Cubes - PSAT Math
Card 1 of 245
If a cube is 3” on all sides, what is the length of the diagonal of the cube?
If a cube is 3” on all sides, what is the length of the diagonal of the cube?
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General formula for the diagonal of a cube if each side of the cube = s
Use Pythagorean Theorem to get the diagonal across the base:
s2 + s2 = h2
And again use Pythagorean Theorem to get cube’s diagonal, then solve for d:
h2 + s2 = d2
s2 + s2 + s2 = d2
3 * s2 = d2
d = √ (3 * s2) = s √3
So, if s = 3 then the answer is 3√3
General formula for the diagonal of a cube if each side of the cube = s
Use Pythagorean Theorem to get the diagonal across the base:
s2 + s2 = h2
And again use Pythagorean Theorem to get cube’s diagonal, then solve for d:
h2 + s2 = d2
s2 + s2 + s2 = d2
3 * s2 = d2
d = √ (3 * s2) = s √3
So, if s = 3 then the answer is 3√3
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A cube is inscribed in a sphere of radius 1 such that all 8 vertices of the cube are on the surface of the sphere. What is the length of the diagonal of the cube?
A cube is inscribed in a sphere of radius 1 such that all 8 vertices of the cube are on the surface of the sphere. What is the length of the diagonal of the cube?
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Since the diagonal of the cube is a line segment that goes through the center of the cube (and also the circumscribed sphere), it is clear that the diagonal of the cube is also the diameter of the sphere. Since the radius = 1, the diameter = 2.
Since the diagonal of the cube is a line segment that goes through the center of the cube (and also the circumscribed sphere), it is clear that the diagonal of the cube is also the diameter of the sphere. Since the radius = 1, the diameter = 2.
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What is the length of the diagonal of a cube with volume of 512 in3?
What is the length of the diagonal of a cube with volume of 512 in3?
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The first thing necessary is to determine the dimensions of the cube. This can be done using the volume formula for cubes: V = _s_3, where s is the length of the cube. For our data, this is:
_s_3 = 512, or (taking the cube root of both sides), s = 8.
The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (8,8,8). The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean Theorem):
d = √( (_x_1 – _x_2)2 + (_y_1 – _y_2)2 + (_z_1 – _z_2)2)
Or for our simpler case:
d = √( (x)2 + (y)2 + (z)2) = √( (s)2 + (s)2 + (s)2) = √( (8)2 + (8)2 + (8)2) = √( 64 + 64 + 64) = √(64 * 3) = 8√(3)
The first thing necessary is to determine the dimensions of the cube. This can be done using the volume formula for cubes: V = _s_3, where s is the length of the cube. For our data, this is:
_s_3 = 512, or (taking the cube root of both sides), s = 8.
The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (8,8,8). The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean Theorem):
d = √( (_x_1 – _x_2)2 + (_y_1 – _y_2)2 + (_z_1 – _z_2)2)
Or for our simpler case:
d = √( (x)2 + (y)2 + (z)2) = √( (s)2 + (s)2 + (s)2) = √( (8)2 + (8)2 + (8)2) = √( 64 + 64 + 64) = √(64 * 3) = 8√(3)
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What is the length of the diagonal of a cube with volume of 1728 in3?
What is the length of the diagonal of a cube with volume of 1728 in3?
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The first thing necessary is to determine the dimensions of the cube. This can be done using the volume formula for cubes: V = _s_3, where s is the length of the cube. For our data, this is:
_s_3 = 1728, or (taking the cube root of both sides), s = 12.
The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (12,12,12). The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean theorem):
d = √( (_x_1 – _x_2)2 + (_y_1 – _y_2)2 + (_z_1 – _z_2)2)
Or, for our simpler case:
d = √( (x)2 + (y)2 + (z)2) = √( (s)2 + (s)2 + (s)2) = √( (12)2 + (12)2 + (12)2) = √( 144 + 144 + 144) = √(3 * 144) = 12√(3) = 12√(3)
The first thing necessary is to determine the dimensions of the cube. This can be done using the volume formula for cubes: V = _s_3, where s is the length of the cube. For our data, this is:
_s_3 = 1728, or (taking the cube root of both sides), s = 12.
The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (12,12,12). The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean theorem):
d = √( (_x_1 – _x_2)2 + (_y_1 – _y_2)2 + (_z_1 – _z_2)2)
Or, for our simpler case:
d = √( (x)2 + (y)2 + (z)2) = √( (s)2 + (s)2 + (s)2) = √( (12)2 + (12)2 + (12)2) = √( 144 + 144 + 144) = √(3 * 144) = 12√(3) = 12√(3)
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What is the length of the diagonal of a cube with surface area of 294 in2?
What is the length of the diagonal of a cube with surface area of 294 in2?
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The first thing necessary is to determine the dimensions of the cube. This can be done using the surface area formula for cubes: A = 6_s_2, where s is the length of the cube. For our data, this is:
6_s_2 = 294
_s_2 = 49
(taking the square root of both sides) s = 7
The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (7,7,7). The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean Theorem):
d = √((_x_1 – _x_2)2 + (_y_1 – _y_2)2 + (_z_1 – _z_2)2)
Or for our simpler case:
d = √((x)2 + (y)2 + (z)2) = √( (s)2 + (s)2 + (s)2) = √( (7)2 + (7)2 + (7)2) = √( 49 + 49 + 49) = √(49 * 3) = 7√(3)
The first thing necessary is to determine the dimensions of the cube. This can be done using the surface area formula for cubes: A = 6_s_2, where s is the length of the cube. For our data, this is:
6_s_2 = 294
_s_2 = 49
(taking the square root of both sides) s = 7
The distance from corner to corner of the cube will be equal to the distance between (0,0,0) and (7,7,7). The distance formula for three dimensions is very similar to that of 2 dimensions (and hence like the Pythagorean Theorem):
d = √((_x_1 – _x_2)2 + (_y_1 – _y_2)2 + (_z_1 – _z_2)2)
Or for our simpler case:
d = √((x)2 + (y)2 + (z)2) = √( (s)2 + (s)2 + (s)2) = √( (7)2 + (7)2 + (7)2) = √( 49 + 49 + 49) = √(49 * 3) = 7√(3)
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A rectangular prism has a volume of 144 and a surface area of 192. If the shortest edge is 3, what is the length of the longest diagonal through the prism?
A rectangular prism has a volume of 144 and a surface area of 192. If the shortest edge is 3, what is the length of the longest diagonal through the prism?
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The volume of a rectangular prism is 
.
We are told that the shortest edge is 3. Let us call this the height.
We now have 
, or 
.
Now we replace variables by known values:
Now we have:
We have thus determined that the other two edges of the rectangular prism will be 4 and 12. We now need to find the longest diagonal. This is equal to:
If you do not remember how to find this directly, you can also do it in steps. You first find the diagonal across one of the sides (in the plane), by using the Pythagorean Theorem. For example, we choose the side with edges 3 and 4. This diagonal will be:
We then use a plane with one side given by the diagonal we just found (length 5) and the other given by the distance of the 3rd edge (length 12).
This diagonal is then 
.
The volume of a rectangular prism is
We are told that the shortest edge is 3. Let us call this the height.
We now have
Now we replace variables by known values:
Now we have:
We have thus determined that the other two edges of the rectangular prism will be 4 and 12. We now need to find the longest diagonal. This is equal to:
If you do not remember how to find this directly, you can also do it in steps. You first find the diagonal across one of the sides (in the plane), by using the Pythagorean Theorem. For example, we choose the side with edges 3 and 4. This diagonal will be:
We then use a plane with one side given by the diagonal we just found (length 5) and the other given by the distance of the 3rd edge (length 12).
This diagonal is then
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The number of square units in the surface area of a cube is twice as large as the number of cubic units in its volume. What is the cube's volume, in cubic units?
The number of square units in the surface area of a cube is twice as large as the number of cubic units in its volume. What is the cube's volume, in cubic units?
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The number of square units in the surface area of a cube is given by the formula 6s2, where s is the length of the side of the cube in units. Moreover, the number of cubic units in a cube's volume is equal to s3.
Since the number of square units in the surface area is twice as large as the cubic units of the volume, we can write the following equation to solve for s:
6s2 = 2s3
Subtract 6s2 from both sides.
2s3 – 6s2 = 0
Factor out 2s2 from both terms.
2s2(s – 3) = 0
We must set each factor equal to zero.
2s2 = 0, only if s = 0; however, no cube has a side length of zero, so s can't be zero.
Set the other factor, s – 3, equal to zero.
s – 3 = 0
Add three to both sides.
s = 3
This means that the side length of the cube is 3 units. The volume, which we previously stated was equal to s3, must then be 33, or 27 cubic units.
The answer is 27.
The number of square units in the surface area of a cube is given by the formula 6s2, where s is the length of the side of the cube in units. Moreover, the number of cubic units in a cube's volume is equal to s3.
Since the number of square units in the surface area is twice as large as the cubic units of the volume, we can write the following equation to solve for s:
6s2 = 2s3
Subtract 6s2 from both sides.
2s3 – 6s2 = 0
Factor out 2s2 from both terms.
2s2(s – 3) = 0
We must set each factor equal to zero.
2s2 = 0, only if s = 0; however, no cube has a side length of zero, so s can't be zero.
Set the other factor, s – 3, equal to zero.
s – 3 = 0
Add three to both sides.
s = 3
This means that the side length of the cube is 3 units. The volume, which we previously stated was equal to s3, must then be 33, or 27 cubic units.
The answer is 27.
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What is the surface area of a cube whose volume is 512 cubic feet?
What is the surface area of a cube whose volume is 512 cubic feet?
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In order to find the surface area of a cube, we need to solve for the length of each side, 
.
Recall the formula for volume: 
Plug in what we know and solve for 
:
Now plug this value into the surface area formula:
In order to find the surface area of a cube, we need to solve for the length of each side,
Recall the formula for volume:
Plug in what we know and solve for
Now plug this value into the surface area formula:
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If the volume of a cube is 64 cubic inches, then it has an edge length of .
If the volume of a cube is 64 cubic inches, then it has an edge length of .
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If the volume of a cube is 512 units, what is the length of one edge of the cube?
If the volume of a cube is 512 units, what is the length of one edge of the cube?
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The volume of a cube is length x width x height. Since it's a cube, though, the length, width, and height are all equal, and equivalent to the length of one edge of the cube. Therefore, to find the lenght of an edge of the cube, just find the cube root of the volume. In this case, the cube root of 512 is equal to 8.
The volume of a cube is length x width x height. Since it's a cube, though, the length, width, and height are all equal, and equivalent to the length of one edge of the cube. Therefore, to find the lenght of an edge of the cube, just find the cube root of the volume. In this case, the cube root of 512 is equal to 8.
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Find the length of an edge of a cube that has a volume of 
.
Find the length of an edge of a cube that has a volume of
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All the edges of a cube have the same length, and the volume of a cube is the length of an edge taken to the third power.
So if we take the edge of the cube to be of length x, then:
So the length of the edge of a cube with a volume of 125 is 5.
All the edges of a cube have the same length, and the volume of a cube is the length of an edge taken to the third power.
So if we take the edge of the cube to be of length x, then:
So the length of the edge of a cube with a volume of 125 is 5.
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A certain shipping company has cubic boxes. One of these boxes has a volume of 
. How long are each of the sides of the box in feet?
A certain shipping company has cubic boxes. One of these boxes has a volume of
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The formula for the volume of a cube is
where 
is the length of a side.
Here, the volume is 729. To find the side length, take the cube root of both sides:
The cube root of 729 is 9, so the length of each side of the cube is 9 feet.
The formula for the volume of a cube is
where
Here, the volume is 729. To find the side length, take the cube root of both sides:
The cube root of 729 is 9, so the length of each side of the cube is 9 feet.
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If a cube has a volume of 
cubic inches, approximately how long, in feet, is one edge of the cube?
If a cube has a volume of
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The formula for the volume of a cube is 
where s is any edge.
This means one edge of the cube is 
. We then divide 8.5 by 12 to convert to feet.
feet.
The formula for the volume of a cube is
This means one edge of the cube is
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A cube has a surface area of 10m2. If a cube's sides all double in length, what is the new surface area?
A cube has a surface area of 10m2. If a cube's sides all double in length, what is the new surface area?
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The equation for surface area of the original cube is 6s2. If the sides all double in length the new equation is 6(2s)2 or 6 * 4s2. This makes the new surface area 4x that of the old. 4x10 = 40m2
The equation for surface area of the original cube is 6s2. If the sides all double in length the new equation is 6(2s)2 or 6 * 4s2. This makes the new surface area 4x that of the old. 4x10 = 40m2
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What is the surface area of a cube with a volume of 1728 in3?
What is the surface area of a cube with a volume of 1728 in3?
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This problem is relatively simple. We know that the volume of a cube is equal to s3, where s is the length of a given side of the cube. Therefore, to find our dimensions, we merely have to solve s3 = 1728. Taking the cubed root, we get s = 12.
Since the sides of a cube are all the same, the surface area of the cube is equal to 6 times the area of one face. For our dimensions, one face has an area of 12 * 12 or 144 in2. Therefore, the total surface area is 6 * 144 = 864 in2.
This problem is relatively simple. We know that the volume of a cube is equal to s3, where s is the length of a given side of the cube. Therefore, to find our dimensions, we merely have to solve s3 = 1728. Taking the cubed root, we get s = 12.
Since the sides of a cube are all the same, the surface area of the cube is equal to 6 times the area of one face. For our dimensions, one face has an area of 12 * 12 or 144 in2. Therefore, the total surface area is 6 * 144 = 864 in2.
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A room has dimensions of 18ft by 15ft by 9ft. The last dimension is the height of the room. It has one door that is 3ft by 7ft and two windows, each 2ft by 5ft. There is no trim to the floor, wall, doors, or windows. What is the total exposed wall space?
A room has dimensions of 18ft by 15ft by 9ft. The last dimension is the height of the room. It has one door that is 3ft by 7ft and two windows, each 2ft by 5ft. There is no trim to the floor, wall, doors, or windows. What is the total exposed wall space?
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If broken down into parts, this is an easy problem. It is first necessary to isolate the dimensions of the walls. If the room is 9 ft high, we know 18 x 15 designates the area of the floor and ceiling. Based on this, we know that the room has the following dimensions for the walls: 18 x 9 and 15 x 9. Since there are two of each, we can calculate the total area of walls - ignoring doors and windows - by doubling the sum of these two areas:
2 * (18 * 9 + 15 * 9) = 2 * (162 + 135) = 2 * 297 = 594 ft2
Now, we merely need to calculate the area "taken out" of the walls:
For the door: 3 * 7 = 21 ft2
For the windows: 2 * (2 * 5) = 20 ft2
The total wall space is therefore: 594 – 21 – 20 = 553 ft2
If broken down into parts, this is an easy problem. It is first necessary to isolate the dimensions of the walls. If the room is 9 ft high, we know 18 x 15 designates the area of the floor and ceiling. Based on this, we know that the room has the following dimensions for the walls: 18 x 9 and 15 x 9. Since there are two of each, we can calculate the total area of walls - ignoring doors and windows - by doubling the sum of these two areas:
2 * (18 * 9 + 15 * 9) = 2 * (162 + 135) = 2 * 297 = 594 ft2
Now, we merely need to calculate the area "taken out" of the walls:
For the door: 3 * 7 = 21 ft2
For the windows: 2 * (2 * 5) = 20 ft2
The total wall space is therefore: 594 – 21 – 20 = 553 ft2
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A room has dimensions of 23ft by 17ft by 10ft. The last dimension is the height of the room. It has one door that is 2.5ft by 8ft and one window, 3ft by 6ft. There is no trim to the floor, wall, doors, or windows. If one can of paint covers 57 ft2 of surface area. How many cans of paint must be bought to paint the walls of the room.
A room has dimensions of 23ft by 17ft by 10ft. The last dimension is the height of the room. It has one door that is 2.5ft by 8ft and one window, 3ft by 6ft. There is no trim to the floor, wall, doors, or windows. If one can of paint covers 57 ft2 of surface area. How many cans of paint must be bought to paint the walls of the room.
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If broken down into parts, this is an easy problem. It is first necessary to isolate the dimensions of the walls. If the room is 10ft high, we know 23 x 17 designates the area of the floor and ceiling. Based on this, we know that the room has the following dimensions for the walls: 23 x 10 and 17 x 10. Since there are two of each, we can calculate the total area of walls - ignoring doors and windows - by doubling the sum of these two areas:
2 * (23 * 10 + 17 * 10) = 2 * (230 + 170) = 2 * 400 = 800 ft2
Now, we merely need to calculate the area "taken out" of the walls:
For the door: 2.5 * 8 = 20 ft2
For the windows: 3 * 6 = 18 ft2
The total wall space is therefore: 800 – 20 – 18 = 762 ft2
Now, if one can of paint covers 57 ft2, we calculate the number of cans necessary by dividing the total exposed area by 57: 762/57 = (approx.) 13.37.
Since we cannot buy partial cans, we must purchase 14 cans.
If broken down into parts, this is an easy problem. It is first necessary to isolate the dimensions of the walls. If the room is 10ft high, we know 23 x 17 designates the area of the floor and ceiling. Based on this, we know that the room has the following dimensions for the walls: 23 x 10 and 17 x 10. Since there are two of each, we can calculate the total area of walls - ignoring doors and windows - by doubling the sum of these two areas:
2 * (23 * 10 + 17 * 10) = 2 * (230 + 170) = 2 * 400 = 800 ft2
Now, we merely need to calculate the area "taken out" of the walls:
For the door: 2.5 * 8 = 20 ft2
For the windows: 3 * 6 = 18 ft2
The total wall space is therefore: 800 – 20 – 18 = 762 ft2
Now, if one can of paint covers 57 ft2, we calculate the number of cans necessary by dividing the total exposed area by 57: 762/57 = (approx.) 13.37.
Since we cannot buy partial cans, we must purchase 14 cans.
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A certain cube has a side length of 25 m. How many square tiles, each with an area of 5 m2, are needed to fully cover the surface of the cube?
A certain cube has a side length of 25 m. How many square tiles, each with an area of 5 m2, are needed to fully cover the surface of the cube?
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A cube with a side length of 25m has a surface area of:
25m * 25m * 6 = 3,750 m2
(The surface area of a cube is equal to the area of one face of the cube multiplied by 6 sides. In other words, if the side of a cube is s, then the surface area of the cube is 6_s_2.)
Each square tile has an area of 5 m2.
Therefore, the total number of square tiles needed to fully cover the surface of the cube is:
3,750m2/5m2 = 750
Note: the volume of a cube with side length s is equal to _s_3. Therefore, if asked how many mini-cubes with side length n are needed to fill the original cube, the answer would be:
s3/n3
A cube with a side length of 25m has a surface area of:
25m * 25m * 6 = 3,750 m2
(The surface area of a cube is equal to the area of one face of the cube multiplied by 6 sides. In other words, if the side of a cube is s, then the surface area of the cube is 6_s_2.)
Each square tile has an area of 5 m2.
Therefore, the total number of square tiles needed to fully cover the surface of the cube is:
3,750m2/5m2 = 750
Note: the volume of a cube with side length s is equal to _s_3. Therefore, if asked how many mini-cubes with side length n are needed to fill the original cube, the answer would be:
s3/n3
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A company wants to build a cubical room around a cone so that the cone's height and diameter are 3 inch less than the dimensions of the cube. If the volume of the cone is 486π ft3, what is the surface area of the cube?
A company wants to build a cubical room around a cone so that the cone's height and diameter are 3 inch less than the dimensions of the cube. If the volume of the cone is 486π ft3, what is the surface area of the cube?
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To begin, we need to solve for the dimensions of the cone.
The basic form for the volume of a cone is: V = (1/3)πr_2_h
Using our data, we know that h = 2r because the height of the cone matches its diameter (based on the prompt).
486_π_ = (1/3)πr_2 * 2_r = (2/3)_πr_3
Multiply both sides by (3/2_π_): 729 = _r_3
Take the cube root of both sides: r = 9
Note that this is in feet. The answers are in square inches. Therefore, convert your units to inches: 9 * 12 = 108, then add 3 inches to this: 108 + 3 = 111 inches.
The surface area of the cube is defined by: A = 6 * _s_2, or for our data, A = 6 * 1112 = 73,926 in2
To begin, we need to solve for the dimensions of the cone.
The basic form for the volume of a cone is: V = (1/3)πr_2_h
Using our data, we know that h = 2r because the height of the cone matches its diameter (based on the prompt).
486_π_ = (1/3)πr_2 * 2_r = (2/3)_πr_3
Multiply both sides by (3/2_π_): 729 = _r_3
Take the cube root of both sides: r = 9
Note that this is in feet. The answers are in square inches. Therefore, convert your units to inches: 9 * 12 = 108, then add 3 inches to this: 108 + 3 = 111 inches.
The surface area of the cube is defined by: A = 6 * _s_2, or for our data, A = 6 * 1112 = 73,926 in2
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If the volume of a cube is 216 cubic units, then what is its surface area in square units?
If the volume of a cube is 216 cubic units, then what is its surface area in square units?
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The volume of a cube is given by the formula V = 
, where V is the volume, and s is the length of each side. We can set V to 216 and then solve for s.
In order to find s, we can find the cube root of both sides of the equaton. Finding the cube root of a number is the same as raising that number to the one-third power.
![\sqrt[3]{216}= 216^{1/3}=6=s](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/17880/gif.latex)
This means the length of the side of the cube is 6. We can use this information to find the surface area of the cube, which is equal to 
. The formula for surface area comes from the fact that each face of the cube has an area of 
, and there are 6 faces in a cube.
surface area = 
The surface area of the square is 216 square units.
The answer is 216.
The volume of a cube is given by the formula V =
In order to find s, we can find the cube root of both sides of the equaton. Finding the cube root of a number is the same as raising that number to the one-third power.
This means the length of the side of the cube is 6. We can use this information to find the surface area of the cube, which is equal to
surface area =
The surface area of the square is 216 square units.
The answer is 216.
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