Grade 9 Surface Area and Volume

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

1 ID : sg-9-surface-area-and-volume [1] Grade 9 Surface Area and Volume For more such worksheets visit Answer t he quest ions (1) The heights of two cylinders are in the ratio of 7:2 and their radii are in the ratio of 1:7. Find the ratio of their volumes. (2) A sphere is just enclosed inside a right circular cylinder. If total surf ace area of cylinder is 270 cm 2, f ind surf ace area of sphere. (3) If radius of a sphere is 4b, f ind its surf ace area. (4) An sphere is expanded to a bigger sphere such that its radius increases by a f actor of 2, f ind the change in its surf ace area. (5) Find the volume of the biggest cone that can f it inside a cube of side 4 cm. (6) If radius of two hemispheres are in ratio 1:2, f ind the ratio of their volumes. (7) A sphere is just enclosed inside a cube of volume 96 cm 3. Find the volume of the sphere. (8) If a cylinder and hemisphere stands on equal bases, and have the same height. Find the ratio of their volumes. Choose correct answer(s) f rom given choice (9) Find the volume the biggest hemisphere, which can f it in a cube of side 4b. a. 2/3 π b 3 b. 18 π b 3 c. 128/3 π b 3 d. 16/3 π b 3 (10) If the radii of two spheres are in ratio 5:3, f ind the ratio of their surf ace area. a. 3:5 b. 25:9 c. 27:125 d. 5:3 (11) A sphere and a cone have the same radii. If the volume of the sphere is triple of the volume of the cone, f ind the ratio of the cone's height and radius. a. 1:2 b. 4:3 c. 2:1 d. 3:1

2 (12) If the radius of a hemisphere is 2x, f ind its total surf ace area. ID : sg-9-surface-area-and-volume [2] a. 2 π x 2 b. 12 π x 2 c. 3 π x 2 d. 18 π x 2 (13) A sphere is just enclosed inside a right circular cylinder. If volume of the cylinder is 90 cm 3, f ind volume of the sphere a. 75 cm 3 b. 120 cm 3 c. 60 cm 3 d. 30 cm 3 (14) If radius of a sphere is 2a, f ind its volume. a. 256/3 π a 3 b. 4/3 π a 3 c. 32/3 π a 3 d. 36 π a 3 Fill in the blanks (15) If volume of a cube is 8 cm 3, its surf ace area = cm Edugain ( All Rights Reserved Many more such worksheets can be generated at

3 Answers ID : sg-9-surface-area-and-volume [3] (1) 1:14 The volume of a cylinders is π r 2 h We can see f rom this that to compare ratios we can ignore the constant multipiler π as it will be present in both the volumes We are told that the ratio of the heights of two cylinders is 7:2. So let's represent the height of the f irst one as 7h, and the second one as 2h Similarly their radii are in the ration 1:7. Let's represent the radius of the f irst one as r and the second one as 7r So the ratio of the volumes is (r) 2 x 7h:(7r) 2 x 2h This can be simplif ied to r x 7: 49r x 2h Step 6 Simplif ying this f urther, we get 1:14

4 (2) 180 cm 2 ID : sg-9-surface-area-and-volume [4] 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 surf ace area of a sphere The curved surf ace area of a cylinder of radius 'r' and height 'h' is 2πrh. Here we know the sphere will f it in exactly in the cylinder, so h=2r, and the f ormula now becomes 4πr 2 The total surf ace area of the same cylinder will be the sum of the curved area and the surf ace area of the two circles at top and bottom. So 4πr 2 + 2πr 2 = 6πr 2 And of course, the sphere will have the radius r too, so it's surf ace area is 4πr 2 From these equations, we see that f or this case, the surf ace area of the sphere is the same as the curved surf ace are of the cylinder, and 2/3 of the total surf ace area of the cylinder Step 6 Here we know that total surf ace area of cylinder is 270 cm 2, and need to f ind surf ace area of sphere Step 7 Substituting f rom the equation above, we get surf ace area of sphere = 180 cm 2 (3) 64 π b 2 The surf ace area of a sphere of radius x is given by 4πx 2 Here the radius is specif ied as 4b. Substituting this into the f ormula, we get the answer is (4π) x (4b) 2 This gives us the answer 64 π b 2

5 (4) 4 times ID : sg-9-surface-area-and-volume [5] The volume of a sphere of radius x = 4 3 π x 3 The surf ace area of a sphere of radius x = 4π x 2 This means that the surf ace 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 increased by a f actor of 2 This means that the surf ace area would have increased by a square of this value i.e. by 2 2 Solving this, we get 4 times (5) 16 π 3 cm 3 The volume of a cone is of radius r and height h = 1 3 π r 2 h Since we have to f it 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 f it inside the cube So the radius of this cone is 4 2 = 2 Putting these values into the equation of the volume, we get the volume of the cone = 1 3 x π x x 4 Solving we get the volume of the cone = 16 π 3 cm 3

6 (6) 1:8 ID : sg-9-surface-area-and-volume [6] The volume of a hemispheres of radius x is given by (4/3)πx 3 The volume of a hemisphere is half that i.e. (2/3)πx 3 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 hemispheres are 1x and 2x (note that this allows us to get the ratio of 1:2, which is the only thing we know about these radii) The volume of the f irst one then is 2/3π1x 3, and the volume of the second one is 2/3π2x 3 The ratio of the the volumes is theref ore 2/3π(1x) 3 :2/3π(2x) 3 This simplif ies to (1x) 3 :(2x) 3, and f urther to 1 3 :2 3 Theref ore the answer is 1:8 (7) 16 π cm 3 The volume of a sphere is of radius r = 4 3 π r 3 If it f its 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 The volume of a cube of side 2r = (2r) 3 = 8r 3 8r 3 = 96 cm 3 This means r = 12 Putting this in the f ormula f or the volume of the sphere, we get the volume = 16 π cm 3

7 (8) 3:2 ID : sg-9-surface-area-and-volume [7] The volume of a cylinder of radius 'r' and height 'h' is πr 2 h. The volume of a hemishpere of radius 'r' is 2 3 πr 3. From these equations we can cancel out the equal terms (remember the heights are also equal) to f ind the ratio as 3:2 (9) d. 16/3 π b 3 The biggest hemisphere that can f it inside a cube of side 4b will have a diameter of 4b (anything larger will not f it in, as opposite sides are separated by a distance of 4b. This means that the radius of this sphere is (1/2)4b The volume of a hemisphere of radius x is (2/3)πx 3 Theref ore the volume of this hemisphere is (2/3)π((1/2)4b) 3 Solving f or this gives us 16/3 π b 3 (10) b. 25:9 The surf ace area of a sphere of radius x is given by 4πx 2 We see that the surf ace is proportional to the 2rd 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 surf ace area of the f irst one then is 4π5x 2, and the surf ace area of the second one is 4π3x 2 The ratio of the the surf ace areas is theref ore 4π(5x) 2 :4π(3x) 2 This simplif ies to (5x) 2 :(3x) 2, and f urther to 5 2 :3 2 Theref ore the answer is 25:9

8 (11) b. 4:3 ID : sg-9-surface-area-and-volume [8] We know that the volume of a cone with radius r and height h = 1/3 π r 2 h. We also know that the volume of a sphere with radius r = 4/3 π r 3. We have been told that the volume of the sphere in question is triple of the volume of the cone in question. Theref ore, 4/3 π r 3 = 3 x (1/3 π r 2 h) h/r = 4:3 Thus, the ratio of the cone's height and radius is 4:3. (12) b. 12 π x 2 The surf ace area of a sphere of radius x is given by 4πx 2 The curved surf ace area of a hemisphere is half of that i.e. 2πx 2 To get the total surf ace area, we need to add to this the area of the circle at the base i.e. πx 2 br> Adding them, we get area of hemisphere = 3πx 2 Here the radius is specif ied as 2x. Substituting this into the f ormula, we get the answer is (3π) x (2x) 2 Step 6 This gives us the answer 12 π x 2

9 (13) c. 60 cm 3 ID : sg-9-surface-area-and-volume [9] There are three equations we need to know this type of question - the total volume of a cylinder, the volume a sphere, and the remaining volume af ter the sphere f its in the cylinder The volume of a cylinder of radius 'r' and height 'h' is πr 2 h. Here we know the sphere will f it in exactly in the cylinder, so h=2r, and the f ormula now becomes 2πr 3 The sphere will have the radius r too (see the f igure here), so it's volume is 4 3 πr 3 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 dif f erence between the volume of the cylinder and the volume of the sphere. i.e volume of the gap = 2πr πr 3 Simplif ying, volume of the gap = 2 3 πr 3 So we have 3 equations Volume(cylinder) = 2πr 3 Volume(sphere) = 4 3 πr 3 Volume(gap) = 2 3 πr 3 Step 6 Here we know that volume of the cylinder is 90 cm 3, and need to f ind volume of the sphere Step 7 Substituting f rom the equation above, we get volume of the sphere = 60 cm 3

10 (14) c. 32/3 π a 3 ID : sg-9-surface-area-and-volume [10] The volume of a sphere of radius x is given by (4/3)πx 3 Here the radius is specif ied as 2a. Substituting this into the f ormula, we get the answer is (4/3π) x (2a) 3 This gives us the answer 32/3 π a 3 (15) 24 We know the f ollowing with regards to a cube with length m - The surf ace area of the cube = 6 x m 2 - The volume of the cube = m 3 We are given that the volume of the cube is 8 cm 3 This means m 3 =8 cm 3 Solving this, we get m = 2 And theref ore surf ace area of the cube = 6 x m 2 = 24 cm 2

Grade 9 Surface Area and Volume

ID : th-9-surface-area-and-volume [1] Grade 9 Surface Area and Volume For more such worksheets visit www.edugain.com Answer t he quest ions (1) Find the volume the biggest sphere which can f it in a cube