Discover why a pyramid is always exactly one-third the volume of the prism with the same base and height, and meet the four volume formulas that flow from this single idea.
Lesson 1.4 of Measurement in Math 10C: Alberta curriculum lessons.
Two Related Ideas. Volume is the amount of space an object occupies, measured in cubic units such as cm^3 or m^3.
Capacity is how much a container can hold, whether solid, liquid, or gas. Volume and capacity use the same cubic units.
The One-Third Rule. Here is the insight that makes this whole lesson click:
A pyramid is exactly (1)/(3) of the prism with the same base and height.
A cone is exactly (1)/(3) of the cylinder with the same base and height.
Picture filling a pyramid-shaped cup with water and pouring it into a prism-shaped box with the same base and height. It would take exactly 3 pours to fill the box. That is where the (1)/(3) comes from.
Memory Tip. You only need to remember the prism and cylinder formulas.
To get the pyramid or cone formula, place (1)/(3) in front of the matching prism or cylinder formula.
Same base, same height, one-third the volume.
Base Area Formulas. In the formula V = (1)/(3)Ah, the value of A depends on the shape of the base.
Rectangular base: A = l × w
Square base: A = s^2
Triangular base: A = (1)/(2) × b × h_base
Always find A as its own step first, then multiply by h and (1)/(3).
Determine the volume of a right rectangular pyramid with base dimensions 3.6 m by 4.7 m and a height of 6.9 m. Round to the nearest cubic metre.
Solution. Step 1: Find the base area. The base is a rectangle:
A = 3.6 × 4.7 = 16.92 m^2
Step 2: Apply the pyramid formula V = (1)/(3)Ah:
V = (1)/(3) × 16.92 × 6.9
V = (1)/(3) × 116.748
V = 38.916 m^3
Answer: V ≈ 39 m^3
Strategy Tip. Find the base area as its own step first, then multiply by the height and (1)/(3).
Breaking it into steps makes mistakes easier to catch.
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