Grade 9 Surface Area and Volume
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1 ID : ph-9-surface-area-and-volume [1] Grade 9 Surface Area and Volume For more such worksheets visit Answer t he quest ions (1) A sphere and a cone have the same radii. If the volume of the sphere is double of the volume of the cone, f ind the ratio of the cone's height and radius. (2) If volume of a cube is 216 cm 3, f ind its surf ace area. (3) A sphere is just enclosed inside a right circular cylinder. If curved surf ace area of cylinder is 10 cm 2, f ind surf ace area of sphere. (4) The radius of a cylinder is halved and the height is tripled. What is the area of the curved surf ace when compared to the same area previously? (5) A sphere is just enclosed inside a cube of volume 36 cm 3. Find the volume of the sphere. (6) If a hemisphere and cone stands on equal bases, and have the same height. Find the ratio of their volumes. Choose correct answer(s) f rom given choice (7) Find the volume the biggest sphere, which can f it in a cube of side 6r. a. 4/3 π r 3 b. 256/3 π r 3 c. 32/3 π r 3 d. 36 π r 3 (8) A cone made completely of metal (i.e. it is not hollow) has a base radius of 10 cm, and height of 40 cm. If we melt it and recast it into a sphere, what will be the radius of sphere? a. 20 cm b. 11 cm c. 10 cm d. 5 cm
2 (9) A sphere is just enclosed inside a right circular cylinder. If volume of the gap between cylinder and sphere is 40 cm 3, f ind volume of the sphere ID : ph-9-surface-area-and-volume [2] a. 40 cm 3 b. 160 cm 3 c. 100 cm 3 d. 80 cm 3 (10) The heights of two cones are in the ratio of 2:3 and their radii are in the ratio of 3:1. Find the ratio of their volumes. a. 6:1 b. 2:1 c. 1:6 d. 1:2 (11) Find the surf ace area of the biggest sphere, which can f it in a cube of side 4x. a. 64 π x 2 b. 36 π x 2 c. 4 π x 2 d. 16 π x 2 (12) Find the volume of the biggest cone that can f it inside a cube of side 1 cm. a. π 12 cm 3 b. π 3 cm 3 c. π 6 cm 3 (13) If the radius of a hemisphere is 3a, f ind its curved surf ace area. a. 18 π a 2 b. 48 π a 2 c. 12 π a 2 d. 8 π a 2 (14) If the radii of two spheres are in ratio 3:5, f ind the ratio of their surf ace area. a. 9:25 b. 125:27 c. 3:5 d. 5:3 (15) If radius of a sphere is 4r, f ind its volume. a. 256/3 π r 3 b. 4/3 π r 3 c. 36 π r 3 d. 32/3 π r Edugain ( All Rights Reserved Many more such worksheets can be generated at
3 Answers ID : ph-9-surface-area-and-volume [3] (1) 2:1 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 double of the volume of the cone in question. Theref ore, 4/3 π r 3 = 2 x (1/3 π r 2 h) h/r = 2:1 Thus, the ratio of the cone's height and radius is 2:1. (2) 216 cm 2 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 216 cm 3 This means m 3 =216 cm 3 Solving this, we get m = 6 And theref ore surf ace area of the cube = 6 x m 2 = 216 cm 2
4 (3) 10 cm 2 ID : ph-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 curved surf ace area of cylinder is 10 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 = 10 cm 2 (4) 1.5 times The curved surf ace area of a cylinder is 2πrh Here we halved the radius and tripled the height Putting this into the f ormula, we see that the curved surf ace area becomes 1.5 times
5 (5) 6 π cm 3 ID : ph-9-surface-area-and-volume [5] 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 = 36 cm 3 This means r = 4.5 Putting this in the f ormula f or the volume of the sphere, we get the volume = 6 π cm 3 (6) 2:1 The volume of a hemishpere of radius 'r' is 2 3 πr 3. The volume of a cone of radius 'r' and height 'h' is 1 3 π r 2 h From these equations we can cancel out the equal terms (remember the heights are also equal) to f ind the ratio as 2:1
6 (7) d. 36 π r 3 ID : ph-9-surface-area-and-volume [6] The biggest sphere that can f it inside a cube of side 6r will have a diameter of 6r (anything larger will not f it in, as opposite sides are separated by a distance of 6r. This means that the radius of this sphere is (1/2)6r The volume of a sphere of radius x is (4/3)πx 3 Theref ore the volume of this sphere is (4/3)π((1/2)6r) 3 Solving f or this gives us 36 π r 3 (8) c. 10 cm The volume of a cone is of radius r and height h = 1 3 π r 2 h The volume of a sphere of radius x = 4 3 π x 3 We know that the cone of base radius 10 cm and height 40 was melted down The volume of metal resulting f rom this = 1 3 π When we recast it into a sphere, we get a sphere of this volume Step 6 That is to say 4 3 π x 3 = 1 3 π Step 7 Solving f or x, we get x = 10 cm
7 (9) d. 80 cm 3 ID : ph-9-surface-area-and-volume [7] 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 gap between cylinder and sphere is 40 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 = 80 cm 3
8 (10) a. 6:1 ID : ph-9-surface-area-and-volume [8] The volume of a cones is 1 3 π r 2 h We can see f rom this that to compare ratios we can ignore the constant multipiler 1 3 π as it will be present in both the volumes We are told that the ratio of the heights of two cones is 2:3. So let's represent the height of the f irst one as 2h, and the second one as 3h Similarly their radii are in the ration 3:1. Let's represent the radius of the f irst one as 3r and the second one as r So the ratio of the volumes is (3r) 2 x 2h:(r) 2 x 3h This can be simplif ied to 9r x 2: r x 3h Step 6 Simplif ying this f urther, we get 6:1 (11) d. 16 π x 2 The biggest sphere that can f it inside a cube of side 4x will have a diameter of 4x (anything larger will not f it in, as opposite sides are separated by a distance of 4x. This means that the radius of this sphere is 1 2 4x The surf ace area of a sphere of radius x is 4πx 2 Theref ore the surf ace area of this sphere is 4π( 1 2 4x) 2 The answer is 16 π x 2
9 (12) a. π 12 cm 3 ID : ph-9-surface-area-and-volume [9] 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 1 cm, we see that the diameter of the cone will be 1 cm, and the height will be 1 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 1 2 = 0.5 Putting these values into the equation of the volume, we get the volume of the cone = 1 3 x π x x 1 Solving we get the volume of the cone = π 12 cm 3 (13) a. 18 π a 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 Here the radius is specif ied as 3a. Substituting this into the f ormula, we get the answer is (2π) x (3a) 2 This gives us the answer 18 π a 2
10 (14) a. 9:25 ID : ph-9-surface-area-and-volume [10] 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 3x and 5x (note that this allows us to get the ratio of 3:5, which is the only thing we know about these radii) The surf ace area of the f irst one then is 4π3x 2, and the surf ace area of the second one is 4π5x 2 The ratio of the the surf ace areas is theref ore 4π(3x) 2 :4π(5x) 2 This simplif ies to (3x) 2 :(5x) 2, and f urther to 3 2 :5 2 Theref ore the answer is 9:25 (15) a. 256/3 π r 3 The volume of a sphere of radius x is given by (4/3)πx 3 Here the radius is specif ied as 4r. Substituting this into the f ormula, we get the answer is (4/3π) x (4r) 3 This gives us the answer 256/3 π r 3
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