# ISEE Upper Level Quantitative : Solid Geometry

## Example Questions

### Example Question #332 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Which is the greater quantity?

(a) The surface area of a sphere with radius 1

(b) 12

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

(a) is greater.

Explanation:

The surface area of a sphere can be found using the formula

.

The surface area of the given sphere can be found by substituting :

so , or

This makes (a) greater.

### Example Question #3 : How To Find The Surface Area Of A Sphere

Sphere A has volume . Sphere B has surface area . Which is the greater quantity?

(a) The radius of Sphere A

(b) The radius of Sphere B

It is impossible to tell from the information given

(b) is greater

(a) is greater

(a) and (b) are equal

(b) is greater

Explanation:

(a) Substitute  in the formula for the volume of a sphere:

inches

(b) Substitute  in the formula for the surface area of a sphere:

inches

(b) is greater.

### Example Question #4 : How To Find The Surface Area Of A Sphere

is a positive number. Which is the greater quantity?

(A) The surface area of a sphere with radius

(B) The surface area of a cube with edges of length

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

(A) is greater

(A) and (B) are equal

(B) is greater

(B) is greater

Explanation:

The surface area of a sphere is  times the square of its radius, which here is ; the surface area of the sphere in (A) is .

The area of one face of a cube is the square of the length of an edge, which here is , so the area of one face of the cube in (B) is . The cube has six faces so the total surface area is .

, so , giving the sphere less surface area. (B) is greater.

### Example Question #1 : Cones

The height of Cone B is three times that of Cone A. The radius of the base of Cone B is one-half the radius of the base of Cone A.

Which is the greater quantity?

(a) The volume of Cone A

(b) The volume of Cone B

(b) is greater.

(a) and (b) are equal.

(a) is greater.

It is impossible to tell from the information given.

(a) is greater.

Explanation:

Let  be the radius and height of Cone A, respectively. Then the radius and height of Cone B are  and , respectively.

(a) The volume of Cone A is .

(b) The volume of Cone B is

.

Since , the cone in (a) has the greater volume.

### Example Question #11 : Volume

The volume of a cone whose height is three times the radius of its base is one cubic yard. Give its radius in inches.

Explanation:

The volume of a cone with base radius  and height  is

The height  is three times this, or . Therefore, the formula becomes

Set this volume equal to one and solve for :

This is the radius in yards; since the radius in inches is requested, multiply by 36.

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

The height of a given cylinder is one half the height of a given cone. The radii of their bases are equal.

Which of the following is the greater quantity?

(a) The volume of the cone

(b) The volume of the cylinder

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(b) is the greater quantity

Explanation:

Call  the radius of the base of the cone and  the height of the cone. The cylinder will have bases of radius  and height .

In the formula for the volume of a cylinder, set  and :

In the formula for the volume of a cone, set  and :

, so

,

meaning that the cylinder has the greater volume.

### Example Question #341 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

The radius of the base of a given cone is three times that of each base of a given cylinder. The heights of the cone and the cylinder are equal.

Which of the following is the greater quantity?

(a) The volume of the cone

(b) The volume of the cylinder

It cannot be determined which of (a) and (b) is greater

(a) is the greater quantity

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

Explanation:

If we let  be the radius of each base of the cylinder, then  is the radius of the base of the cone. We can let  be their common height.

In the formula for the volume of a cylinder, set  and :

In the formula for the volume of a cone, set  and :

, so . The cone has the greater volume.

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

What is the volume of a cylinder with a radius of 6 meters and a height of 11 meters? Use 3.14 for .

Note: The formula for the volume of a cylinder is:

Explanation:

To calculate the volume, you must plug into the formula given in the problem. When you plug in, it should look like this: . Multiply all of these out and you get . The units are cubed because volume is always cubed.

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

The volume of a cylinder whose height is twice the diameter of its base is one cubic yard. Give its radius in inches.

Explanation:

The volume of a cylinder with base radius  and height  is

The diameter of this circle is ; its height is twice this, or . Therefore, the formula becomes

Set this volume equal to one and solve for :

This is the radius in yards; multiply by 36 to get the radius in inches.

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

What is the height of a cylinder with a volume of   and a radius of  ?