The volume of an object is defined as the three-dimensional space it occupies, but it might be easier to think of it as the amount of water, gas or any other substance that said object would hold. Either way, when faced with a square-based pyramid – think the pyramids of Egypt as an example – you can find its volume using a simple formula that requires the height of the pyramid and the length of one side along its base.
TL;DR (Too Long; Didn't Read)
To find the volume of a square-based pyramid, use the formula V = A(h/3), where V is the volume and A is the area of the base.
Collect the Essential Information
Both measurements must be made in the same units. Also, to use this formula, the height must be the distance from the uppermost vertex of the pyramid (its peak) straight down to the middle of the base, not the slant height from the peak of the pyramid to one of its lower vertexes. If you're given the slant height of the pyramid, it represents the hypotenuse of a right triangle formed by itself, the height of the pyramid, and 1/2 the length of the pyramid's base. Use the Pythagorean theorem, a^2 + b^2 = c^2, to find the height of the pyramid. In this case c is the slant height of the pyramid, a is 1/2 the length of the base, and b will be the height of the pyramid.
Find the Area of the Base
Multiply by h/3
You can use the same procedure to find the volume of a pyramid with a rectangular base, with one small modification: Instead of finding the area of the base by squaring one side of its length, you must find both the length and width of the base, then multiply them together to find the area of the base. So if the base of the pyramid measures 5 inches by 4 inches, the area of its base would be 20 inches squared.
Collect, measure or calculate the height of the pyramid and the length of one side along its base. Consider the example of a square pyramid where one side of the pyramid's base measures 5 inches, and the height of the pyramid is 6 inches.
Square the length of the pyramid's base or, in other words, multiply the length by itself. This gives you the area of the pyramid's base in square units. To continue the example, this would be 5 inches × 5 inches = 25 inches squared.
Multiply the area of the pyramid's base by the pyramid's height, then divide the answer by 3. The result is the volume of your pyramid, written in units cubed. To continue the example, you have 25 inches squared × 6 inches = 150. Divide this by three to get the pyramid's volume: 150 ÷ 3 = 50 inches cubed.