Volumes of Cylinders, Pyramids, Cones, and Spheres by Matt Mondares

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Video Introduction of Cylinders, Cones, and Spheres comparing and illustrating the peculiar qualities of shapes and their respective volumes.

Key Questions and Objectives

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How do the volumes of complex 3D shapes differ in quantities and calculations?
How can we make sense of the volumetric equations in relation to their 3D shapes?

Key Objective 1: Discerning Volumes

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Briefly explain that the volume of a 3D object measures the area inside the object and is related to calculating the area of 2D objects, such as rectangles and triangles.

Review the volumetric formulas for rectangular and triangular prisms and provide a few open-ended examples for guided practice.

We outline the volumetric formulas for cylinders, pyramids, cones, and spheres for a class discussion. We then dissect student reasoning and strategies for remembering and understanding 3D basic object volumes.

Students then solve word problems independently for rectangular and triangular prisms.

Finally, we discuss Cavalieri's principle, which states that the volumes of two or more solids are equal when they share the same areas and lengths for their cross-sections and heights, respectively using the volume of a sphere as an example.

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Problem 1: Open-ended

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What can the length of the edges be for a rectangular prism with a volume of 100 cm^3?

Problem 2 Open-ended

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What base length, length, and height do we need to calculate the volume of a triangular prism to equal at least 20 cubic inches?

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Write your answer in "cubic inches" as the unit.

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Students are tasked with creating one concrete problem per 3D shape independently or collaboratively, employing several previously discussed strategies, and representing multiple open-ended approaches, solutions, and representations to test another student with their questions.

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Once we have walked through concrete examples through guided practice, students are instructed to create increasingly challenging, abstract word problems. The bonus question challenges students to employ strategies and reasoning independently or collaboratively to come up with solutions to solve the compound figure volume. The instructor mediates independent practice with assistance and academic interventions to address the diverse needs of learners. Finally, the instructor performs guided practice, sharing the solution to the compound figure problem.

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Given a radius of 4 in, calculate the volume of a sphere. Use pi = 3.14 and round your answer to the nearest hundredth.

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An Egyptian pyramid has a base measuring 500 m per edge length with a height of 800 m. Calculate the volume using appropriate units to express your answer.

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A tepee tent has a circular base with a diameter of 2 m. A pole holds up the tent that stands 2 m tall. Calculate the volume inside the tent.

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A soda can has a base that measures 3 inches in diameter while its height is 5 inches. Calculate the amount of liquid the container can hold.

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BONUS QUESTION! Work independently or collaboratively in small groups to devise strategies to solve the presented problem. You must explain your reasoning, not just provide an answer.

The following questions represent examples for students to scaffold their created math problems on.

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Use the following volumetric equations to create one problem per 3D shape. Refer to concrete examples below as reference.

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Use the following concrete examples as reference to create your own word problems:

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