1. Introduction
Think about the dissection arguments used to develop the area of a circle formula. These same arguments can be used to develop the volume formulas for cylinders, pyramids, cones, and spheres. You have already used volume formulas for prisms, cylinders, pyramids, cones, and spheres, but how are the formulas derived? In this lesson, you will learn how to find the volumes of cylinders, pyramids, cones, and spheres. As part of learning the formulas, you will see proofs of why the formulas work for those objects. The real world is filled with these objects. Using the formulas of volume for these objects expands your problem-solving skills.
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

2. Key Concepts
The formula for finding the volume of a prism is V = length • width • height. This can also be shown as V = area of base • height.
Remember to use cubic units or volume measures when calculating volume. Some examples are cubic feet (ft3), cubic meters (m3), liters (L), and gallons (gal).
If the dimensions of an object are all multiplied by a scale factor, k, then the volume will be multiplied by a scale factor of k3. For example, if the dimensions are enlarged by a scale factor of 5, then the volume will be enlarged by a scale factor of 125.
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

3. Key Concepts, continued
Cylinders
A cylinder has two bases that are parallel. This is also true of a prism.
Bonaventura Cavalieri, an Italian mathematician, formulated Cavalieri’s Principle. This principle states that the volumes of two objects are equal if the areas of their corresponding cross sections are in all cases equal.
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

4. Key Concepts, continued
This principle is illustrated by the diagram below. A rectangular prism has been sliced into six pieces and is shown in three different ways.
The six pieces maintain their same volume regardless of how they are moved.
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

5. Key Concepts, continued
Cavalieri’s Principle describes how each piece is a thin slice in the plane of the prism.
If each thin slice in each object has the same area, then the volumes of the objects are the same.
The following diagram shows a prism, a prism at an oblique angle, and a cylinder.
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

6. Key Concepts, continued
The three objects meet the two criteria of Cavalieri’s Principle. First, the objects have the same height. Secondly, the areas of the objects are the same when a plane slices them at corresponding heights. Therefore, the three objects have the same volume.
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

7. Key Concepts, continued
A square prism that has side lengths of will have a base area of on every plane that cuts through it. The same is true of a cylinder, which has a radius, r.
The base area of the cylinder will be This shows how a square prism and a cylinder can have the same areas at each plane.
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

8. Key Concepts, continued
Another way of thinking about the relationship of a polygonal prism and the cylinder is to remember the earlier proof about the area of a circle.
A cylinder can be thought of as a prism with an infinite number of sides.
The diagram on the next slide shows a polygonal prism with 200 sides.
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

9. Key Concepts, continued
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

10. Key Concepts, continued
The area of its base can be calculated by dividing it into quadrilaterals and triangles, but its base is approaching the limit of being a circle.
If the base area and the height of a prism and a cylinder are the same, the prism and cylinder will have the same volume.
The formula for finding the volume of a cylinder is
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

11. Key Concepts, continued
Pyramids
A pyramid is a solid or hollow polyhedron object that has three or more triangular faces that converge at a single vertex at the top; the base may be any polygon.
A polyhedron is a three-dimensional object that has faces made of polygons.
A triangular prism can be cut into three equal triangular pyramids.
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

12. Key Concepts, continued
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

13. Key Concepts, continued
Triangular prisms
Corresponding triangular pyramids
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

14. Key Concepts, continued
A cube can be cut into three equal square pyramids.
This dissection proves that the volume of a pyramid is one-third the volume of a prism:
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

15. Key Concepts, continued
Cones
A cone is a solid or hollow object that tapers from a circular base to a point.
A cone and a pyramid use the same formula for finding volume.
This can be seen by increasing the number of sides of a pyramid.
The limit approaches that of being a cone.
A pyramid with 100 sides follows on the next slide. With such a large number of sides, it looks like a cone.
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

16. Key Concepts, continued
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

17. Key Concepts, continued
The formula for the volume of a cone is
Cavalieri’s Principle shows how pyramids and cones have the same volume.
The diagram that follows shows cross sections of areas with the same planes.
Each object has the same area at each cross section.
Therefore, the volumes of both objects are the same.
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

18. Key Concepts, continued
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

19. Key Concepts, continued
Spheres
A sphere is a three-dimensional surface that has all its points the same distance from its center.
The volume of a sphere can be derived in several ways, including by using Cavalieri’s Principle and a limit process.
The formula for the volume of a sphere is
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

20. Common Errors/Misconceptions
confusing formulas; for example, using the prism volume formula for a pyramid or a cone
calculating the wrong base area for a pyramid due to not noticing the shape of the pyramid’s base (e.g., triangle, square)
performing miscalculations with the number
making arithmetic errors when dealing with the formulas, especially with fractions
forgetting to express a calculation in the correct units for volume (e.g., m3, liters)
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

21. Guided Practice Example 2
Find the dimensions for a cone that has the same volume as a pyramid of the same height as the cone. Both the cone and the pyramid have a height of 2 meters. The volume of the pyramid is 3 cubic meters. A cone and a pyramid both taper to a point or vertex at the top. The “slant” of the taper is linear, meaning it is a straight line. The dimensions of both the cone and the pyramid change at a constant rate from base to tip.
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

22. Guided Practice: Example 2, continued
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

23. Guided Practice: Example 2, continued
Cavalieri’s Principle states that the pyramid and cone will have the same volume if the area of each cross section of a plane is the same at every height of the two objects. This means that if the cone and pyramid have bases of equal area, then their volumes will also be equal.
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

24. Guided Practice: Example 2, continued
Set up an equation to find the radius of the cone.
The volume of the pyramid is 3 m3.
Both objects must have bases of equal area to have the same volume.
The area of the base of the pyramid can be found by solving for B in the formula for the volume of a pyramid.
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

25. Guided Practice: Example 2, continued
The area of the base of the pyramid is 4.5 square meters.
Substitute the volume and the height of the pyramid.
Simplify.
Multiply both sides by .
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

26. Guided Practice: Example 2, continued
Set up the equation.
Substitute the value found for the area of the base of the pyramid.
Divide both sides by .
Take the square root of both sides.
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

27. ✔ Guided Practice: Example 2, continued
The radius of the cone is approximately meters. ✔
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

28. Guided Practice: Example 2, continued
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

29. Guided Practice Example 5
A teenager buying some chewing gum is comparing packages of gum in order to get the most gum possible. Each package costs the same amount. Package 1 has 20 pieces of gum shaped like spheres. Each piece has a radius of 5 mm. Package 2 has 5 pieces of gum shaped like spheres. Each piece has a radius of 10 mm. Which package should the teenager buy? Round to the nearest millimeter.
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

30. Guided Practice: Example 5, continued
Find the volume of a piece of gum in package 1 by using the volume formula for a sphere.
Formula for volume of a sphere
Substitute 5 for r.
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

31. Guided Practice: Example 5, continued
Multiply the volume of the single piece of gum in package 1 by 20 to get the total volume of the gum in the package.
(524)(20) = 10,480 mm3
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

32. Guided Practice: Example 5, continued
The radius of each piece of gum in package 2 is 10 mm. Find the volume of a piece of gum in package 2 by using the volume formula for a sphere.
Formula for volume of a sphere
Substitute 10 for r.
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

33. Guided Practice: Example 5, continued
Multiply the volume of a single piece of gum in package 2 by 5 to get the total volume of gum in the package.
(4189)(5) = 20,945 mm3
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

34. ✔ Guided Practice: Example 5, continued
Compare the volumes of the two packages to determine the best purchase.
10,480 < 20,945
The teenager should buy package 2. ✔
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres

35. Guided Practice: Example 5, continued
6.5.2: Volumes of Cylinders, Pyramids, Cones, and Spheres