Rectangular Solids & Cylinders - GMAT Quantitative Reasoning
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is a cube. What is the length of diagonal 
?
(1) The area of a face of the cube is 
.
(2) The length of 
is 
.
(1) The area of a face of the cube is
(2) The length of
To know the diagonal HB of the cube, we need to have information about the length of the edges of the cube. Knowing the area of a face would allow us to find the length of an edge; therefore statement 1 alone is sufficient.
Statment 2 alone is sufficient as well since the length of a diagonal of a square is given by 
, where 
is the length of a side of the square and ultimately we can find the length of an edge of the square.
Therefore each statement alone is sufficient.
To know the diagonal HB of the cube, we need to have information about the length of the edges of the cube. Knowing the area of a face would allow us to find the length of an edge; therefore statement 1 alone is sufficient.
Statment 2 alone is sufficient as well since the length of a diagonal of a square is given by
Therefore each statement alone is sufficient.
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Calculate the diagonal of a cube.
- The surface area of the cube is
.
- The volume of the cube is
.
Calculate the diagonal of a cube.
- The surface area of the cube is
- The volume of the cube is
Statement 1: We can use the surface area to find the length of the cube's edge and then determine the length of the diagonal.
where 
represents the length of the cube's edge
Now that we know the length of the edge, we can find the length of the diagonal: 
.
Statement 2: Once again, we can use the provided information to find the length of the cube's edge.
Which means the length of the diagonal is 
.
Thus, each statement alone is sufficient to answer the question.
Statement 1: We can use the surface area to find the length of the cube's edge and then determine the length of the diagonal.
where
Now that we know the length of the edge, we can find the length of the diagonal:
Statement 2: Once again, we can use the provided information to find the length of the cube's edge.
Which means the length of the diagonal is
Thus, each statement alone is sufficient to answer the question.
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Give the length of the diagonal of the cube.
- The surface area of the cube is
.
- The volume of the cube is
.
Give the length of the diagonal of the cube.
- The surface area of the cube is
- The volume of the cube is
Statement 1: To find the length of the diagonal of the cube we need to find the length of the cube's edge. We can use the given surface area value to do so:
Now that we know the length of the cube's edge, we can calculate the diagonal:
Statement 2: We're given the volume of the cube which we can also use to solve for the length of the cube's edge.
Knowing the length of the cube's edge allows us to calculate the diagonal:
Each statement alone is sufficient to answer the question.
Statement 1: To find the length of the diagonal of the cube we need to find the length of the cube's edge. We can use the given surface area value to do so:
Now that we know the length of the cube's edge, we can calculate the diagonal:
Statement 2: We're given the volume of the cube which we can also use to solve for the length of the cube's edge.
Knowing the length of the cube's edge allows us to calculate the diagonal:
Each statement alone is sufficient to answer the question.
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Find the diagonal of the cube.
- The surface area of the cube is
.
- The length of an edge of the cube measures
.
Find the diagonal of the cube.
- The surface area of the cube is
- The length of an edge of the cube measures
Statement 1: We need the length of an edge in order to find the diagonal of the cube. Luckily, we can find the length using the information provided:
Now that we know the length of an edge is 11 cm we can find the length of the diagonal.
Statement 2: We're given the information we need to find the diagonal, all we need to do is plug it into the equation
Statement 1: We need the length of an edge in order to find the diagonal of the cube. Luckily, we can find the length using the information provided:
Now that we know the length of an edge is 11 cm we can find the length of the diagonal.
Statement 2: We're given the information we need to find the diagonal, all we need to do is plug it into the equation
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What is the length of the diagonal of the cube A if the diagonal of cube B is 
?
- The lengths of the edges of cube A to cube B is a ratio of 1:2.
- The surface area of cube A is
.
What is the length of the diagonal of the cube A if the diagonal of cube B is
- The lengths of the edges of cube A to cube B is a ratio of 1:2.
- The surface area of cube A is
Statement 1: The information provided in the question is only useful if we're given a relationship between cube A and cube B. Since this statement does provide us with the ratio of 1:2,we can answer the question.
where 
represents the length of the cube's edge
we can easily see the length measures 
Remember the ratio of cube A to cube B is 1:2.
Now that we know the length, we can find the diagonal of the cube:
Statement 2: We're given information about cube A so we don't need to worry about cube B. Using this information we can solve for the edge length of cube A and then calculate the diagonal.
Knowing the length of the edge allows us to find the diagonal of the cube
Statement 1: The information provided in the question is only useful if we're given a relationship between cube A and cube B. Since this statement does provide us with the ratio of 1:2,we can answer the question.
where
we can easily see the length measures
Remember the ratio of cube A to cube B is 1:2.
Now that we know the length, we can find the diagonal of the cube:
Statement 2: We're given information about cube A so we don't need to worry about cube B. Using this information we can solve for the edge length of cube A and then calculate the diagonal.
Knowing the length of the edge allows us to find the diagonal of the cube
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Find the length of the diagonal of cube 
.
I) 
has a volume of 
.
II) 
has a surface area of 
.
Find the length of the diagonal of cube
I)
II)
To find the length of the diagonal, we need the side length.
I) Gives us the volume of the cube. Take the cubed root to find the side length.
II) Gives us the surface area, divide by 6 (the number of faces in a cube) and take the square root to find the side length.
Diagonal of a cube is the side length times the square root of three. Alternatively, this can be found using the Pythagorean Theorem twice.
Either way, we can use I) or II) to find our side lenght and then our diagonal.
To find the length of the diagonal, we need the side length.
I) Gives us the volume of the cube. Take the cubed root to find the side length.
II) Gives us the surface area, divide by 6 (the number of faces in a cube) and take the square root to find the side length.
Diagonal of a cube is the side length times the square root of three. Alternatively, this can be found using the Pythagorean Theorem twice.
Either way, we can use I) or II) to find our side lenght and then our diagonal.
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What is the diagonal of a rectangular prism?
-
Its surface area is 
-
Its height = twice its width = thrice its length.
What is the diagonal of a rectangular prism?
-
Its surface area is
-
Its height = twice its width = thrice its length.
The diagonal of a rectangular prism is found via the formula 
The second statement reduces this to 
However, the actual length is unknown. Statement 1 allows the calculation of a numerical value:
The diagonal of a rectangular prism is found via the formula
The second statement reduces this to
However, the actual length is unknown. Statement 1 allows the calculation of a numerical value:
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What is the length of the edge of a cube?
-
Its volume is 1,728 cubic meters.
-
Its surface area is 864 square meters
What is the length of the edge of a cube?
-
Its volume is 1,728 cubic meters.
-
Its surface area is 864 square meters
Call the sidelength, surface area, and volume of the cube 
, 
, and 
, respectively.
Then
or, equivalently,
So, given statement 1 alone - that is, given only the volume, you can demonstrate the sidelength to be
Also,
or, equivalently,
Given statement 2 alone - that is, given only the surface area, you can demonstrate the sidelength to be
Therefore, the answer is that either statement alone is sufficient.
Call the sidelength, surface area, and volume of the cube
Then
or, equivalently,
So, given statement 1 alone - that is, given only the volume, you can demonstrate the sidelength to be
Also,
or, equivalently,
Given statement 2 alone - that is, given only the surface area, you can demonstrate the sidelength to be
Therefore, the answer is that either statement alone is sufficient.
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A sphere is inscribed inside a cube. What is the volume of the sphere?
Statement 1: The surface area of the cube is 216.
Statement 2: The volume of the cube is 216.
A sphere is inscribed inside a cube. What is the volume of the sphere?
Statement 1: The surface area of the cube is 216.
Statement 2: The volume of the cube is 216.
The diameter of a sphere inscribed inside a cube is equal to the length of one of the edges of a cube. From either the surface area or the volume of a cube, the appropriate formula can be used to calculate this length. Half this is the radius, from which the formula 
can be used to find the volume of the sphere.
The diameter of a sphere inscribed inside a cube is equal to the length of one of the edges of a cube. From either the surface area or the volume of a cube, the appropriate formula can be used to calculate this length. Half this is the radius, from which the formula
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What is the length of edge 
of cube 
?
(1) 
.
(2) 
.
What is the length of edge
(1)
(2)
In order to find the length of an edge, we would need any information about one of the faces of the cube or about the diagonal of the cube.
Statement 1 gives us the length of the diagonal of the cube, since the formula for the diagonal is 
where 
is the length of an edge of the cube and 
is the length of the diagonal we are able to find the length of the edge. Therefore statement 1 alone is sufficient.
Statement 2 alones is insufficient, it gives us something we can already tell knowing that ABCDEFGH is a cube.
Statement 1 alone is sufficient.
In order to find the length of an edge, we would need any information about one of the faces of the cube or about the diagonal of the cube.
Statement 1 gives us the length of the diagonal of the cube, since the formula for the diagonal is
Statement 2 alones is insufficient, it gives us something we can already tell knowing that ABCDEFGH is a cube.
Statement 1 alone is sufficient.
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is a cube. What is the length of edge 
?
(1) The volume of the cube is 
.
(2) The area of face 
is 
.
(1) The volume of the cube is
(2) The area of face
Like we have previously seen, to find the length of an edge, we need to have information about the other faces or anything else within the cube.
Statement 1 tells us that the volume of the cube is 
, from this we can find the length of the side of the cube. Statement 1 alone is sufficient.
Statement 2, tells us that the area of ABCD is 
, similarily, by taking the square root of this number, we can find the length of the edge of the cube.
Therefore each statement alone is sufficient.
Like we have previously seen, to find the length of an edge, we need to have information about the other faces or anything else within the cube.
Statement 1 tells us that the volume of the cube is
Statement 2, tells us that the area of ABCD is
Therefore each statement alone is sufficient.
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Find the length of an edge of the cube.
- The volume of the cube is
.
- The surface area of the cube is
.
Find the length of an edge of the cube.
- The volume of the cube is
- The surface area of the cube is
Statement 1: Use the volume formula for a cube to solve for the side length.
where 
represents the length of the edge
Statement 2: Use the surface area formula for a cube to solve for the side length.
Each statement alone is sufficient to answer the question.
Statement 1: Use the volume formula for a cube to solve for the side length.
Statement 2: Use the surface area formula for a cube to solve for the side length.
Each statement alone is sufficient to answer the question.
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A carpenter is building a box to hold his tools. Find the legth of the second smallest side of the box.
I) The box will have a volume of 
.
II) The smallest side is half the length of the longest side and the middle side is three-quarters of the length of the longest side.
A carpenter is building a box to hold his tools. Find the legth of the second smallest side of the box.
I) The box will have a volume of
II) The smallest side is half the length of the longest side and the middle side is three-quarters of the length of the longest side.
Volume of a prism is found by:
We are given the volume in statement I.
We are told how the sides relate in statement II.
Put together these two statements will allow us to set up an equation to find the middle side.
Volume of a prism is found by:
We are given the volume in statement I.
We are told how the sides relate in statement II.
Put together these two statements will allow us to set up an equation to find the middle side.
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A pirate wants to hide all of his treasure. He commisions a local woodworker to build him a series of wooden chests of volume of 
. Find the length of the longest side given the following:
I) The shortest side will be 
the length of the medium side.
II) The middle side will be 2 feet long.
A pirate wants to hide all of his treasure. He commisions a local woodworker to build him a series of wooden chests of volume of
I) The shortest side will be
II) The middle side will be 2 feet long.
A pirate wants to hide all of his treasure. He commisions a local woodworker to build him a series of wooden chests of volume of 
. Find the length of the longest side given the following:
I) The shortest side will be 
the length of the medium side
II) The middle side will be 2 feet long
Use I) and II) to find the length of the smallest side
Next, use the short and medium sides, along with info in the prompt, to find the last side:
A pirate wants to hide all of his treasure. He commisions a local woodworker to build him a series of wooden chests of volume of
I) The shortest side will be
II) The middle side will be 2 feet long
Use I) and II) to find the length of the smallest side
Next, use the short and medium sides, along with info in the prompt, to find the last side:
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Find the height of a box used to ship a computer, given the following:
I) The computer, along with all the cushioning, will take up a space of 
.
II) The width of the box will be 
the length of the box.
Find the height of a box used to ship a computer, given the following:
I) The computer, along with all the cushioning, will take up a space of
II) The width of the box will be
Find the height of a box used to ship a computer, given the following:
I) The computer, along with all the cushioning, will take up a space of 
II) The width of the box will be 
the length of the box
Begin by recalling the volume of a rectangular prism formula:
Where l,w and h are length, width and height.
Next, use II) to set up a relationship between w and l
Then, use the voume formula:
As you can see, we still have two unknowns, and no way of finding either.
Therefore, we do not have information.
Find the height of a box used to ship a computer, given the following:
I) The computer, along with all the cushioning, will take up a space of
II) The width of the box will be
Begin by recalling the volume of a rectangular prism formula:
Where l,w and h are length, width and height.
Next, use II) to set up a relationship between w and l
Then, use the voume formula:
As you can see, we still have two unknowns, and no way of finding either.
Therefore, we do not have information.
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Find the length of the edge of a tetrahedron.
Statement 1: The volume is 6.
Statement 2: The surface area is 6.
Find the length of the edge of a tetrahedron.
Statement 1: The volume is 6.
Statement 2: The surface area is 6.
Statement 1:) The volume is 6.
Write the formula to find the edge of the tetrahedron given the volume.
Given the volume, it is possible to find the edge of the tetrahedron.
Statement 2:) The surface area is 6.
Write the formula to find the edge of the tetrahedron given the surface.
Substitute the surface area to find the edge.
Therefore:
Statement 1:) The volume is 6.
Write the formula to find the edge of the tetrahedron given the volume.
Given the volume, it is possible to find the edge of the tetrahedron.
Statement 2:) The surface area is 6.
Write the formula to find the edge of the tetrahedron given the surface.
Substitute the surface area to find the edge.
Therefore:
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Ron is making a box in the shape of a cube. He needs to know how much wood he needs. Find the surface area of the box.
I) The diagonal distance across the box will be equivalent to 
.
II) Half the length of one side is 
.
Ron is making a box in the shape of a cube. He needs to know how much wood he needs. Find the surface area of the box.
I) The diagonal distance across the box will be equivalent to
II) Half the length of one side is
To find the surface area of a cube, we need the length of one side.
Statement I gives the diagonal, we can use this to find the length of one side.
Statement II gives us a clue about the length of one side; we can use that to find the full length of one side.
The following formula gives us the surface area of a cube:
Use Statement I to find the length of the side with the following formula, where 
is the diagonal and 
is the side length:
So, using Statement I, we find the surface area to be
Using Statement, we get that the length of one side is two times two:
Again, use the surface area formula to get the following:
To find the surface area of a cube, we need the length of one side.
Statement I gives the diagonal, we can use this to find the length of one side.
Statement II gives us a clue about the length of one side; we can use that to find the full length of one side.
The following formula gives us the surface area of a cube:
Use Statement I to find the length of the side with the following formula, where
So, using Statement I, we find the surface area to be
Using Statement, we get that the length of one side is two times two:
Again, use the surface area formula to get the following:
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Of a given cylinder and a given cube, which, if either, has the greater surface area?
Statement 1: Both the height of the cylinder and the diameter of its bases are equal to the length of one edge of the cube.
Statement 2: Each face of the cube has as its area four times the square of the radius of the bases of the cylinder.
Of a given cylinder and a given cube, which, if either, has the greater surface area?
Statement 1: Both the height of the cylinder and the diameter of its bases are equal to the length of one edge of the cube.
Statement 2: Each face of the cube has as its area four times the square of the radius of the bases of the cylinder.
The surface area of a cylinder, given height 
and radius of the bases 
, is given by the formula
The surface area of a cube, given the length 
of each edge, is given by the formula
.
Assume Statement 1 alone. Then 
and, since the diameter of a base is 
, the radius is half this, or 
. The surface area of the cylinder is, in terms of 
, equal to
.
Since 
, the cylinder has the greater area regardless of the actual measurements.
Assume Statement 2 alone. The cube has six faces with area 
, so its surface area is six times this, or 
. The surface area of the cylinder is 
; however, without knowing anything aobout the height of the cylinder, we cannot compare the two surface areas.
The surface area of a cylinder, given height
The surface area of a cube, given the length
Assume Statement 1 alone. Then
Since
Assume Statement 2 alone. The cube has six faces with area
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Jiminy wants to paint one of his silos. One gallon of this paint covers about 
square feet. How many gallons will he need?
I) The radius of the silo is 
feet.
II) The height is 
times longer the radius.
Jiminy wants to paint one of his silos. One gallon of this paint covers about
I) The radius of the silo is
II) The height is
Review our statements:
I) The radius of the silo is 
feet.
II) The height is 
times longer the radius
We need to find our surface area in order to find how many gallons we need. Surface area is given by:
So to find the surface area, we need the radius and the height, so both statments are needed here.
Review our statements:
I) The radius of the silo is
II) The height is
We need to find our surface area in order to find how many gallons we need. Surface area is given by:
So to find the surface area, we need the radius and the height, so both statments are needed here.
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A tin can has a volume of 
.
I) The height of the can is 
inches.
II) The radius of the base of the can is 
inches.
What is the surface area of the can? (Assume it is a perfect cylinder)
A tin can has a volume of
I) The height of the can is
II) The radius of the base of the can is
What is the surface area of the can? (Assume it is a perfect cylinder)
To find surface area of a cylinder we need the radius and the height.
If we are given the volume, and either the radius or the height, we can work backwards to find the other dimension.
Since I and II give us the height and the radius, either statement can be used to find the surface area.
To find surface area of a cylinder we need the radius and the height.
If we are given the volume, and either the radius or the height, we can work backwards to find the other dimension.
Since I and II give us the height and the radius, either statement can be used to find the surface area.
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