# ISEE Upper Level Quantitative : Solid Geometry

## Example Questions

### Example Question #1 : How To Find The Surface Area Of A Pyramid

A pyramid with a square base has height equal to the perimeter of its base. Which is the greater quantity?

(A) Twice the area of its base

(B) The area of one of its triangular faces

(A) and (B) are equal

It is impossible to determine which is greater from the information given

(B) is greater

(A) is greater

(B) is greater

Explanation:

Since the answer is not dependent on the actual dimensions, for the sake of simplicity, we assume that the base has sidelength 2. Then the area of the base is the square of this, or 4.

The height of the pyramid is equal to the perimeter of the base, or . A right triangle can be formed with the lengths of its legs equal to the height of the pyramid, or 8, and one half the length of a side, or 1; the length of its hypotenuse, which is the slant height, is

This is the height of one triangular face; its base is a side of the square, so the length of the base is 2. The area of a face is half the product of these dimensions, or

Since twice the area of the base is , the problem comes down to comparing  and ; the latter, which is (B), is greater.

### Example Question #11 : Solid Geometry

A pyramid with a square base and a cone have the same height and the same volume. Which is the greater quantity?

(A) The perimeter of the base of the pyramid

(B) The circumference of the base of the cone

(B) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

(A) is greater

(A) is greater

Explanation:

The volume of a pyramid or a cone with height  and base of area  is

so in both cases, the area of the base is

Since the pyramid and the cone have the same volume and height, their bases has the same area .

The length of one side of the square base of the pyramid is the square root of this, or , and the perimeter is four times this, or .

The radius and the area of the base of the cone are related as follows:

Multiply both sides by  to get:

, so

, and

The perimeter of the base of the pyramid, which is (A), is greater than the circumference of the base of the cone.

### Example Question #1 : How To Find The Volume Of A Cube

Which is the greater quantity?

(a) The volume of a cube with surface area  inches

(b) The volume of a cube with diagonal  inches

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(b) is greater.

(b) is greater.

Explanation:

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.

### Example Question #2 : Cubes

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

It cannot be determined from the information given.

(a) is greater.

(b) is greater.

(a) and (b) are equal.

(a) is greater.

Explanation:

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.

### Example Question #3 : Cubes

What is the volume of a cube with side length  ? Round your answer to the nearest hundredth.

Explanation:

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:"

### Example Question #4 : Cubes

What is the volume of a cube on which one face has a diagonal of  ?

Explanation:

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:

### Example Question #1 : How To Find The Surface Area Of A Cube

The volume of a cube is 343 cubic inches. Give its surface area.

Explanation:

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.

### Example Question #6 : Cubes

What is the surface area of a cube with side length  ?

Explanation:

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:

### Example Question #11 : Solid Geometry

What is the surface area of a cube with a volume  ?

Explanation:

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:

### Example Question #1 : How To Find The Surface Area Of A Cube

What is the surface area of a cube with a volume  ?

Explanation:

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: