How to find the volume of a cube - ISEE Upper Level (grades 9-12) Quantitative Reasoning
Card 0 of 4
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
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
Compare your answer with the correct one above
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
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
Compare your answer with the correct one above
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
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:
Compare your answer with the correct one above
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
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:"
Compare your answer with the correct one above