how to find the surface area of a square pyramid

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The surface area of any pyramid can be found by adding the surface area of the base to the surface area of the lateral faces. When working with regular pyramids, you can find the surface area using a formula, as long as you know how to find the area of the base of the pyramid. Since the base can be any polygon, it is helpful to know how to find the area of shapes such as pentagons and hexagons. When working with the common, regular square pyramid, however, calculating the total surface area is a simple calculation, provided you know the slant height of the pyramid and the side length of the square base.

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Plug the perimeter of the base into the formula. If you aren't given the perimeter but know the length of one edge of the base, you can calculate the perimeter by multiplying the length of one edge by the number of edges.
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Plug the value of the slant height into the formula. Make sure you are using the slant height, not the perpendicular height. The problem should provide the slant height. If you don't know the slant height, you cannot use this method.
- For example, if the slant height of a hexagonal pyramid is 12 cm, your formula will look like this: $SA={\frac {24\times 12}{2}}+B$ .
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Calculate the area of the base. How you do this will depend on the shape of the base. To learn more about finding the area of a polygon, read Find the Area of Regular Polygons.

EXPERT TIP

Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University.

Grace Imson, MA
Math Instructor, City College of San Francisco

Our Expert Agrees: The surface area of a pyramid is equal to the sum of the areas of all of the faces. First, you have to get the area of the base, then add the area of the lateral sides, which is one face times the number of sides.
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Plug the area of the base into the formula. Make sure you substitute for the variable $B$ .
- For example, if the area of the hexagonal base is 41.57 sq. cm., your formula for surface area will now look like this: $SA={\frac {24\times 12}{2}}+41.57$ .
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Multiply the perimeter of the base and the slant height of the pyramid. Then, divide by two. This will give you the lateral surface area of the pyramid.
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Add the two values together. The sum will be the lateral surface area, plus the base surface area, providing you with the total surface area for the pyramid, in square units.

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Square the side length of the base. This will give you the surface area of the base.
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Multiply the side length of the base by the slant height and divide by two. Then, multiply by 4. This will give you the lateral surface area of the pyramid.
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Add the base surface area and the lateral surface area. This will give you the total surface area of the pyramid, in square units.

Add New Question

Question

How do you find the lateral area of hexagonal pyramid, given the height and length of each side?

Use the base times height for the rectangles, and the altitude times base of the hexagonal face times three.
Question

How would you calculate the surface area of pyramid that does not have a square base?

Use the formula (p x h/2) + (B), where p is the perimeter of the base, h is the slant height of the pyramid, and B is the area of the base. Below are some articles on finding the area of a pentagon and hexagon, two common pyramid bases: http://www.wikihow.com/Find-the-Area-of-a-Pentagon http://www.wikihow.com/Calculate-the-Area-of-a-Hexagon
Question

How do I double the lateral surface area of a square pyramid?

One way would be to double either the length of the sides of the base or the slant height (but not both).
Question

Where does the height fit in?

The actual height of a pyramid does not factor in to the calculation of its surface area. However, the "slant height" does, being one of the dimensions of a pyramid's slanted surface (along with the width of the base).
Question

A regular square pyramid is 3 m. Height and the perimeter of its base is 16 m. How do I find the volume of the pyramid?

V = (s)²(h) / 3, where s is the length of one side of the base, and h is the pyramid's height. The side of a square is 1/4 of the perimeter, so this pyramid has a base side of 4. Therefore, V = (4)²(3) / 3 = 16 cubic meters.
Question

What if I'm only given the lateral surface area?

That's not enough information to find the total surface area (including the base area).
Question

What if I do not know the height?

If you don't know either the height or the slant height of a pyramid, you cannot determine the surface area.
Question

How would I write the total surface area of a square-based pyramid having base side x cm and slant height y cm?

See Method 2 Step 1 above. Substitute x for b and y for h.
Question

How do I calculate the surface area of a triangle pyramid? Is it the same kind of mathematical equation, or something different?

If it's an equilateral triangle, it qualifies for Method 1 above. Otherwise, you would have to calculate the areas of each of the surfaces separately and add them together.
Question

How do I find the surface area of a heptagonal prism?

Assuming regular heptagonal bases, and lateral sides perpendicular to the bases, you would multiply an edge of the base by the height, and multiply by seven to find the total lateral surface area. If desired, add the areas of the bases. (The area of one base is half the product of the perimeter and the apothem.)

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Article SummaryX

To find the surface area of a pyramid, start by multiplying the perimeter of the pyramid by its slant height. Then, divide that number by 2. Finally, add the number you get to the area of the pyramid's base to find the surface area. To learn how to find the surface area of a square pyramid, scroll down!

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