Cubes - Math
Card 0 of 524
What is the surface area of a cube on which one face has a diagonal of 
?
What is the surface area of a cube on which one face has a diagonal of
Tap to see back →
One of the faces of the cube could be drawn like this:
Notice that this makes a 
triangle.
This means that we can create a proportion for the sides. On the standard triangle, the non-hypotenuse sides are both 
, and the hypotenuse is 
. This will allow us to make the proportion:
Multiplying both sides by 
, you get:
To find the area of the square, you need to square this value:
Now, since there are 
sides to the cube, multiply this by 
to get the total surface area:
One of the faces of the cube could be drawn like this:
Notice that this makes a
This means that we can create a proportion for the sides. On the standard triangle, the non-hypotenuse sides are both
Multiplying both sides by
To find the area of the square, you need to square this value:
Now, since there are
Which is the greater quantity?
(a) The volume of a cube with surface area 
inches
(b) The volume of a cube with diagonal 
inches
Which is the greater quantity?
(a) The volume of a cube with surface area inches
(b) The volume of a cube with diagonal inches
Tap to see back →
The cube with the greater sidelength has the greater volume, so we need only calculate and compare sidelengths.
(a) 
, so the sidelength of the first cube can be found as follows:
inches
(b) 
by an extension of the Pythagorean Theorem, so the sidelength of the second cube can be found as follows:
Since 
, 
. The second cube has the greater sidelength and, subsequently, the greater volume. This makes (b) greater.
The cube with the greater sidelength has the greater volume, so we need only calculate and compare sidelengths.
(a)
(b)
Since
Cube 2 has twice the sidelength of Cube 1; Cube 3 has twice the sidelength of Cube 2; Cube 4 has twice the sidelength of Cube 3.
Which is the greater quantity?
(a) The mean of the volumes of Cube 1 and Cube 4
(b) The mean of the volumes of Cube 2 and Cube 3
Cube 2 has twice the sidelength of Cube 1; Cube 3 has twice the sidelength of Cube 2; Cube 4 has twice the sidelength of Cube 3.
Which is the greater quantity?
(a) The mean of the volumes of Cube 1 and Cube 4
(b) The mean of the volumes of Cube 2 and Cube 3
Tap to see back →
The sidelengths of Cubes 1, 2, 3, and 4 can be given values 
, respectively.
Then the volumes of the cubes are as follows:
Cube 1: 
Cube 2: 
Cube 3: 
Cube 4: 
In both answer choices ask for a mean, so we can determine which answer (mean) is greater simply by comparing the sums of volumes.
(a) The sum of the volumes of Cubes 1 and 4 is 
.
(b) The sum of the volumes of Cubes 2 and 3 is 
.
Regardless of 
, the sum of the volumes of Cubes 1 and 4 is greater, and therefore, so is their mean.
The sidelengths of Cubes 1, 2, 3, and 4 can be given values
Then the volumes of the cubes are as follows:
Cube 1:
Cube 2:
Cube 3:
Cube 4:
In both answer choices ask for a mean, so we can determine which answer (mean) is greater simply by comparing the sums of volumes.
(a) The sum of the volumes of Cubes 1 and 4 is
(b) The sum of the volumes of Cubes 2 and 3 is
Regardless of
What is the length of the diagonal of a cube with a side length of 
? Round to the nearest hundreth.
What is the length of the diagonal of a cube with a side length of
Tap to see back →
It is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:
(Note that not all points are drawn in on the cube.)
The two points we are looking at are:
and 
Solve this by using the distance formula. This is very easy since one point is all 
s. It is merely:
This is approximately 
.
It is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:
(Note that not all points are drawn in on the cube.)
The two points we are looking at are:
Solve this by using the distance formula. This is very easy since one point is all
This is approximately
What is the length of the diagonal of a cube with a side length of 
? Round to the nearest hundreth.
What is the length of the diagonal of a cube with a side length of
Tap to see back →
It is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:
(Note that not all points are drawn in on the cube.)
The two points we are looking at are:
and 
Solve this by using the distance formula. This is very easy since one point is all 
s. It is merely:
This is approximately 
.
It is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:
(Note that not all points are drawn in on the cube.)
The two points we are looking at are:
Solve this by using the distance formula. This is very easy since one point is all
This is approximately
What is the length of the diagonal of a cube with a volume of 
? Round to the nearest hundredth.
What is the length of the diagonal of a cube with a volume of
Tap to see back →
First, you need to find the side length of this cube. We know that the volume is:
, where 
is the side length.
Therefore, based on our data, we can say:
Solving for 
by taking the cube-root of both sides, we get:
Now, it is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:
(Note that not all points are drawn in on the cube.)
The two points we are looking at are:
and 
Solve this by using the distance formula. This is very easy since one point is all 
s. It is merely:
This is approximately 
.
First, you need to find the side length of this cube. We know that the volume is:
Therefore, based on our data, we can say:
Solving for
Now, it is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:
(Note that not all points are drawn in on the cube.)
The two points we are looking at are:
Solve this by using the distance formula. This is very easy since one point is all
This is approximately
What is the length of the diagonal of a cube with a surface area of 
? Round your answer to the nearest hundredth.
What is the length of the diagonal of a cube with a surface area of
Tap to see back →
First, you need to find the side length of this cube. We know that the surface area is defined by:
, where 
is the side length. (This is because the cube is 
sides of equal area).
Therefore, based on our data, we can say:
Take the square root of both sides and get:
Now, it is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:
(Note that not all points are drawn in on the cube.)
The two points we are looking at are:
and 
Solve this by using the distance formula. This is very easy since one point is all 
s. It is merely:
This is approximately 
.
First, you need to find the side length of this cube. We know that the surface area is defined by:
Therefore, based on our data, we can say:
Take the square root of both sides and get:
Now, it is easiest to think about diagonals like this by considering two points on a cube (if it were drawn in three dimensions). We could draw it like this:
(Note that not all points are drawn in on the cube.)
The two points we are looking at are:
Solve this by using the distance formula. This is very easy since one point is all
This is approximately
What is the volume of a cube on which one face has a diagonal of 
?
What is the volume of a cube on which one face has a diagonal of
Tap to see back →
One of the faces of the cube could be drawn like this:
Notice that this makes a 
triangle.
This means that we can create a proportion for the sides. On the standard triangle, the non-hypotenuse sides are both 
, and the hypotenuse is 
. This will allow us to make the proportion:
Multiplying both sides by 
, you get:
Recall that the formula for the volume of a cube is:
Therefore, we can compute the volume using the side found above:
Now, rationalize the denominator:
One of the faces of the cube could be drawn like this:
Notice that this makes a
This means that we can create a proportion for the sides. On the standard triangle, the non-hypotenuse sides are both
Multiplying both sides by
Recall that the formula for the volume of a cube is:
Therefore, we can compute the volume using the side found above:
Now, rationalize the denominator:
The volume of a cube is 343 cubic inches. Give its surface area.
The volume of a cube is 343 cubic inches. Give its surface area.
Tap to see back →
The volume of a cube is defined by the formula
where 
is the length of one side.
If 
, then
and
So one side measures 7 inches.
The surface area of a cube is defined by the formula
, so
The surface area is 294 square inches.
The volume of a cube is defined by the formula
where
If
and
So one side measures 7 inches.
The surface area of a cube is defined by the formula
The surface area is 294 square inches.
What is the surface area of a cube with a volume of 
?
What is the surface area of a cube with a volume of
Tap to see back →
We know that the volume of a cube can be found with the equation:
, where 
is the side length of the cube.
Now, if the volume is 
, then we know:
Either with your calculator or with careful math, you can solve by taking the cube-root of both sides. This gives you:
This means that each side of the cube is 
long; therefore, each face has an area of 
, or 
. Since there are 
sides to a cube, this means the total surface area is 
, or 
.
We know that the volume of a cube can be found with the equation:
Now, if the volume is
Either with your calculator or with careful math, you can solve by taking the cube-root of both sides. This gives you:
This means that each side of the cube is
What is the surface area of a cube that has a side length of 
?
What is the surface area of a cube that has a side length of
Tap to see back →
This question is very easy. Do not over-think it! All you need to do is calculate the area of one side of the cube. Then, multiply that by 
(since the cube has 
sides). Each side of a cube is, of course, a square; therefore, the area of one side of this cube is 
, or 
. This means that the whole cube has a surface area of 
or 
.
This question is very easy. Do not over-think it! All you need to do is calculate the area of one side of the cube. Then, multiply that by
What is the surface area of a cube with side length 
?
What is the surface area of a cube with side length
Tap to see back →
Recall that the formula for the surface area of a cube is:
, where 
is the length of a side of the cube. This equation is easy to memorize because it is merely a multiplication of a single side (
) by 
because a cube has 
equal sides.
For our data, we know that 
; therefore, our equation is:
Recall that the formula for the surface area of a cube is:
For our data, we know that
What is the surface area of a cube with a volume 
?
What is the surface area of a cube with a volume
Tap to see back →
To solve this, first calculate the side length based on the volume given. Recall that the equation for the volume of a cube is:
, where 
is the side length.
For our data, this gives us:
Taking the cube-root of both sides, we get:
Now, use the surface area formula to compute the total surface area:
, where 
is the length of a side of the cube. This equation is easy to memorize because it is merely a multiplication of a single side (
) by 
because a cube has 
equal sides.
For our data, this gives us:
To solve this, first calculate the side length based on the volume given. Recall that the equation for the volume of a cube is:
For our data, this gives us:
Taking the cube-root of both sides, we get:
Now, use the surface area formula to compute the total surface area:
For our data, this gives us:
What is the surface area of a cube with a volume 
?
What is the surface area of a cube with a volume
Tap to see back →
To solve this, first calculate the side length based on the volume given. Recall that the equation for the volume of a cube is:
, where 
is the side length.
For our data, this gives us:
Taking the cube-root of both sides, we get:
(You will need to use a calculator for this. If your calculator gives you something like 
. . . it is okay to round. This is just the nature of taking roots!).
Now, use the surface area formula to compute the total surface area:
, where 
is the length of a side of the cube. This equation is easy to memorize because it is merely a multiplication of a single side (
) by 
because a cube has 
equal sides.
For our data, this gives us:
To solve this, first calculate the side length based on the volume given. Recall that the equation for the volume of a cube is:
For our data, this gives us:
Taking the cube-root of both sides, we get:
(You will need to use a calculator for this. If your calculator gives you something like
Now, use the surface area formula to compute the total surface area:
For our data, this gives us:
What is the surface area for a cube with a diagonal length of 
?
What is the surface area for a cube with a diagonal length of
Tap to see back →
Now, this could look like a difficult problem; however, think of the equation for finding the length of the diagonal of a cube. It is like the Pythagorean Theorem, just adding an additional dimension:
(It is very easy, because the three lengths are all the same: 
).
So, we know this, then:
To solve, you can factor out an 
from the root on the right side of the equation:
Just by looking at this, you can tell that the answer is:
Now, use the surface area formula to compute the total surface area:
, where is the length of a side of the cube. This equation is easy to memorize because it is merely a multiplication of a single side () by because a cube has equal sides.
For our data, this is:
Now, this could look like a difficult problem; however, think of the equation for finding the length of the diagonal of a cube. It is like the Pythagorean Theorem, just adding an additional dimension:
(It is very easy, because the three lengths are all the same:
So, we know this, then:
To solve, you can factor out an
Just by looking at this, you can tell that the answer is:
Now, use the surface area formula to compute the total surface area:
For our data, this is:
What is the volume of a cube with a diagonal length of 
?
What is the volume of a cube with a diagonal length of
Tap to see back →
Now, this could look like a difficult problem. However, think of the equation for finding the length of the diagonal of a cube. It is like the Pythagorean Theorem, just adding an additional dimension:
(It is very easy, because the three lengths are all the same: 
).
So, we know this, then:
To solve, you can factor out an 
from the root on the right side of the equation:
Just by looking at this, you can tell that the answer is:
Now, use the equation for the volume of a cube:
(It is like doing the area of a square, then adding another dimension!).
For our data, it is:
Now, this could look like a difficult problem. However, think of the equation for finding the length of the diagonal of a cube. It is like the Pythagorean Theorem, just adding an additional dimension:
(It is very easy, because the three lengths are all the same:
So, we know this, then:
To solve, you can factor out an
Just by looking at this, you can tell that the answer is:
Now, use the equation for the volume of a cube:
(It is like doing the area of a square, then adding another dimension!).
For our data, it is:
What is the volume of a cube with side length 
? Round your answer to the nearest hundredth.
What is the volume of a cube with side length
Tap to see back →
This question is relatively straightforward. The equation for the volume of a cube is:
(It is like doing the area of a square, then adding another dimension!)
Now, for our data, we merely need to "plug and chug:"
This question is relatively straightforward. The equation for the volume of a cube is:
(It is like doing the area of a square, then adding another dimension!)
Now, for our data, we merely need to "plug and chug:"
Suppose the volume of a cube is 
. What is the length of the diagonal?
Suppose the volume of a cube is
Tap to see back →
Write the equation for the volume of a cube. Substitute the volume to find the side length, 
.
Write the equation for finding diagonals given an edge length for a cube.
Substitute the side length to find the diagonal length.
The answer is 
.
Write the equation for the volume of a cube. Substitute the volume to find the side length,
Write the equation for finding diagonals given an edge length for a cube.
Substitute the side length to find the diagonal length.
The answer is
If the surface area of a cube is 
, what is the length of one side of the cube?
If the surface area of a cube is
Tap to see back →
Recall how to find the surface area of a cube:
Since the question asks you to find the length of a side of this cube, rearrange the equation.
Substitute in the given surface area to find the side length.
Simplify.
Reduce.
Recall how to find the surface area of a cube:
Since the question asks you to find the length of a side of this cube, rearrange the equation.
Substitute in the given surface area to find the side length.
Simplify.
Reduce.
If the surface area of a cube is 
, find the length of one side of the cube.
If the surface area of a cube is
Tap to see back →
Recall how to find the surface area of a cube:
Since the question asks you to find the length of a side of this cube, rearrange the equation.
Substitute in the given surface area to find the side length.
Simplify.
Reduce.
Recall how to find the surface area of a cube:
Since the question asks you to find the length of a side of this cube, rearrange the equation.
Substitute in the given surface area to find the side length.
Simplify.
Reduce.