Activity 6.3.3 Prisms and Cylinders Postulates for Volume In elementary or middle school you may have learned to find the volume of a rectangular prism, by counting the number of unit cubes that fit in the solid. find the volume of this prism: Let’s Find the area of its base (B) in square units. Find the height (h) of the prism. Find the volume (V) of the prism. Explain why the formulas V = and = both work for rectangular length width height V Bh prisms. We prefer the formula = since as we will show it applies to every prism. V Bh Recall from Unit 3 the Rectangle Area Postulate: If is the base and is the height of a rectangle, then Rectangle Area Postulate: b h = Area bh. By analogy we have a postulate for volume: Rectangular Prism Volume Postulate: If is the area of the base and is the height of a right B h rectangular prism, then Volume = Bh and both refer to something called the Explain the difference between b “B” “b” “base.” in the Rectangle Area postulate and B in the Rectangular Prism Volume Postulate. To continue our work with prisms we need to you discovered make Cavalieri’s Principle, which in Activity 6.3.2 into another postulate: Postulate: If two solid figures lie between two parallel planes and every plane Cavalieri’s parallel to these two planes intersects both solids in cross-sections of equal area, then the two solid figures have equal volumes. Triangular Prisms We can now show that the formula = applies to right triangular prisms as well as right V Bh rectangular prisms.
Activity 6.3.3 Prisms and Cylinders Postulates for Volume In elementary or middle school you may have learned to find the volume of a rectangular prism, by counting the number of unit cubes that fit in the solid. Let’s find the volume of this prism: Find the area of its base (B) in square units. Find the height (h) of the prism. Find the volume (V) of the prism. Explain why the formulas V = length width height and V = Bh both work for rectangular prisms. We prefer the formula V = Bh since as we will show it applies to every prism. Recall from Unit 3 the Rectangle Area Postulate: Rectangle Area Postulate: If b is the base and h is the height of a rectangle, then Area = bh. By analogy we have a postulate for volume: Rectangular Prism Volume Postulate: If B is the area of the base and h is the height of a right rectangular prism, then Volume = Bh “B” and “b” both refer to something called the “base.” Explain the difference between b in the Rectangle Area postulate and B in the Rectangular Prism Volume Postulate. To continue our work with prisms we need to make Cavalieri’s Principle, which you discovered in Activity 6.3.2 into another postulate: Cavalieri’s Postulate: If two solid figures lie between two parallel planes and every plane parallel to these two planes intersects both solids in cross-sections of equal area, then the two solid figures have equal volumes. Triangular Prisms We can now show that the formula V = Bh applies to right triangular prisms as well as right rectangular prisms.
Activity_6_3_3.docx