Lesson 11: The Volume Formula of a Pyramid and Cone
Students use Cavalieri’s principle and the cone cross-section theorem to show that a general pyramid or cone has volume 13Bh, where B is the area of the base and h is the height by comparing it with a right rectangular pyramid with base area B and height h.
The Exploratory Challenge is debriefed in the first discussion. The Exploratory Challenge and Discussion are the springboard for the main points of the lesson: (1) explaining why the formula for finding the volume of a cone or pyramid includes multiplying by one-third and (2) applying knowledge of the cone cross-section theorem and Cavalieri’s principle from previous lessons to show why a general pyramid or cone has volume 13area of baseheight. Provide manipulatives to students, 3 congruent pyramids and 6 congruent pyramids, to explore the volume formulas in a hands-on manner; that is, attempt to construct a cube from 3 congruent pyramids or 6 congruent pyramids such as in the image below.
Exploratory Challenge (5 minutes)
Exploratory Challenge * Use the provided manipulatives to aid you in answering the questions below. a. i. What is the formula to find the area of a triangle?
ii. Explain why the formula works.
The formula works because the area of a triangle is half the area of a rectangle with the same base and same height. * b. iii. What is the formula to find the volume of a triangular prism?
For base area B and height of prism h,
* iv. Explain why the formula works.
A triangular prism is essentially a stack of congruent triangles. Taking the area of a triangle, repeatedly, is like multiplying by the height of the prism. Then the volume of the prism would be the sum of the areas of all of the congruent triangles, which is