume inside the cylinder not taken up by the sphere. This is the difference between the volume of the cylinder and the volume of the sphere. i.e. volume of the gap = 2πr - 4 πr Simplifying, volume of the gap = 2 πr So we have equations: Volume of the cylinder = 2πr Volume of the sphere = 4 πr Volume of the gap = 2 πr Step 6 Here, we know that volume of the cylinder is 240 cm. We need to find the volume of the sphere. Step 7 Substituting from the equation above, we get volume of the sphere = 160 cm

Grade 9 Surface Area and Volume

ID : my-9-surface-area-and-volume [1] Grade 9 Surface Area and Volume For more such worksheets visit www.edugain.com Answer the questions (1) If the radii of two spheres are in ratio 5:2, find the ratio