# ISEE Upper Level Quantitative : Solid Geometry

## Example Questions

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### Example Question #1 : Cylinders

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

Explanation:

This is a rather direct question. Recall that the equation of for the volume of a cylinder is:

For our values this is:

This is the volume of the cylinder.

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

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

Explanation:

This is a rather direct question. Recall that the equation of for the volume of a cylinder is:

For our values this is:

This is the volume of the cylinder.

### Example Question #3 : How To Find The Volume Of A Cylinder

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

Explanation:

Recall that the equation of for the volume of a cylinder is:

For our values this is:

Solve for :

Using a calculator to calculate , you will see that

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

What is the surface area of a cylinder of height  in., with a radius of  in?

Explanation:

Recall that to find the surface area of a cylinder, you need to find the surface area of its two bases and then the surface area of its "outer face." The first two are very easy since they are circles. The equation for one base is:

For our problem, this is:

You need to double this for the two bases:

The area of the "outer face" is a little bit trickier, but it is not impossible. It is actually a rectangle that has the height of the cylinder and a width equal to the circumference of the base; therefore, it is:

For our problem, this is:

Therefore, the total surface area is:

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

What is the surface area of a cylinder having a base of radius in and a height of  in?

Explanation:

Recall that to find the surface area of a cylinder, you need to find the surface area of its two bases and then the surface area of its "outer face." The first two are very easy since they are circles. The equation for one base is:

For our problem, this is:

You need to double this for the two bases:

The area of the "outer face" is a little bit trickier, but it is not impossible. It is actually a rectangle that has the height of the cylinder and a width equal to the circumference of the base; therefore, it is:

For our problem, this is:

Therefore, the total surface area is:

### Example Question #2 : Cylinders

What is the surface area of a cylinder with a height of in. and a diameter of in?

Explanation:

Recall that to find the surface area of a cylinder, you need to find the surface area of its two bases and then the surface area of its "outer face." The first two are very easy since they are circles. Notice, however that the diameter is  inches. This means that the radius is . Now, the equation for one base is:

For our problem, this is:

You need to double this for the two bases:

The area of the "outer face" is a little bit trickier, but it is not impossible. It is actually a rectangle that has the height of the cylinder and a width equal to the circumference of the base; therefore, it is:

For our problem, this is:

Therefore, the total surface area is:

### Example Question #51 : Solid Geometry

The volume of a cylinder with height of  is . What is its surface area?

Explanation:

To begin, we must solve for the radius of this cylinder. Recall that the equation of for the volume of a cylinder is:

For our values this is:

Solving for , we get:

Hence,

Now, recall that to find the surface area of a cylinder, you need to find the surface area of its two bases and then the surface area of its "outer face." The first two are very easy since they are circles. The equation for one base is:

For our problem, this is:

You need to double this for the two bases:

The area of the "outer face" is a little bit trickier, but it is not impossible. It is actually a rectangle that has the height of the cylinder and a width equal to the circumference of the base; therefore, it is:

For our problem, this is:

Therefore, the total surface area is:

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

What is the surface area of a cylinder of height  in, with a radius of  in?

Explanation:

Recall that to find the surface area of a cylinder, you need to find the surface area of its two bases and then the surface area of its "outer face." The first two are very easy since they are circles. The equation for one base is:

For our problem, this is:

You need to double this for the two bases:

The area of the "outer face" is a little bit trickier, but it is not impossible. It is actually a rectangle that has the height of the cylinder and a width equal to the circumference of the base; therefore, it is:

For our problem, this is:

Therefore, the total surface area is:

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