Discovering Volume Formulas
1. Pick a two-dimensional shape and make 5 copies of it.
2. Stack your shapes and calculate the volume of this new three-dimensional object. Based
on your example, think of a general formula for the volume of the shape formed.
Most likely at least one student will start with a rectangle, square, and circle and then they get
formulas for the volume of a rectangular prism (length∙width∙height), cube (side3), and cylinder
(𝜋𝑟 2 ℎ).
3. Share your shape and volume formula with the rest of the class.
4. Look at this model where the cube is split into three equal shapes so that each has a full
side of the cube. Find the volume of each one shape.
They should use the fact that the volume of a cube is the side cubed to calculate the volume of
1
each object as 3 the side cubed. At this point, the objects can be referred to as pyramids.
5. How would the volume formula change if instead of having a square base the pyramid had a
rectangular base? Come up with a general formula for a pyramid with any shape base. Use your
formula to find the volume of a cone.
1
The formula for a pyramid with a rectangular base is 3 ∙ length ∙ width ∙ height. The formula
1
for a pyramid with any shape base is 3 ∙ area of base ∙ height. The formula for the volume of a
1
cone, which has a circular base, is 3 𝜋 r2 h.