Master finding volumes of pyramids and cones. Math 10-C Alberta mathematics curriculum.
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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