# ISEE Upper Level Math : Solid Geometry

## Example Questions

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### Example Question #3 : Solve For Surface Area

Find the surface area of a non-cubic prism with the following measurements:

Explanation:

The surface area of a non-cubic prism can be determined using the equation:

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

The above diagram shows a rectangular solid. The shaded side is a square. In terms of , give the surface area of the box.

Explanation:

A square has four sides of equal length, as seen in the diagram below.

All six sides are rectangles, so their areas are equal to the products of their dimensions:

Top, bottom, front, back (four surfaces):

Left, right (two surfaces):

The total surface area:

### Example Question #8 : Finding Volume Of A Rectangular Prism

A rectangular prism has a width of 3 inches, a length of 6 inches, and a height triple its length. Find the volume of the prism.

Explanation:

A rectangular prism has a width of 3 inches, a length of 6 inches, and a height triple its length. Find the volume of the prism.

Find the volume of a rectangular prism via the following:

Where l, w, and h are the length width and height, respectively.

We know our length and width, and we are told that our height is triple the length, so...

Now that we have all our measurements, plug them in and solve:

### Example Question #9 : Finding Volume Of A Rectangular Prism

The above diagram shows a rectangular solid. The shaded side is a square. In terms of , give the volume of the box.

Explanation:

A square has four sides of equal length, as seen in the diagram below.

The volume of the solid is equal to the product of its length, width, and height, as follows:

.

### Example Question #1 : Volume Of A Pyramid

A pyramid has height 4 feet. Its base is a square with sidelength 3 feet. Give its volume in cubic inches.

Explanation:

Convert each measurement from inches to feet by multiplying it by 12:

Height: 4 feet =  inches

Sidelength of the base: 3 feet =  inches

The volume of a pyramid is

Since the base is a square, we can replace :

Substitute

The pyramid has volume 20,736 cubic inches.

### Example Question #1 : Solid Geometry

A foot tall pyramid has a square base measuring  feet on each side. What is the volume of the pyramid?

Explanation:

In order to find the area of a triangle, we use the formula .  In this case, since the base is a square, we can replace with , so our formula for volume is .

Since the length of each side of the base is feet, we can substitute it in for .

We also know that the height is feet, so we can substitute that in for .

This gives us an answer of .

It is important to remember that volume is expressed in units cubed.

### Example Question #4 : Volume

The height of a right pyramid is  feet. Its base is a square with sidelength  feet. Give its volume in cubic inches.

Explanation:

Convert each of the measurements from feet to inches by multiplying by .

Height:  inches

Sidelength of base:  inches

The base of the pyramid has area

square inches.

Substitute   into the volume formula:

cubic inches

### Example Question #2 : Solid Geometry

The height of a right pyramid is  inches. Its base is a square with sidelength  inches. Give its volume in cubic feet.

Explanation:

Convert each of the measurements from inches to feet by dividing by .

Height:  feet

Sidelength:  feet

The base of the pyramid has area

square feet.

Substitute   into the volume formula:

cubic feet

### Example Question #5 : Volume

The height of a right pyramid and the sidelength of its square base are equal. The perimeter of the base is 3 feet. Give its volume in cubic inches.

Explanation:

The perimeter of the square base,  feet, is equivalent to  inches; divide by  to get the sidelength of the base - and the height:  inches.

The area of the base is therefore  square inches.

In the formula for the volume of a pyramid, substitute :

cubic inches.

### Example Question #6 : Volume

What is the volume of a pyramid with the following measurements?

Explanation:

The volume of a pyramid can be determined using the following equation:

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