# ISEE Upper Level Quantitative : Solid Geometry

## Example Questions

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### Example Question #1 : Solid Geometry

is a positive number. Which is the greater quantity?

(A) The surface area of a rectangular prism with length , width , and height

(B) The surface area of a rectangular prism with length , width , and height .

(A) and (B) are equal

(A) is greater

(B) is greater

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

(A) is greater

Explanation:

The surface area of a rectangular prism can be determined using the formula:

Using substitutions, the surface areas of the prisms can be found as follows:

The prism in (A):

Regardless of the value of ,  - that is, the first prism has the greater surface area. (A) is greater.

### Example Question #1 : How To Find The Length Of An Edge

A cube has sidelength one and one-half feet; a rectangular prism of equal volume has length 27 inches and height 9 inches. Give the width of the prism in inches.

Explanation:

One and one half feet is equal to eighteen inches, so the volume of the cube, in cubic inches, is the cube of this, or

cubic inches.

The volume of a rectangular prism is

Since its volume is the same as that of the cube, and its length and height are 27 and 9 inches, respectively,  we can rewrite this as

The width is 24 inches.

### Example Question #2 : How To Find The Length Of An Edge

A cube has sidelength one and one-half feet; a rectangular prism of equal surface area has length 27 inches and height 9 inches. Give the width of the prism in inches.

Explanation:

One and one half feet is equal to eighteen inches, so the surface area of the cube, in square inches, is six times the square of this, or

square inches.

The surface area of a rectangular prism is determined by the formula

.

So, with substitutiton, we can find the width:

inches

### Example Question #3 : How To Find The Length Of An Edge

A rectangular prism has volume one cubic foot; its length and width are, respectively, 9 inches and  inches. Which of the following represents the height of the prism in inches?

Explanation:

The volume of a rectangular prism is the product of its length, its width, and its height. The prism's volume of one cubic foot is equal to  cubic inches.

Therefore,  can be rewritten as .

We can solve for  as follows:

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

A large crate in the shape of a rectangular prism has dimensions 5 feet by 4 feet by 12 feet. Give its volume in cubic yards.

Explanation:

Divide each dimension by 3 to convert feet to yards, then multiply the three dimensions together:

### Example Question #2 : How To Find The Volume Of A Prism

Which is the greater quantity?

(A) The volume of a rectangular solid ten inches by twenty inches by fifteen inches

(B) The volume of a cube with sidelength sixteen inches

(A) is greater

(B) is greater

(A) and (B) are equal

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

(B) is greater

Explanation:

The volume of a rectangular solid ten inches by twenty inches by fifteen inches is

cubic inches.

The volume of a cube with sidelength 13 inches is

cubic inches.

This makes (B) greater

### Example Question #301 : Geometry

Pyramid 1 has a square base with sidelength ; its height is .

Pyramid 2 has a square base with sidelength ; its height is .

Which is the greater quantity?

(a) The volume of Pyramid 1

(b) The volume of Pyramid 2

(a) is greater.

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

Explanation:

Use the formula on each pyramid.

(a)

(b)

Regardless of , (b) is the greater quantity.

### Example Question #302 : Geometry

Which is the greater quantity?

(a) The volume of a pyramid with height 4, the base of which has sidelength 1

(b) The volume of a pyramid with height 1, the base of which has sidelength 2

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) and (b) are equal.

Explanation:

The volume of a pyramid with height  and a square base with sidelength  is

.

(a) Substitute

(b) Substitute

The two pyramids have equal volume.

### Example Question #303 : Geometry

Which is the greater quantity?

(a) The volume of a pyramid whose base is a square with sidelength 8 inches

(b) The volume of a pyramid whose base is an equilateral triangle with sidelength one foot

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(b) is greater.

It is impossible to tell from the information given.

Explanation:

The volume of a pyramid is one-third of the product of the height and the area of the base. The areas of the bases can be calculated, but no information is given about the heights of the pyramids. There is not enough information to determine which one has the greater volume.

### Example Question #304 : Geometry

A pyramid with a square base has height equal to the perimeter of its base. Its volume is . In terms of , what is the length of each side of its base?

Explanation:

The volume of a pyramid is given by the formula

where  is the area of its base and  is its height.

Let  be the length of one side of the square base. Then the height is equal to the perimeter of that square, so

and the area of the base is

So the volume formula becomes

Solve for :

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