Unit 5, Lesson 10
Cross-Sections and Volume
Geometry
Preparation
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, with 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.
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 applet because students will benefit from seeing the relationship in a dynamic way.
- Comprehend that if two solids of equal height 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).