How to Find the Surface Area of a Pyramid

Step-by-Step Guide

1. 1

   Step 1: Set up the formula for the surface area of a regular pyramid.

   The formula is SA=p×h2+B{\displaystyle SA={\frac {p\times h}{2}}+B}, where SA{\displaystyle SA} equals the total surface area of the pyramid, p{\displaystyle p} equals the perimeter of the base, h{\displaystyle h} equals the slant height of the pyramid, and B{\displaystyle B} equals the area of the base.The basic formula for the surface area of any pyramid, regular or irregular, is Total Surface Area = Base Area + Lateral Area.Don’t confuse “slant height” with “height.” The “slant height” is the diagonal distance from the apex of the pyramid to the edge of the base.The “height” is the perpendicular distance from the vertex to the base.
2. 2

   Step 2: 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.
   
   For example, If you are finding the surface area of a hexagonal pyramid, and you know that the length of one edge of the base is 4 cm, you would calculate 4×6=24{\displaystyle 4\times 6=24} to find the perimeter of the base, since a hexagon has six edges, or sides.
   
   Thus, the perimeter of the base is 24 cm, so your surface area formula will look like this:
   SA=24×h2+B{\displaystyle SA={\frac {24\times h}{2}}+B}. , 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=24×122+B{\displaystyle SA={\frac {24\times 12}{2}}+B}. , 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.
   
   For example, if you are working with a hexagonal pyramid, the base is a hexagon.
   
   To find out how to calculate the area of the base, you can read Calculate the Area of a Hexagon.
   
   The formula is A=33×s22{\displaystyle A={\frac {3{\sqrt {3}}\times s^{2}}{2}}}, where s{\displaystyle s} is the length of one side of the hexagon.
   
   Since the length of one side of the hexagon is 4 cm, you would calculate:
   A=33×422{\displaystyle A={\frac {3{\sqrt {3}}\times 4^{2}}{2}}}A=33×162{\displaystyle A={\frac {3{\sqrt {3}}\times 16}{2}}}A=4832{\displaystyle A={\frac {48{\sqrt {3}}}{2}}}A=83.142{\displaystyle A={\frac {83.14}{2}}}A=41.57{\displaystyle A=41.57}.So the area of the base is
   41.57 square centimeters. , Make sure you substitute for the variable B{\displaystyle 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=24×122+41.57{\displaystyle SA={\frac {24\times 12}{2}}+41.57}. , Then, divide by two.
   
   This will give you the lateral surface area of the pyramid.
   
   For example:
   SA=24×122+41.57{\displaystyle SA={\frac {24\times 12}{2}}+41.57}SA=2882+41.57{\displaystyle SA={\frac {288}{2}}+41.57}SA=144+41.57{\displaystyle SA=144+41.57} , 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.
   
   For example:
   SA=144+41.57{\displaystyle SA=144+41.57}SA=185.57{\displaystyle SA=185.57}So, the total surface area of a hexagonal pyramid, given a base edge length of 4 cm and a slant height of 12 cm, is
   185.57 square centimeters.
3. 3

   Step 3: Plug the value of the slant height into the formula.

4. 4

   Step 4: Calculate the area of the base.

5. 5

   Step 5: Plug the area of the base into the formula.

6. 6

   Step 6: Multiply the perimeter of the base and the slant height of the pyramid.

7. 7

   Step 7: Add the two values together.

Detailed Guide

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