Suppose you lean an aluminum extension ladder against an outside wall of your house. Maybe you've been told to clean the gutters, who knows? You are a mathematics student at heart, so you wonder: which is a greater distance, the height of the wall straight up from the ground to where the ladder touches, or the distance from the ground up, along the leaning ladder?

Instinctively you say the leaning, angled ladder covers a greater distance. That is the basic idea behind calculating the surface area of any pyramid.

## How To Find The Surface Area Of A Pyramid

The pyramid has a lot of possible base shapes, from the simple square seen in Egyptian pyramids to hexagonal deck prisms, used in sailing ships of olden days, to cast light into lower decks.

Because you can find pyramids with triangular, rectangular, square, pentagonal, hexagonal and octagonal bases, you could be asked to perform mathematical exercises such as these:

1. Which will always be larger, the lateral area of a pyramid or the total surface area of a pyramid?
2. Calculate the total surface area of a rectangular pyramid with a base 20 meters by 15 meters and a height of 50 meters.
3. What is the total surface area of a hexagonal pyramid with slanted sides 7 yards in length and each side of the base measuring 3 yards?
4. Find the lateral area of a pyramid with a square base, a height of 150 feet, and a slanted side length of 175 feet.
5. Compare the total surface area of a triangular pyramid with base sides 30 cm and slant height of its three sides 60 cm to the total surface area of a square pyramid with base sides 20 cm and slant height of its four sides 50 cm.

These are all challenging problems, becoming more complex as they go. Fortunately, several formulas exist to help you deal with every type of pyramid, no matter the polygon forming its base.

### Slant Height Of A Pyramid

A pyramid has height, which is a perpendicular measurement from base to the top of the pyramid (as if we had cut the pyramid in half to see inside). For measuring surface area, we are uninterested in the height. We need the slant height, or slant length, which is a measurement from any side of the base along a triangular, slanted side to the top.

In most of the formulas for surface area of a pyramid, we need slant height, $l$, not height, $h$.

## Surface Area Of A Pyramid Formula

You may be asked to find two different kinds of area for pyramids:

• Total surface area of a pyramid - The area of the slanting sides and the area of the base
• Lateral area of a pyramid - The area of only the slanting sides of any pyramid

For pyramids, where $A$ is the total surface area, $p$ is the perimeter of the base, $l$ is the slant height, and $B$ is the area of the base, the formula is:

The problem you are attempting to solve has to give you enough information to know the perimeter, $p$, the slant height, $l$, and the area of the base, $B$. It may not provide that information to you directly; you may have multiple steps to solve to get to $A$.

## Surface Area Of A Square Pyramid

Suppose you are making a square pyramid for a fundraiser for your mathematics club at school. Here total surface area really matters; you do not want to waste money buying too much plywood, nor can you take time for multiple trips to get more plywood. You sketch out a drawing, knowing plywood comes in 4' x 8' sheets:

[insert 3D isometric or perspective drawing of a square-based, plywood pyramid with labels showing sides 4' and slant height 6']

The pyramid you need will have four slanted, triangular sides, each a triangle 4' wide and 6' tall. The base will be 4' x 4'. What is the total surface area?

### How To Find The Surface Area Of A Square Pyramid

First, calculate the base area:

Next, calculate the perimeter of the base:

Next, plug in your slant height, $l$, and the other numbers into your formula:

If we don't mind piecing the sides when we build the pyramid, the club can purchase only two sheets of plywood, since one sheet has an area of .

### Lateral Surface Area Of A Pyramid Formula

Now we have our pyramid; we need to paint it. For that, we need only the lateral surface area, since the base will not be visible. To calculate the lateral surface area of a pyramid, use this formula:

Does that look familiar? It should; it is just the total surface area formula without the base! We still need the base perimeter, $p$, and the slant height, $l$. So for our mathematics club pyramid, the lateral surface area would be:

You only need a third of a gallon of paint, but buy two quarts, just to be safe.

## How To Find The Surface Area Of A Pyramid Using Height

What if, though, your problem gives you the height, $h$, of a pyramid, but not the slant height, $l$? Two somewhat more complicated formulas are available, for base width, $w$, height, $h$ (not slant height), and base length, $l$.

For total surface area of a pyramid, the formula looks like this:

### Surface Area Of A Rectangular Pyramid

Let's try that formula with our second task from earlier: "Calculate the total surface area of a rectangular pyramid with a base 20 meters x 15 meters and a height of 50 meters."

For lateral surface area of a pyramid, the formula is similar, but a little simpler, since we are leaving off the base's area:

By the way, the answer to the first of our exercises above should now be obvious: total surface area of a pyramid is always greater than lateral surface area of a pyramid.

### Next Lesson:

Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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