Calculate the Volume of a Pyramid

To calculate the volume of a pyramid, use the formula <math>V = \frac{1}{3}lwh</math>, where l and w are the length and width of the base, and h is the height. You can also use the equivalent formula <math>V = \frac{1}{3}A_{b}h</math>, where <math>A_{b}</math> is the area of the base and h is the height. The method varies slightly depending on whether the pyramid has a triangular or a rectangular base. If you want to know how to calculate the volume of a pyramid, just follow these steps.

Steps

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Doc:Volume of a Pyramid

Pyramid with a Rectangular Base

  1. Find the length and width of the base. In this example, the length of the base is 4 cm and the width is 3 cm. If you're working with a square base, the method is the same, except the length and width of the square base will be equal. Write down these measurements.
    • Remember, <math>V = \frac{1}{3}lwh = \frac{1}{3}A_{b}h</math>, so you need to know <math>l</math> and <math>w</math> first.
    • <math>l = 4\,\text{cm}</math>
    • <math>w = 3\,\text{cm}</math>
  2. Multiply the length and width to find the area of the base. To get the area of the base, simply multiply 3 cm by 4 cm.2[1]
    • Remember, <math>V = \frac{1}{3}A_{b}h</math>, so you need to know <math>A_{b}</math>. You can find this by plugging in <math>l = 4\,\text{cm}</math> and <math>w = 3\,\text{cm}</math> from the previous step.
    • <math>A_{b} = lw</math>
    • <math>A_{b} = (4\,\text{cm})(3\,\text{cm})= 12\,\text{cm}^2</math>
  3. Multiply the area of the base by the height. The area of the base is 12 cm2 and the height is 4 cm, so you can multiply 12 cm2 by 4 cm.
    • Remember, <math>V = \frac{1}{3}A_{b}h</math>, so you need to know <math>A_{b}h</math>. You can find this using <math>A_{b}</math> from the previous step.
    • <math>A_{b} = 12\,\text{cm}^2</math>
    • <math>h = 4\,\text{cm}</math>
    • <math>A_{b}h = (12\,\text{cm}^2)(4\,\text{cm}) = 48\,\text{cm}^3</math>
  4. Multiply your result so far by <math>\frac{1}{3}</math>. Or, in other words, divide by 3. Remember to state your answer in cubic units whenever you're working with three-dimensional space.
    • Remember, <math>V = \frac{1}{3}lwh = \frac{1}{3}A_{b}h</math>. You can plug in <math>A_{b}h = 48\,\text{cm}^3</math> from the previous step.
    • <math>V = \frac{1}{3}A_{b}h</math>
    • <math>V = (\frac{1}{3})(48\,\text{cm}^3)= 16\,\text{cm}^3</math>

Pyramid with a Triangular Base

  1. Find the length and width of the base. The length and width of the base must be perpendicular to each other for this method to work. They can also be considered the base and height of the triangle. In this example, the width of the base is 2 cm and the length of the triangle is 4 cm.[1]
    • If the length and width are not perpendicular and you don't know the height of the triangle, there are a few other methods you can try to calculate the area of a triangle.
    • Remember, <math>V = \frac{1}{3}lwh = \frac{1}{3}A_{b}h</math>, so you need to know <math>l</math> and <math>w</math> first.
    • <math>l = \text{width of pyramid base} = \text{base of triangle, or}\, b = 2\,\text{cm}</math>
    • <math>w = \text{length of pyramid base} = \text{height of triangle, or}\, h = 4\,\text{cm}</math>
  2. Calculate the area of the base. To calculate the area of the base, just plug the base and height of the triangle into the following formula: <math>A_{b} = \frac{1}{2}bh</math>.
    • Remember, <math>V = \frac{1}{3}lwh = \frac{1}{3}A_{b}h</math>, so you need to know <math>A_{b}</math>. You can find this using <math>b</math> and <math>h</math> from the previous step.
    • <math>A_{b} = \frac{1}{2}bh</math>
    • <math>A_{b} = (\frac{1}{2})(2\,\text{cm})(4\,\text{cm})</math>
    • <math>A_{b} = (\frac{1}{2})(8\,\text{cm}^2)</math>
    • <math>A_{b} = 4\,\text{cm}^2</math>
  3. Multiply the area of the base by the height of the pyramid. The area of the base is 4 cm2 and the height is 5 cm.
    • Remember, <math>V = \frac{1}{3}A_{b}h</math>, so you need to know <math>A_{b}h</math>. You can find this using <math>A_{b}</math> from the previous step.
    • <math>A_{b} = \text{area of triangular base} = 4\,\text{cm}^2</math>
    • <math>h = \text{height of pyramid} = 5\,\text{cm}</math>
    • <math>A_{b}h = (4\,\text{cm}^2)(5\,\text{cm}) = 20\,\text{cm}^3</math>
  4. Multiply your result so far by <math>\frac{1}{3}</math>. Or, in other words, divide by 3. Your result will show that the volume of a pyramid with a height of 5 cm and a triangular base with a width of 2 cm and a length of 4 cm is 6.67 cm.3
    • Remember, <math>V = \frac{1}{3}lwh = \frac{1}{3}A_{b}h</math>. You can plug in <math>A_{b}h = 20\,\text{cm}^3</math> from the previous step.
    • <math>V = (\frac{1}{3})A_{b}h</math>
    • <math>V = (\frac{1}{3})(20\,\text{cm}^3)= 6.67\,\text{cm}^3</math>



Tips

  • In a square pyramid, the true height, slant height, and length of the edge of the base face are all related by the Pythagorean theorem: (edge ÷ 2)2 + (true height)2 = (slant height)2
  • In all regular pyramids, the slant height, edge height, and edge length are also related by the Pythagorean theorem: (edge ÷ 2)2 + (slant height)2 = (edge height)2
  • This method can be further generalized to such objects as pentagonal pyramids, hexagonal pyramids, etc. The overall process is: A) calculate the area of the base shape; B) measure the height from the tip of the pyramid to the center of the base shape; C) multiply A with B; D) divide by 3.

Warnings

  • Pyramids have three kinds of height --- a slant height, down the center of the triangular sides; a true height or perpendicular height, that goes from the tip of the pyramid to the center of the base face; and an edge height, that goes down one edge of the triangular sides. For volume, you must use the true height.

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Sources and Citations

  1. ↑ 1.0 1.1 http://www.mathsteacher.com.au/year10/ch14_measurement/25_pyramid/21pyramid.htm
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