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# Surface Area and Volume of Pyramids

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Surface Area of Pyramids

Have you ever tried to calculate the surface area of a pyramid? Take a look at this dilemma.

Find the surface area of the figure below.

You will learn how to solve this dilemma in this Concept.

### Guidance

A pyramid has sides that are triangular faces and a base. The base can be any shape.

Surface area is the total of the areas of each face in a solid figure.

Imagine you could wrap a pyramid in wrapping paper, like a present. The amount of wrapping paper needed to cover the figure represents its surface area. To find the surface area, we must be able to calculate the area of each face and then add these areas together.

We will look at different ways to calculate surface area.

One way is to use a net .

A net is a two-dimensional diagram of a three-dimensional figure.

Imagine you could unfold a pyramid so that it is completely flat.

Here is what the net of a pyramid would look like.

This net is of a square pyramid. You can imagine folding up the sides to create a pyramid. With a net, we can see each face of the pyramid more clearly.

To find the surface area, we need to calculate the area for each face in the net: the sides and the base. The side faces of a pyramid are always triangles, so we use the area formula for triangles to calculate their area: $A = \frac{1}{2} bh$ . For triangles, we will need the height or slant height in the case of a pyramid.

The area of the base depends on what shape it is. Remember, pyramids can have bases in the shape of a triangle, square, rectangle, or any other polygon. We use whichever area formula is appropriate for the shape.

Here are some common area formulas:

Rectangle: $A = lw$

Square: $A = s^2$

Triangle: $A = \frac{1}{2} bh$

In the pyramid of the net above, we can see that it is a square pyramid. Imagine that it has a slant height of 4 cm and a side length of 6 cm. We can use these measurements to find the area of each face of the pyramid.

The base has a side length of 6 cm, so we use the formula for finding the area of a base.

$A&= s^2\\ A&=6^2\\A&=36 \ sq.cm$

The area of the base of the pyramid is 36 sq. cm.

Next, we need to find the area of each triangular side. To find the area of one side, we use the formula for finding the area of a triangle.

$A&=\frac{1}{2} bh\\A&=\frac{1}{2}(6)(4)\\ A&=12 \ sq.cm$

This is the area of one triangle. We have four total, so we can multiply this value by 4.

$12 \times 4 = 48 \ sq. cm$

Next, we add up all of the areas.

$48 + 36 = 84 \ sq. cm$

The surface area of this square pyramid is 84 sq. cm.

Nets let us see each face of a pyramid so that we can calculate its area. However, we can also use a formula to represent the faces as we find their area. The formula is like a short cut, because we can put the measurements in for the appropriate variable in the formula and solve for $SA$ , surface area. Let’s start with pyramids.

Here is the formula for finding the surface area of a pyramid.

$SA = \frac{1}{2} \ \text{perimeter} \times \text{slant height} + B$

Now let’s look at how we can understand this formula.

The first part of the formula, $\frac{1}{2} \ \text{perimeter} \times \text{slant height}$ , is a quick way of finding the area of all of the triangular sides of the pyramid at once. Remember, the area formula for a triangle is $A = \frac{1}{2} bh$ . In the formula, $b$ stands for base. The perimeter of the pyramid’s bottom face represents all of the bases of the triangular faces at once, because it’s their sum. The height of each triangle is always the same, so we can just call this the slant height of the pyramid. Therefore “ $\frac{1}{2} \ \text{perimeter} \times \text{slant height}$ ” is really the same as $\frac{1}{2} bh$ .

The $B$ in the formula represents the base’s area. Remember, pyramids can have bases of different shapes, so the area formula we use to find $B$ varies. We will find the base’s area first and then put it into the formula in place of $B$ .

Sometimes, you will have to find a linear measurement. This means that you will be given the surface area and one other dimension. Then you will need to work backwards to figure out the measurement for the missing dimension. This may seem challenging, but if you think of it as a puzzle, then you will be right on track.

The base of a square pyramid has sides of 4 cm each and a surface area of $96 \ cm^2$ . What is the slant height of the pyramid?

This time we know the surface area, but we need to find the slant height. Let’s find the perimeter and $B$ first so that we can put these into the formula. The base is a square with sides of 4 centimeters, so the perimeter must be $4 \times 4 = 16 \ cm$ . Now we use the square area formula to find $B$ .

$B &= s^2\\ B &= 4^2\\B &= 16 \ cm^2$

$B$ is 16 square centimeters. Let’s put these values in for the appropriate variables in the formula and solve for $s$ , the slant height.

$SA &= \frac{1}{2} \ \text{perimeter} \times \text{slant height} + B \\96 &= \frac{1}{6} (16)s + 16\\96 &= 8s + 16\\96 - 16 &= 8s\\80 &= 8s\\80 \div 8 &= s\\10 \ cm &= s$

A square pyramid with a base of 4 centimeters on each side and a surface area of 96 square centimeters must have a slant height of 10 centimeters.

Write these formulas for finding the surface area of a pyramid in your notebook. Be sure to note that you will need to find the area of the base $(B)$ before putting values into the formula for surface area.

Find the surface area of each pyramid.

#### Example A

A square pyramid with side of 8 in, slant height of 9 in

Solution: $208 \ in^2$

#### Example B

A rectangular pyramid with a length of 6 in, a width of 4 in and a slant height of 3 in

Solution: $54 \ in^2$

#### Example C

A square pyramid with side of 6 cm, slant height of 5 cm

Solution: $96 \ cm^2$

Now let's go back to the dilemma from the beginning of the Concept.

First of all, what kind of pyramid is this? It is a triangular pyramid because its base is a triangle. That means we need to use the area formula for triangles to find $B$ . The base’s sides are all the same length, so we can calculate the perimeter by multiplying $16 \times 3 = 48$ . Now let’s find $B$

$B &= \frac{1}{2} bh\\B &= \frac{1}{2} (16) (13.86)\\B &= 8 (13.86)\\B &= 110.88 \ cm^2$

Now we’re ready to put all of the information into the formula. Let’s see what happens.

$SA &= \frac{1}{2} \ \text{perimeter} \times \text{slant height} + B\\SA &= \left[\frac{1}{2} (48) \times 13.86 \right] + 110.88\\SA &= (24 \times 13.86) + 110.88\\SA &= 332.64 + 110.88\\SA &= 443.52 \ cm^2$

The surface area of this triangular pyramid is 443.52 square centimeters.

### Vocabulary

Pyramid
a three – dimensional solid figure with any polygon as a base and all triangular side faces meeting at one vertex.
Surface area
the measurement of the outer covering of a solid figure.
Net
a diagram that represents what a solid figure would look like in two-dimensions, flattened out as a pattern.

### Guided Practice

Here is one for you to try on your own.

What is the surface area of the pyramid below?

Solution

This is a square pyramid. The four sides of the base are all 8 inches, so the perimeter of the base is $8 \times 4 = 32 \ inches$ . We also know we will need to use the area formula for squares to find $B$ , the base’s area.

$B&=s^2\\B &= s^2\\B &= 8^2\\B &= 64 \ in^2.$

Now that we have the area of the base, we have all the information that we need. We can put it into the formula and solve for $SA$ , surface area.

$SA &= \frac{1}{2} \ \text{perimeter} \times \text{slant height} + B\\ SA &= \left[\frac{1}{2}(32) \times 13.6\right] + 64\\SA &= (16 \times 13.6) + 64\\SA &= 217.6 + 64\\SA &= 281.6 \ in^2.$

The surface area of the pyramid is 281.6 square inches.

### Explore More

1. What is the name of the figure represented by this net?
2. What is the length of each of the sides of the base?
3. What is the surface area of the figure?

1. What is the name of this figure?
2. What is the shape of the base?
3. How many faces does this figure have?
4. What is the surface area of this figure?

1. What is the name of the figure represented by this net?
2. What is the length of each of the sides of the base?
3. What is the surface area of the figure?

1. What is the name of the figure represented by this net?
2. What is the length of each of the sides of the base?
3. What is the surface area of the figure?
1. True or false. The B in the formula for surface area of a pyramid stands for bottom.
2. True or false. You need to know the slant height to figure out the perimeter of a pyramid.