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Three-dimensional figures like cubes, prisms, and pyramids, appear in different forms everywhere in the real world. Some numeric characteristics about these objects can be very useful for various aspects of daily life. This lesson will show how to calculate the volume and surface area of different kinds of prisms.

### Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

## Choosing the Shape of a Dog House

Mark is building a model of the house of his dreams. He enjoys constructing different rooms and furniture from various materials. He also wants to build a dog house for his toy dog Bubbles.
External credits: @pikisuperstar
However, he cannot decide on the shape of the dog house. He knows it will be a prism, but should it be rectangular like a box or triangular like a tent? Given their dimensions, which dog house offers the most space for Bubbles?
Discussion

## Rectangular and Triangular Prisms

Prisms can be categorized by the shape of their bases. For example, a prism with a triangular base is called a triangular prism. A rectangular prism, on the other hand, has a rectangular base.

A rectangular prism has a unique name if its bases are squares and its lateral edges are the same length as the square's sides.
Discussion

## Cube

A cube is a three-dimensional solid object bounded by six square faces with three meeting at each vertex. All square faces have the same side length. A cube is a specific type of rectangular prism.
Cubes have faces, edges, and vertices.
Pop Quiz

## Identifying the Type of a Prism

Determine whether the given solid is a triangular prism, a rectangular prism, or neither.

Discussion

## Volume and Surface Area

There are two important characteristics that give information about objects.

Concept

## Volume

The volume of a solid is the measure of the amount of space inside the solid. It is the three-dimensional equivalent of the area of the figure. Volume is measured using cubic units, such as cubic meters, or The applet below illustrates the volume of some solids. Move the slider to fill the solids.

Discussion

## Surface Area

The surface area of a three-dimensional shape is the total area of all the surfaces of the shape. The lateral area is the total area of all lateral surfaces, the surfaces of the shape that are not bases. Lateral area can be defined for polyhedrons, cones, and cylinders. In these instances, the surface area of the shape can be calculated by the formula below.

The net of a solid can be helpful to visualize the surfaces of a three-dimensional figure.

Discussion

## Volume of a Prism

Consider a prism with a base area and height

The volume of the prism is calculated by multiplying the base's area by its height.

By Cavalieri's principle, this formula holds true for all types of prisms, including skewed ones.

### Proof

Informal Justification

Recall that the volume of a solid is the measure of the amount of space inside the solid. Note that the top and bottom faces of the prism are always the same and parallel to each other.

Additionally, the prism is, so to speak, filled with identical base areas that are stacked on top of each other to the height of the prism.

This means that the volume of the prism can be calculated as the sum of all these base areas. The number of bases is equal to the height of the prism. Therefore, the volume of a prism equals the product of its base area and height.
Discussion

## Surface Area of a Prism

Consider a prism with a height base area and base perimeter

The surface area of the prism is the sum of the two base areas and the lateral area, which can be calculated as the product of the base perimeter and the height of the prism.

### Proof

The surface area of the prism can be seen as the sum of two separate parts: the lateral area and the combined area of the two identical bases.
The base area, often symbolized as can be substituted into the equation.
To determine the lateral area of the prism, consider a net of the given prism. Let be the length of the side of the base, assuming that it is a regular polygon.
Notice that the lateral surface consists of rectangles equal to the number of sides in the base. The pentagonal prism shown here has five lateral faces because a pentagon has five sides. The area of each rectangular lateral face is the product of its sides and
If there are rectangular lateral faces in a prism, then the total lateral area is the product of and the area of one lateral face.
Notice that is the perimeter of the base, which is often denoted by Then, the lateral area can be expressed as follows.
Therefore, the formula for the surface area is obtained.

Note that although this proof is written for a regular prism, it is also true for a non-regular prism.

Discussion

## Volume and Surface Area of a Rectangular Prism

It is possible to derive two specific formulas for the volume and surface area of a rectangular prism.

Rule

## Volume of a Rectangular Prism

Consider a rectangular prism with a width length and height

The volume of the prism is calculated by multiplying the area of the base by the height of the prism.

### Proof

The volume of a prism can found by multiplying the base area and height
The base of a rectangular prism is a rectangle. The area of a rectangle is the product of its width and length
Substitute this expression for in the first formula.
This process produces the formula for the volume of a rectangular prism.
Discussion

## Surface Area of a Rectangular Prism

Consider a rectangular prism with a height base area and base perimeter

The surface area of the rectangular prism can be found using the following formula.

### Proof

Recall that the surface area of a prism is the sum of the lateral area and the combined areas of the two identical bases.
Here is the perimeter of the base, is the height of the prism, and is area of one base. The base of a rectangular prism is a rectangle, so its area is the product of its width and length.
The perimeter of a rectangle can be found using the following formula.
Now the expressions for and can be substituted into the formula presented in the beginning and the whole expression simplified.
This process results in the formula for the surface area of a rectangular prism.
Discussion

## Volume of a Cube

Consider a cube with a side length

The volume of the cube can be calculated by raising the side length to the power of or cubing it.

### Proof

The volume of a prism is calculated by multiplying the area of one base by the height. For rectangular prisms, this value is found by multiplying the area of the rectangular base by the height of the prism.
A square is a special type of rectangle whose length and width are equal, so the area of a square is found by calculating the square of the side length. Similarly, a cube is a special type of rectangular prism whose length, width, and height are equal.
The formula for the volume of a cube can therefore be derived by substituting into the standard equation for the volume of a rectangular prism.
Discussion

## Surface Area of a Cube

Consider a cube with a side length

The surface area of the cube is given by the following formula.

### Proof

The surface area of a prism is the sum of the lateral area and the combined areas of the two identical bases.
Since the base of a cube is a square with a side length its area is equal to the square of
The perimeter of the square base can be calculated as the product of and
Finally, the height of the cube is also equal to the side length
Now these values will be substituted into the formula and the resulting equation simplified.
Example

## Renovations to the Bedroom

Mark wants to start painting the walls in the model bedroom and fill it with furniture. To understand how much paint he needs and how much space is available, Mark has to determine the surface area and the volume of the room. He starts by measuring the dimensions of the room.
External credits: @pikisuperstar
a What is the volume of the bedroom?
b What is the surface area of the bedroom?

### Hint

a Use the formula for the volume of a rectangular prism.
b Use the formula for the surface area of a rectangular prism.

### Solution

a The bedroom has the shape of a rectangular prism, so its volume can be calculated by using the formula for the volume of a rectangular prism.
Here, is the width, is the length, and is the height of the prism. The diagram shows that the width of the room is inches, the length is inches, and the height is inches. Substitute these values into the formula and evaluate.
The volume of the room is cubic inches.
b The surface area of the bedroom can be found by using the formula for the surface area of a rectangular prism.
Substitute with with and with into the formula and simplify.
The surface area of the bedroom is square inches.
Example

## Working on the Closet

Later in the day, Mark decides to work on the bedroom closet, which also has a rectangular prism shape. His measurements show that the length of the closet is inches and its height is inches, but he is not able to measure the width since the closet was already installed in the wall.
External credits: @pikisuperstar
However, Mark remembers that the volume of the closet is cubic inches. What is the surface area of the closet?

### Hint

Find the width of the prism by using the known volume of the prism and the formula for the volume of a rectangular prism. Then, use the formula for the surface area of a rectangular prism.

### Solution

The surface area of a rectangular prism can be found by using the following formula.
Here, is the width, is the length, and is the height of the prism. The length and the height of the prism are known but the width is not. To find the width of the prism, use the fact that the volume of the prism is cubic inches.
Recall the formula for the volume of a rectangular prism.
Substitute with with and with into the formula and solve it for
The width of the closet is inches. Now there is enough information to calculate the surface area of the closet. Substitute the values into the formula for the surface area of a rectangular prism and evaluate
The surface area of the closet is square inches.
Example

## Designing the Attic

After mostly finishing with the first floor, Mark begins brainstorming ideas for the design of the attic. He needs to determine how much space there is in the attic and the areas of the walls, floor, and ceiling.
External credits: @pikisuperstar
a What is the volume of the attic?
b What is the surface area of the attic?

### Hint

a Use the formula for the volume of a prism.
b Use the formula for the surface area of a prism.

### Solution

a The attic has the shape of a triangular prism, which means that its volume can be calculated by using the formula for the volume of a prism.
Here, is the area of one base and is the height of the prism. The base of a triangular prism is a triangle, so recall the formula for the area of a triangle.
The diagram shows that the base of the triangle is inches and its height is inches. Substitute for and for in the formula and solve for
The area of the triangle, and therefore the base area of the prism, is square inches. Now that this value is known, substitute and in the formula for the volume of a prism.
The volume of the attic is cubic inches.
b The surface area of the attic can be found by using the formula for the surface area of a prism.
In this formula, is the area of one base, is the perimeter of the base, and is the height of the prism. The lengths of all the sides of the triangular base are given on the diagram. Find the perimeter of the triangle by adding all the side lengths together.
In Part B it was found that the base area is square inches. Substitute and into the formula for the surface area of a prism and evaluate
The surface area of the attic is square inches.
Example

## The Space Next to the Bathroom

Finally, Mark noticed that there is some empty space above the living room next to the upstairs bathroom. He wants to potentially turn in into another room. Mark knows that the surface area of that room would be square inches.
External credits: @pikisuperstar
What would the volume of this potential room be?

### Hint

Use the formula for the surface area of a prism and the given surface area to find the height of the prism. Then, find the volume of the prism.

### Solution

The potential room has the shape of a triangular prism. Recall the formula for the surface area of a prism.
Here, is the area of one base, is the perimeter of the base, and is the height of the prism. The base of the prism is a right triangle, so recall the formula for the area of a triangle.
Here is the base of the triangle and is its height. Substitute for and for into the formula and find the area of the triangle.
The area of the triangular base of the prism is square inches. Next, add the side lengths of the triangle to calculate the perimeter of the base of the prism.
Now every value in the formula for the surface area of a triangular prism is known except for its height Mark knows that the surface area of the potential room is square inches. Substitute for for and for into the formula and solve for
The height of the prism is inches. Finally, find the volume of the prism by using the following formula.
Substitute for and for then evaluate for
The volume of the potential room is cubic inches.
Pop Quiz

## Calculating the Volume or Surface Area of a Prism

Calculate the volume or surface area of the triangular or rectangular prism given the base area the height, and the side length of the base. The base is always a regular polygon. Round the answer to one decimal place if necessary.

Closure

## Calculating the Volume of the Dog House

As mentioned before, Mark is building a model of the house of his dreams. He also wants to build a dog house for his toy dog Bubbles.