How to find the volume of a figure - HSPT Math
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I have a hollow cube with 3” sides suspended inside a larger cube of 9” sides. If I fill the larger cube with water and the hollow cube remains empty yet suspended inside, what volume of water was used to fill the larger cube?
I have a hollow cube with 3” sides suspended inside a larger cube of 9” sides. If I fill the larger cube with water and the hollow cube remains empty yet suspended inside, what volume of water was used to fill the larger cube?
Determine the volume of both cubes and then subtract the smaller from the larger. The large cube volume is 9” * 9” * 9” = 729 in3 and the small cube is 3” * 3” * 3” = 27 in3. The difference is 702 in3.
Determine the volume of both cubes and then subtract the smaller from the larger. The large cube volume is 9” * 9” * 9” = 729 in3 and the small cube is 3” * 3” * 3” = 27 in3. The difference is 702 in3.
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A cube weighs 5 pounds. How much will a different cube of the same material weigh if the sides are 3 times as long?
A cube weighs 5 pounds. How much will a different cube of the same material weigh if the sides are 3 times as long?
A cube that has three times as long sides is 3x3x3=27 times bigger than the original. Therefore, the answer is 5x27= 135.
A cube that has three times as long sides is 3x3x3=27 times bigger than the original. Therefore, the answer is 5x27= 135.
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The volume of a cylinder is 36π. If the cylinder’s height is 4, what is the cylinder’s diameter?
The volume of a cylinder is 36π. If the cylinder’s height is 4, what is the cylinder’s diameter?
Volume of a cylinder? V = πr2h. Rewritten as a diameter equation, this is:
V = π(d/2)2h = πd2h/4
Sub in h and V: 36p = πd2(4)/4 so 36p = πd2
Thus d = 6
Volume of a cylinder? V = πr2h. Rewritten as a diameter equation, this is:
V = π(d/2)2h = πd2h/4
Sub in h and V: 36p = πd2(4)/4 so 36p = πd2
Thus d = 6
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An empty tank in the shape of a right solid circular cone has a radius of r feet and a height of h feet. The tank is filled with water at a rate of w cubic feet per second. Which of the following expressions, in terms of r, h, and w, represents the number of minutes until the tank is completely filled?
An empty tank in the shape of a right solid circular cone has a radius of r feet and a height of h feet. The tank is filled with water at a rate of w cubic feet per second. Which of the following expressions, in terms of r, h, and w, represents the number of minutes until the tank is completely filled?
The volume of a cone is given by the formula V = (πr2)/3. In order to determine how many seconds it will take for the tank to fill, we must divide the volume by the rate of flow of the water.
time in seconds = (πr2)/(3w)
In order to convert from seconds to minutes, we must divide the number of seconds by sixty. Dividing by sixty is the same is multiplying by 1/60.
(πr2)/(3w) * (1/60) = π(r2)(h)/(180w)
The volume of a cone is given by the formula V = (πr2)/3. In order to determine how many seconds it will take for the tank to fill, we must divide the volume by the rate of flow of the water.
time in seconds = (πr2)/(3w)
In order to convert from seconds to minutes, we must divide the number of seconds by sixty. Dividing by sixty is the same is multiplying by 1/60.
(πr2)/(3w) * (1/60) = π(r2)(h)/(180w)
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What is the volume of a hollow cylinder whose inner radius is 2 cm and outer radius is 4 cm, with a height of 5 cm?
What is the volume of a hollow cylinder whose inner radius is 2 cm and outer radius is 4 cm, with a height of 5 cm?
The volume is found by subtracting the inner cylinder from the outer cylinder as given by V = πrout2 h – πrin2 h. The area of the cylinder using the outer radius is 80π cm3, and resulting hole is given by the volume from the inner radius, 20π cm3. The difference between the two gives the volume of the resulting hollow cylinder, 60π cm3.
The volume is found by subtracting the inner cylinder from the outer cylinder as given by V = πrout2 h – πrin2 h. The area of the cylinder using the outer radius is 80π cm3, and resulting hole is given by the volume from the inner radius, 20π cm3. The difference between the two gives the volume of the resulting hollow cylinder, 60π cm3.
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An 8-inch cube has a cylinder drilled out of it. The cylinder has a radius of 2.5 inches. To the nearest hundredth, approximately what is the remaining volume of the cube?
An 8-inch cube has a cylinder drilled out of it. The cylinder has a radius of 2.5 inches. To the nearest hundredth, approximately what is the remaining volume of the cube?
We must calculate our two volumes and subtract them. The volume of the cube is very simple: 8 * 8 * 8, or 512 in3.
The volume of the cylinder is calculated by multiplying the area of its base by its height. The height of the cylinder is 8 inches (the height of the cube through which it is being drilled). Therefore, its volume is πr2h = π * 2.52 * 8 = 50π in3
The volume remaining in the cube after the drilling is: 512 – 50π, or approximately 512 – 157.0795 = 354.9205, or 354.92 in3.
We must calculate our two volumes and subtract them. The volume of the cube is very simple: 8 * 8 * 8, or 512 in3.
The volume of the cylinder is calculated by multiplying the area of its base by its height. The height of the cylinder is 8 inches (the height of the cube through which it is being drilled). Therefore, its volume is πr2h = π * 2.52 * 8 = 50π in3
The volume remaining in the cube after the drilling is: 512 – 50π, or approximately 512 – 157.0795 = 354.9205, or 354.92 in3.
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What is the difference between the volume and surface area of a sphere with a radius of 6?
What is the difference between the volume and surface area of a sphere with a radius of 6?
Surface Area = 4_πr_2 = 4 * π * 62 = 144_π_
Volume = 4_πr_3/3 = 4 * π * 63 / 3 = 288_π_
Volume – Surface Area = 288_π_ – 144_π_ = 144_π_
Surface Area = 4_πr_2 = 4 * π * 62 = 144_π_
Volume = 4_πr_3/3 = 4 * π * 63 / 3 = 288_π_
Volume – Surface Area = 288_π_ – 144_π_ = 144_π_
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A cubic box has sides of length x. Another cubic box has sides of length 4_x_. How many of the boxes with length x could fit in one of the larger boxes with side length 4_x_?
A cubic box has sides of length x. Another cubic box has sides of length 4_x_. How many of the boxes with length x could fit in one of the larger boxes with side length 4_x_?
The volume of a cubic box is given by (side length)3. Thus, the volume of the larger box is (4_x_)3 = 64_x_3, while the volume of the smaller box is _x_3. Divide the volume of the larger box by that of the smaller box, (64_x_3)/(_x_3) = 64.
The volume of a cubic box is given by (side length)3. Thus, the volume of the larger box is (4_x_)3 = 64_x_3, while the volume of the smaller box is _x_3. Divide the volume of the larger box by that of the smaller box, (64_x_3)/(_x_3) = 64.
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Two cylinders are full of milk. The first cylinder is 9” tall and has a base diameter of 3”. The second cylinder is 9” tall and has a base diameter of 4”. If you are going to pour both cylinders of milk into a single cylinder with a base diameter of 6”, how tall must that cylinder be for the milk to fill it to the top?
Two cylinders are full of milk. The first cylinder is 9” tall and has a base diameter of 3”. The second cylinder is 9” tall and has a base diameter of 4”. If you are going to pour both cylinders of milk into a single cylinder with a base diameter of 6”, how tall must that cylinder be for the milk to fill it to the top?
Volume of cylinder = π * (base radius)2 x height = π * (base diameter / 2 )2 x height
Volume Cylinder 1 = π * (3 / 2 )2 x 9 = π * (1.5 )2 x 9 = π * 20.25
Volume Cylinder 2 = π * (4 / 2 )2 x 9 = π * (2 )2 x 9 = π * 36
Total Volume = π * 20.25 + π * 36
Volume of Cylinder 3 = π * (6 / 2 )2 x H = π * (3 )2 x H = π * 9 x H
Set Total Volume equal to the Volume of Cylinder 3 and solve for H
π * 20.25 + π * 36 = π * 9 x H
20.25 + 36 = 9 x H
H = (20.25 + 36) / 9 = 6.25”
Volume of cylinder = π * (base radius)2 x height = π * (base diameter / 2 )2 x height
Volume Cylinder 1 = π * (3 / 2 )2 x 9 = π * (1.5 )2 x 9 = π * 20.25
Volume Cylinder 2 = π * (4 / 2 )2 x 9 = π * (2 )2 x 9 = π * 36
Total Volume = π * 20.25 + π * 36
Volume of Cylinder 3 = π * (6 / 2 )2 x H = π * (3 )2 x H = π * 9 x H
Set Total Volume equal to the Volume of Cylinder 3 and solve for H
π * 20.25 + π * 36 = π * 9 x H
20.25 + 36 = 9 x H
H = (20.25 + 36) / 9 = 6.25”
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A cube with sides of 4” each contains a floating sphere with a radius of 1”. What is the volume of the space outside of the sphere, within the cube?
A cube with sides of 4” each contains a floating sphere with a radius of 1”. What is the volume of the space outside of the sphere, within the cube?
Volume of Cube = side3 = (4”)3 = 64 in3
Volume of Sphere = (4/3) * π * r3 = (4/3) * π * 13 = (4/3) * π * 13 = (4/3) * π = 4.187 in3
Difference = Volume of Cube – Volume of Sphere = 64 – 4.187 = 59.813 in3
Volume of Cube = side3 = (4”)3 = 64 in3
Volume of Sphere = (4/3) * π * r3 = (4/3) * π * 13 = (4/3) * π * 13 = (4/3) * π = 4.187 in3
Difference = Volume of Cube – Volume of Sphere = 64 – 4.187 = 59.813 in3
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If the volume of a cube is 50 cubic feet, what is the volume when the sides double in length?
If the volume of a cube is 50 cubic feet, what is the volume when the sides double in length?
Using S as the side length in the original cube, the original is s*s*s. Doubling one side and tripling the other gives 2s*2s*2s for a new volume formula for 8s*s*s, showing that the new volume is 8x greater than the original.
Using S as the side length in the original cube, the original is s*s*s. Doubling one side and tripling the other gives 2s*2s*2s for a new volume formula for 8s*s*s, showing that the new volume is 8x greater than the original.
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A cube is inscribed inside a sphere of radius 1 such that each of the eight vertices of the cube lie on the surface of the sphere. What is the volume of the cube?
A cube is inscribed inside a sphere of radius 1 such that each of the eight vertices of the cube lie on the surface of the sphere. What is the volume of the cube?

To make this problem easier to solve, we can inscribe a smaller square in the cube. In the diagram above, points
are midpoints of the edges of the inscribed cube. Therefore point
, a vertex of the smaller cube, is also the center of the sphere. Point
lies on the circumference of the sphere, so
.
is also the hypotenuse of right triangle
. Similarly,
is the hypotenuse of right triangle
. If we let
, then, by the properties of a right triangle,
.
Using the Pythagorean Theorem, we can now solve for
:






Since the side of the inscribed cube is
, the volume is
.
To make this problem easier to solve, we can inscribe a smaller square in the cube. In the diagram above, points are midpoints of the edges of the inscribed cube. Therefore point
, a vertex of the smaller cube, is also the center of the sphere. Point
lies on the circumference of the sphere, so
.
is also the hypotenuse of right triangle
. Similarly,
is the hypotenuse of right triangle
. If we let
, then, by the properties of a right triangle,
.
Using the Pythagorean Theorem, we can now solve for :
Since the side of the inscribed cube is , the volume is
.
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The surface area of a sphere is
. Find the volume of the sphere in cubic millimeters.
The surface area of a sphere is . Find the volume of the sphere in cubic millimeters.
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A cylinder has a height of 5 inches and a radius of 3 inches. Find the lateral area of the cylinder.
A cylinder has a height of 5 inches and a radius of 3 inches. Find the lateral area of the cylinder.
LA = 2π(r)(h) = 2π(3)(5) = 30π
LA = 2π(r)(h) = 2π(3)(5) = 30π
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If a sphere's diameter is doubled, by what factor is its volume increased?
If a sphere's diameter is doubled, by what factor is its volume increased?
The formula for the volume of a sphere is 4πr3/3. Because this formula is in terms of the radius, it would be easier for us to determine how the change in the radius affects the volume. Since we are told the diameter is doubled, we need to first determine how the change in the diameter affects the change in radius.
Let us call the sphere's original diameter d and its original radius r. We know that d = 2r.
Let's call the final diameter of the sphere D. Because the diameter is doubled, we know that D = 2d. We can substitute the value of d and obtain D = 2(2r) = 4r.
Let's call the final radius of the sphere R. We know that D = 2R, so we can now substituate this into the previous equation and write 2R = 4r. If we simplify this, we see that R = 2r. This means that the final radius is twice as large as the initial radius.
The initial volume of the sphere is 4πr3/3. The final volume of the sphere is 4πR3/3. Because R = 2r, we can substitute the value of R to obtain 4π(2r)3/3. When we simplify this, we get 4π(8r3)/3 = 32πr3/3.
In order to determine the factor by which the volume has increased, we need to find the ratio of the final volume to the initial volume.
32πr3/3 divided by 4πr3/3 = 8
The volume has increased by a factor of 8.
The formula for the volume of a sphere is 4πr3/3. Because this formula is in terms of the radius, it would be easier for us to determine how the change in the radius affects the volume. Since we are told the diameter is doubled, we need to first determine how the change in the diameter affects the change in radius.
Let us call the sphere's original diameter d and its original radius r. We know that d = 2r.
Let's call the final diameter of the sphere D. Because the diameter is doubled, we know that D = 2d. We can substitute the value of d and obtain D = 2(2r) = 4r.
Let's call the final radius of the sphere R. We know that D = 2R, so we can now substituate this into the previous equation and write 2R = 4r. If we simplify this, we see that R = 2r. This means that the final radius is twice as large as the initial radius.
The initial volume of the sphere is 4πr3/3. The final volume of the sphere is 4πR3/3. Because R = 2r, we can substitute the value of R to obtain 4π(2r)3/3. When we simplify this, we get 4π(8r3)/3 = 32πr3/3.
In order to determine the factor by which the volume has increased, we need to find the ratio of the final volume to the initial volume.
32πr3/3 divided by 4πr3/3 = 8
The volume has increased by a factor of 8.
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A cube has 2 faces painted red and the remaining faces painted green. The total area of the green faces is 36 square inches. What is the volume of the cube in cubic inches?
A cube has 2 faces painted red and the remaining faces painted green. The total area of the green faces is 36 square inches. What is the volume of the cube in cubic inches?
Cubes have 6 faces. If 2 are red, then 4 must be green. We are told that the total area of the green faces is 36 square inches, so we divide the total area of the green faces by the number of green faces (which is 4) to get the area of each green face: 36/4 = 9 square inches. Since each of the 6 faces of a cube have the same size, we know that each edge of the cube is √9 = 3 inches. Therefore the volume of the cube is 3 in x 3 in x 3 in = 27 cubic inches.
Cubes have 6 faces. If 2 are red, then 4 must be green. We are told that the total area of the green faces is 36 square inches, so we divide the total area of the green faces by the number of green faces (which is 4) to get the area of each green face: 36/4 = 9 square inches. Since each of the 6 faces of a cube have the same size, we know that each edge of the cube is √9 = 3 inches. Therefore the volume of the cube is 3 in x 3 in x 3 in = 27 cubic inches.
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A cylinder has a volume of 20. If the radius doubles, what is the new volume?
A cylinder has a volume of 20. If the radius doubles, what is the new volume?
The equation for the volume of the cylinder is πr2h. When the radius doubles (r becomes 2r) you get π(2r)2h = 4πr2h. So when the radius doubles, the volume quadruples, giving a new volume of 80.
The equation for the volume of the cylinder is πr2h. When the radius doubles (r becomes 2r) you get π(2r)2h = 4πr2h. So when the radius doubles, the volume quadruples, giving a new volume of 80.
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A sphere increases in volume by a factor of 8. By what factor does the radius change?
A sphere increases in volume by a factor of 8. By what factor does the radius change?
Volume of a sphere is 4Πr3/3. Setting that equation equal to the original volume, the new volume is given as 8*4Πr3/3, which can be rewritten as 4Π8r3/3, and can be added to the radius value by 4Π(2r)3/3 since 8 si the cube of 2. This means the radius goes up by a factor of 2
Volume of a sphere is 4Πr3/3. Setting that equation equal to the original volume, the new volume is given as 8*4Πr3/3, which can be rewritten as 4Π8r3/3, and can be added to the radius value by 4Π(2r)3/3 since 8 si the cube of 2. This means the radius goes up by a factor of 2
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A rectangular box has a length of 2 meters, a width of 0.5 meters, and a height of 3.2 meters. How many cubes with a volume of one cubic centimeter could fit into this rectangular box?
A rectangular box has a length of 2 meters, a width of 0.5 meters, and a height of 3.2 meters. How many cubes with a volume of one cubic centimeter could fit into this rectangular box?
In order to figure out how many cubic centimeters can fit into the box, we need to figure out the volume of the box in terms of cubic centimeters. However, the measurements of the box are given in meters. Therefore, we need to convert these measurements to centimeters and then determine the volume of the box.
There are 100 centimeters in one meter. This means that in order to convert from meters to centimeters, we must multiply by 100.
The length of the box is 2 meters, which is equal to 2 x 100, or 200, centimeters.
The width of the box is 0.5(100) = 50 centimeters.
The height of the box is 3.2(100) = 320 centimeters.
Now that all of our measurements are in centimeters, we can calculate the volume of the box in cubic centimeters. Remember that the volume of a rectangular box (or prism) is equal to the product of the length, width, and height.
V = length x width x height
V = (200 cm)(50 cm)(320 cm) = 3,200,000 cm3
To rewrite this in scientific notation, we must move the decimal six places to the left.
V = 3.2 x 106 cm3
The answer is 3.2 x 106.
In order to figure out how many cubic centimeters can fit into the box, we need to figure out the volume of the box in terms of cubic centimeters. However, the measurements of the box are given in meters. Therefore, we need to convert these measurements to centimeters and then determine the volume of the box.
There are 100 centimeters in one meter. This means that in order to convert from meters to centimeters, we must multiply by 100.
The length of the box is 2 meters, which is equal to 2 x 100, or 200, centimeters.
The width of the box is 0.5(100) = 50 centimeters.
The height of the box is 3.2(100) = 320 centimeters.
Now that all of our measurements are in centimeters, we can calculate the volume of the box in cubic centimeters. Remember that the volume of a rectangular box (or prism) is equal to the product of the length, width, and height.
V = length x width x height
V = (200 cm)(50 cm)(320 cm) = 3,200,000 cm3
To rewrite this in scientific notation, we must move the decimal six places to the left.
V = 3.2 x 106 cm3
The answer is 3.2 x 106.
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A cylinder has a height that is three times as long as its radius. If the lateral surface area of the cylinder is 54π square units, then what is its volume in cubic units?
A cylinder has a height that is three times as long as its radius. If the lateral surface area of the cylinder is 54π square units, then what is its volume in cubic units?
Let us call r the radius and h the height of the cylinder. We are told that the height is three times the radius, which we can represent as h = 3r.
We are also told that the lateral surface area is equal to 54π. The lateral surface area is the surface area that does not include the bases. The formula for the lateral surface area is equal to the circumference of the cylinder times its height, or 2πrh. We set this equal to 54π,
2πrh = 54π
Now we substitute 3r in for h.
2πr(3r) = 54π
6πr2 = 54π
Divide by 6π
r2 = 9.
Take the square root.
r = 3.
h = 3r = 3(3) = 9.
Now that we have the radius and the height of the cylinder, we can find its volume, which is given by πr2h.
V = πr2h
V = π(3)2(9) = 81π
The answer is 81π.
Let us call r the radius and h the height of the cylinder. We are told that the height is three times the radius, which we can represent as h = 3r.
We are also told that the lateral surface area is equal to 54π. The lateral surface area is the surface area that does not include the bases. The formula for the lateral surface area is equal to the circumference of the cylinder times its height, or 2πrh. We set this equal to 54π,
2πrh = 54π
Now we substitute 3r in for h.
2πr(3r) = 54π
6πr2 = 54π
Divide by 6π
r2 = 9.
Take the square root.
r = 3.
h = 3r = 3(3) = 9.
Now that we have the radius and the height of the cylinder, we can find its volume, which is given by πr2h.
V = πr2h
V = π(3)2(9) = 81π
The answer is 81π.
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