# 7.3: Base, Lateral and Surface Areas of Prisms

## Learning Objectives

- Use nets to represent prisms.
- Find the surface area of a prism.

## Prisms

A **prism** is a 3-dimensional figure with a pair of *parallel* and *congruent* ends, or **bases**. The *sides* of a prism are **parallelograms**. **Prisms** are *identified* by their **bases**.

Look at the *names* of the prisms above. Each type of prism is named after its **base**. In the figure on the left, the base is a *rectangle*, so it is called a *rectangular prism*. In the middle, the base is a *triangle*, so the shape is a *triangular prism*. On the right, the base is a *hexagon* (6-sided figure), so the prism is called a *hexagonal prism*.

As you can see, the base is not always on the “bottom” of the prism!

**Prisms** are named by their ______________________.

If a prism is called a *pentagonal prism*, then its base is a __________________________.

## Surface Area of a Prism Using Nets

The **prisms** in the picture above are **right prisms**. In a **right prism**, the lateral sides are *perpendicular* to the **bases** of prism. In the picture below, compare a right prism to an **oblique prism**, in which sides and bases are *not perpendicular*:

**Reading Check:**

1. *Fill in the blanks:*

A **prism** is a _____-dimensional figure with parallel and congruent ________________ and sides that are __________________________________.

2. *In a right prism, what is the relationship between the sides and the bases?*

3. *What is the difference between a right prism and an oblique prism?*

## Surface Area

**Surface area** is the exposed area of a 3-dimensional or solid figure. This means that **surface area** is a measurement of the *outside* area of an object. For example, the surface area of a soccer ball is the outside part of the ball that touches the air. The inside of the ball is not included in surface area. **Surface area** is, as its name describes, the *area* of the *surface* of a solid object.

**Surface area** is the area of the _________________________ of a 3-dimensional object.

**Area Addition Postulate**

The **surface area** of a 3-dimensional figure is the sum of the areas of all of its non-overlapping parts.

This means that we find the **surface area** by adding up all of the face areas of the figure. Since the **faces** are on the *surface* of the object, if we add up all of the faces, we will have the entire surface area!

To find the **surface area** of a shape, you add the areas of its _______________________.

You can use a **net** and the **Area Addition Postulate** to find the **surface area** of a **right prism**:

From the **net** of the **prism**, you can see that that the **surface area** of the entire prism equals the *sum* of the shapes that make up the net. Since there are 6 faces of the prism and 6 shapes in the net, there are 6 areas that you add together:

You may notice that the faces labeled and ____ are called the **bases** of the prism, and the faces labeled ____ , and ____ are called the **sides** of the prism. The **surface area** is the sum of the areas of the **bases** and the areas of the **sides**.

To find the areas of shapes in the **net**, we must use the formula for the area of a rectangle:

Find the areas of the other rectangles in the **net**:

Area of shape square units

Area of shape square units

Area of shape square units

Area of shape square units

Area of shape square units

Then insert these areas back into the **surface area** equation above:

If 2 polygons (or plane figures) are congruent, then their areas are congruent.

This means that if 2 shapes are the same, then they have the same area! This may seem simple, and it is!

**Example 1**

*Use a net to find the surface area of the prism.*

The area of the **net** is equal to the **surface area** of the figure:

To find the area of the triangles (shapes and ), we use the formula:

, where is the base of the triangle and is its height

Since triangles and are *congruent*, their areas are the *same*:

So the area of shapes is 54 square units *and* the area of shape is 54 square units.

For the areas of shapes and , we use the formula for the area of a rectangle.

Now insert the areas you found back into the **surface area** equation on the previous page:

**Reading Check:**

1. *True/False: The surface area of a 3-D figure is the same as the area of its net.*

2. *True/False: You can figure out the area of a net by finding the area of each of the shapes in it (rectangles, triangles, etc.) and multiplying them together.*

## Surface Area of a Prism Using Perimeter

The hexagonal prism below has 2 regular hexagons for **bases** and 6 **sides**. Since all sides of the hexagon are *congruent*, all of the rectangles that make up the **lateral sides** of the 3-dimensional figure are also *congruent*.

You can break down the figure like this:

*The word “lateral” means “on the side.”*

*For example, in football, a lateral pass is when the quarterback throws the ball sideways to a receiver: the ball is passed to the side instead of forwards.*

*Lateral sides**of a solid are the faces on the sides of the shape (not the bases).*

*Lateral area**, which you will learn next, is the area of the sides of the shape.*

The **surface area** of the *rectangular sides* of the figure is called the **lateral area** of the figure. The **lateral area** does *not* include the area of the **bases**. To find the **lateral area**, you can add up all of the areas of the 6 rectangles:

You can also see that the **perimeter** of the **base** of the hexagonal prism on the previous page is or .

Another way to find the **lateral area** of the figure is to *multiply* the perimeter of the base by , which is the height of the figure:

Substituting , the **perimeter** of the **base**, for , we get the formula for *any* **lateral area** of a **right prism:**

The **lateral area** is the surface area of the rectangular ____________________ of a right prism.

The **lateral area** does not include the area of the _____________________.

Find the **lateral area** of a right prism by multiplying the ________________________ of the base by the ________________________ of the prism.

We can use the **lateral area** formula to calculate the *total* **surface area** of the prism. Remember, lateral area does *not* include the area of the bases, but we must include the bases to find the *total* surface area of the prism!

Using for the **perimeter** of the base and for the **area** of the base:

You can use this formula to find the **surface area** of *any* **right prism.**

**Reading Check:**

1. *True/False: The lateral area includes the area of the sides and the area of the bases.*

2. *True/False: The perimeter of the base and the area of the base are the same thing.*

3. *How could you find the total surface area of a 3-dimensional shape if you know the height of the shape and the perimeter of its base? What other information would you need to know? Explain.*

**Example 2**

*Use the formula to find the total surface area of the trapezoidal prism below.*

The dimensions of the trapezoidal base are shown in the diagram, but we should list our information so it is easy to see:

Height of the entire prism (we will call this )

Base 1 of the trapezoid

Base 2 of the trapezoid

height of the trapezoid (call this so it is not confused with above)

Use the formula for **total surface area** from the previous page:

(where is *perimeter of the base, is height,* and is *area of the base*)

Find the **area** of each trapezoidal base. Do this with the formula for area of a trapezoid. Remember that the height of the trapezoid is small :

Now find the **perimeter** of the base:

Use these values to find the total surface area of the solid:

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Feb 23, 2012## Last Modified:

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