Lesson 3 Take Another Spin Solidify Understanding

Learning Focus

Calculate the volume of solids of revolution that can be approximated by cylinders and portions of cones.

How do I find the volume of a solid of revolution that isn’t a cylinder or part of a cone?

How can I approximate the volume of a solid of revolution that has a curved silhouette like a vase or a bottle?

Open Up the Math: Launch, Explore, Discuss

The trapezoid shown is revolved about the -axis to form a frustum (e.g., bottom slice) of a cone.


Draw a sketch of the 3-D object formed by rotating the trapezoid about the -axis.

Quadrilateral with points (0,0), (0,2), (8,0), (7,2)x–5–5–5555y555101010151515202020000


Find the volume of the object formed. Explain how you used the diagram to help you find the volume.

You have used the formulas for cylinders and cones in your work with solids of revolution. Sometimes a solid of revolution cannot be decomposed exactly into cylinders and cones. We can approximate the volume of solids of revolution whose cross-sections include curved edges by replacing them with line segments.


The following diagram shows the cross-section of a flower vase. Approximate the volume of the vase by using line segments to approximate the curved edges. (Show the line segments you used to approximate the figure on the diagram.)

Cross section with multiple points indicatedx555101010y–5–5–5555000


Describe and carry out a strategy that will improve your approximation for the volume of the vase.

Ready for More?

Your strategy for approximating the volume of the vase may have overestimated the volume on some intervals, and underestimated it on other intervals. Therefore, it may be difficult to know if your final estimate is too large or too small.

Devise a strategy for finding the volume of the vase where each section underestimates the volume of the vase, and a strategy where each section overestimates the volume of the vase. You will then know that the actual volume of the vase lies between these two estimates.

Based on all of your approximations, what would you guess for the actual volume of the vase? What might you do to improve your guess?


Strategic ideas for approximating the volume of a solid of revolution:

Adding Notation, Vocabulary, and Conventions

Defining frustum as the “bottom slice” of a cone is not very precise and needs to be revised and refined.

Our definition of a frustum:

Our volume formula for a frustum:


  • frustum
  • Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned how to approximate the volume of a solid of revolution like a vase whose silhouette contained curves, rather than straight lines. By decomposing the shape into smaller pieces, we could approximate the volume of each piece using formulas for cylinders, cones, and frustums.



A fair two-sided coin is flipped and then a fair six-sided number cube is tossed.


What is the probability of a head on the coin and an odd number on the cube?


How many possible outcomes are there for these two events?


Represent the sample space.


Find the measure of each angle in the triangle.

Triangle ABC with Angle A=2x 28, Angle C=4x-15, Angle B=x 13