Solid Geometry - ISEE Upper Level: Quantitative Reasoning

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:

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:

.

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:

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.

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.

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

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

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.

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

The pyramid has a square base that is units by units, and its height is units, as can be seen from this diagram,

The square base has area ; the pyramid has volume

Since the volume is 1,000, we can set this equal to 1,000 and solve for :

To find the volume of a pyramid, we will use the following formula:

where l is the length, w is the width, and h is the height of the pyramid.

Now, we know the base of the pyramid has a length of 4in. We also know the base of the pyramid has a width of 3in. We also know the pyramid has a height of 5in.

Knowing this, we can substitute into the formula. We get

To find the volume of a pyramid, we will use the following formula:

where l is the length, w is the width, and h is the height of the pyramid.

Now, we know the following measurements:

• length = 4cm
• width = 9cm
• height = 8cm

Knowing this, we can substitute into the formula. We get

To find the volume of a pyramid, we will use the following formula:

where l is the length, w is the width_,_ and h is the height of the pyramid.

Now, we know the following measurements:

• length: 7in
• width: 6in
• height: 8in

So, we get

To find the volume of a pyramid, we will use the following formula:

where l is the length, w is the width_,_ and h is the height of the pyramid.

Now, we know the following measurements:

• length: 7in
• width: 6in
• height: 8in

So, we get

Let be the length of an edge of the cube. The volume of a cube can be determined by the equation:

The surface area of a cube is six times the square of the length of each edge, which here is . Therefore,

Substituting, then solving for :

Since the sidelength is positive,

There is a sculpture in front of town hall which is shaped like a cube. If it has a volume of , what is the length of one side of the cube?

To find the side length of a cube from its volume, simply use the following formula:

Plug in what is known and use some algebra to get our answer:

You have a cube with a volume of . What is the cube's side length?

If we begin with the formula for volume of a cube, we can work backwards to find the side length.

Making our answer:

You have a crate with equal dimensions (height, length and width). If the volume of the cube is , what is the length of one of the crate's dimension?

Let's begin with realizing that we are dealing with a cube. A crate with equal dimensions will have equal height, length, and width, so it must be a cube.

With that in mind, we can find our side length by starting with the volume and working backward.

So, to find our side length, we just need to take the cubed root of the volume.

So, our answer is 6 meters, a fairly large crate!