Lesson 10

Cross Sections and Volume

Lesson Narrative

In this lesson, students develop the idea of oblique versus right solids. They analyze volumes of two prisms: one right and one oblique, but of equal height and with bases that have equal area. They conclude the volumes of the two prisms are equal. This leads to the introduction of Cavalieri’s Principle, or the idea that solids of equal height have the same volume if their cross sections have equal area at all heights. This concept is needed for understanding volumes of oblique solids and will also be used to develop the formula for the volume of a pyramid in upcoming lessons.

Students have the opportunity to look for and make use of structure (MP7) as they identify fundamental characteristics of these solids regardless of their obliqueness or cross-sectional shape.

One of the activities in this lesson works best when each student has access to devices that can run the GeoGebra applet because students will benefit from seeing the relationship in a dynamic way.

Learning Goals

Teacher Facing

  • Comprehend that if two solids are cut into cross sections by parallel planes, and the corresponding cross sections on each plane always have equal areas, then the two solids have the same volume (Cavalieri’s Principle).

Student Facing

  • Let’s look at how cross sections and volume are related.

Required Materials

Required Preparation

You will need a tall stack of index cards to demonstrate shifting the cross-sectional slices of a rectangular prism. You may consider also using stacks of CDs, coins, decks of cards, paper, or other items that are thin, flat, and congruent.

Devices are required for the digital version of the activity, Rectangular Prism, Shifted. The digital version is recommended for all classes over the paper and pencil version. Acquire devices that can run dynamic geometry software, one for every 2–3 students. If technology is not available, there is a paper and pencil alternative.

Learning Targets

Student Facing

  • I know that if two solids have equal-area cross sections at all heights, they have the same volumes.

CCSS Standards

Building On

Addressing

Building Towards

Glossary Entries

  • Cavalieri’s Principle

    If two solids are cut into cross sections by parallel planes, and the corresponding cross sections on each plane always have equal areas, then the two solids have the same volume.

  • oblique (solid)

    Prisms and cylinders are said to be oblique if when one base is translated to coincide with the other, the directed line segment that defines the translation is not perpendicular to the bases.

    A cone is said to be oblique if a line drawn from its apex at a right angle to the plane of its base does not intersect the center of the base. The same definition applies to pyramids whose bases are figures with a center point, such as a square or a regular pentagon.

  • right (solid)

    Prisms or cylinders are said to be right if when one base is translated to coincide with the other, the directed line segment that defines the translation is perpendicular to the bases.

    A cone is said to be right if a line drawn from its apex at a right angle to the plane of its base passes through the center of the base. The same definition applies to pyramids whose bases are figures with a center point, such as a square or a regular pentagon.