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Word Lesson: Volume and Surface Area of Pyramids
We will work with regular pyramids, in which the triangular sides are congruent and the base is a regular polygon. In order to solve problems which require application of the volume and surface area for pyramids, it is necessary to
  1. Surface area of a regular pyramid = area of base +

  2. Volume of a pyramid = .

  • know how to find the area of a base:

  1. for a regular polygon: A =

  2. for a hexagonal base: A =

A typical problem involving the volume or surface area of a pyramid gives us one or more of the volume, lateral area, area of a base, height and/or radius of the pyramid. We need to calculate some of these quantities given information about the others.
Suppose that the height of a regular square pyramid is 3 cm and the length of one edge is 5 cm. What are the surface area and volume of this pyramid? A diagram shown below includes the length of the radius CB.
To get started, we must determine which quantities are known and look at the formulas available for our use to to see which additional variables we need to calculate.
For both surface area and volume, we need the area of the square base.
It is usually easier to find the area of the base if we first know the length of one side. Since triangle ACB is a right triangle with hypotenuse 5 and one leg 3, our square base has a radius of 4 . We can now use the formula
along with the knowledge that the radius r = 4 and the number of sides n = 4 to find the length of one side of the square,
We now know the area of the square base is
A = s2 =
We can use this to find the volume which is
V =
To find the surface area, we need the perimeter of the base, P, and the slant height, s
The perimeter of our square base is just four times the length of one side:
perimeter = 4()
The slant height is the length of AM in triangle ABD. Using the Pythagorean Theorem and triangle ABM,
where MB = because it is half of one side
The surface area equals .

Example Trumpet A regular triangular pyramid has a height of 4 inches and a radius of 3 inches. What are the surface area and volume?
What is your answer?
Example Trumpet A regular pentagonal pyramid has height 4 cm and a side of the base 5 cm. What are the volume and surface area?
What is your answer?

Example Trumpet A regular pyramid has a regular hexagon of radius 2 as a base. If the height is 5, what are the volume and surface area of this pyramid? Give the exact volume and the surface area correct to two decimal places.
  1. V = and SA = 42.14

  2. V = and SA = 36.95

  3. V = and SA = 42.14
What is your answer?
Example Trumpet A regular pyramid has a height of 14 and a regular octagonal base of radius 9. What are the volume and surface area of this pyramid? Give answers correct to one decimal place. (Note that student answers may vary slightly due to rounding).
  1. V = 3207.4 and SA = 677.7
  2. V = 1069.1 and SA = 677.7
  3. V = 1286.6 and SA is a negative value
What is your answer?

This type of problem involves the use of several formulas. If the base of the pyramid is neither an equilateral triangle nor a regular hexagon, we must use formulas involving sine and cosine to get the area of the base and the length of one side of the base. If the base is square, we can use we can use the square of a side to find the base area, but the side may not be given. In that case, we go back to trigonometric formulas relating the side and radius.
In any case, the base area is used to get the volume by multiplying by 1/3 and the height. The side of the base is then used to get the perimeter which is subsequently used with the slant height to calculate the surface area.
Since sine and cosine are sometimes used, it is important to have your calculator set for either radians or degrees depending on your given information. Usually calculations can left in the degree MODE since our basic formulas involve 180/n where 180 is in degrees.
It is difficult to check these answers for reasonableness and therefore very important to double check all arithmetic carefully, particularly arithmetic involving radicals when using the Pythagorean Theorem repeatedly.

M Ransom

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