G.3.9

Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

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Formative Assessments
Formative Assessments
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CPALMS: Students are asked to solve a problem that requires calculating the volume of a sphere.

Burning Sphere

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CPALMS: Students are asked to find the height of a square pyramid given the length of a base edge and its volume.

Louvre Pyramid

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CPALMS: Students are asked to write the formula for the volume of a sphere, explain what each variable represents, and label the variables on a diagram.

Sphere Formula

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CPALMS: Students are asked to write the formula for the volume of a cone, explain what each variable represents, and label the variables on a diagram.

Cone Formula

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CPALMS: Students are asked to solve a problem that requires calculating the volume of a cylinder.

Platinum Cylinder

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CPALMS: Students are asked to solve a problem that requires calculating the volume of a cone.

Sugar Cone

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CPALMS: Students are asked to write the formula for the volume of a cylinder, explain what each variable represents, and label the variables on a diagram.

Cylinder Formula

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CPALMS: Students are asked to write the formula for the volume of a pyramid, explain what each variable represents, and label the variables on a diagram.

Pyramid Formula

Virtual/Hands-on Manipulatives
Virtual/Hands-on Manipulatives
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This page gives students the opportunity to see a virtual animation for how the formula for finding the volume of a cone is derived. The simulation demonstrates the cone’s volume compared to that of a cylinder.
Formula Volume of a Cone by Math Warehouse
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Students can drag the orange dots on the virtual image in order to adjust the radius and/or the height of the cylinder and note how the formula is used to calculate the volume. The details can be adjusted to practice calculating volume and then checking the work.
Volume of a Cylinder by Math Open Resource
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Students drag orange dots on image of a cone in order to adjust the radius and height and note how volume changes. It explains how volume of cones and pyramids are calculated in a similar way, in addition to the cone’s relation to a cylinder.
Volume of a Cone by Math Open Resource
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Students can drag the orange dot on the virtual image in order to adjust the radius of the sphere and note how the formula is used in order to calculate the volume. The details can be hidden and radius can be adjusted to practice calculating volume and check their work.
Volume of a Sphere by Math Open Resource
Assessment Questions
Assessment Questions
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Videos of Expert Instruction
Videos of Expert Instruction
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In this video (approx. 8:00 min.), Sal uses pictorial examples to model and explain how to use the cylinder formulas volume is π r² h, and its surface area is 2π r h + 2π r² to solve problems.
Cylinder Volume and Surface Area by Khan Academy
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This video, approx.. 12:30 min. has quite a number of illustrations about area, surface area, and volume of solid figures (rectangular, triangular, and cones, cylinders, and spheres). Depending on the course and concepts, students could view parts and class discussion could follow.
Volume by Math Antics
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In this tutorial video (approx. 4:30 min), the instructor models how to solve a problem using the formula for volume of a cylinder. Relevant vocabulary is used and she explains how the base of the cylinder is associated with the area of a circle and its formula.
How do you find the volume of a cylinder?  by Virtual Nerd
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In this video (approx. 5:45 min), Sal uses pictorial examples to model and explain how a cone's volume formula V=1/3hπr² to solve a problem. He explains how this is related to the formula for volume of a cylinder.
Volume of a Cone by Khan Academy
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As students work with cylinders, they understand that in many cases, volume is the area of 2D objects stacked upon each other until they reach the desired height. A conceptual challenge students may encounter is that increasing the size of the radius has a greater impact on the volume of a cylinder than increasing the height, since the radius is squared (used twice) and the height is not. Requires Free Account.
Extending the Understanding of Volume of Prisms to Volume of Cylinders by LearnZillion
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In this video (approx. 2:35 min), Sal uses the formula for the volume of a sphere is V = 4/3 πr³ to solve an example problem with pictorial representation where the measurement of the diameter is given.
Volume of a Sphere by Khan Academy
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In this video (approx. 2:15 min), watch an animated demonstration of finding the volume of a cone and cylinder with the same base and height. There is an accompanying class activity where students estimate the relative volume of a cone and cylinder with the same base and height. They can test estimates with hands-on activity.
Volume of a Cone by PBS LearningMedia
Hook/Performance Tasks
Hook/Performance Tasks
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Illustrative Mathematics: With similarity being new to 8th grade, students may not recognize that it is needed in this context. There is opportunity for discussion about parts of the task involving percentage increase and volume.

Comparing Snow Cones

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Illustrative Mathematics: In this task, students should think of different ways the cylindrical containers can be set up in a rectangular box. In this process, they should realize that although some setups may seem different, they result in a box with the same volume.

Shipping Rolled Oats

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Illustrative Mathematics: The purpose of this task is to give students practice working the formulas for the volume of cylinders, cones and spheres, in an engaging context that provides and opportunity to attach meaning to the answers.

Flower Vases

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Illustrative Mathematics: Students work with volumes of cylinders, spheres and cones. The insight required increases as one moves across the glasses, from simple application of the formula for volume, to decomposition of volume into two, to calculating height.

Glasses

Real-World Application
Real-World Application
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Did you know that altering computer code can increase 3D printing efficiency? Check it out!
Mathematically Optimizing 3D Printing
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How many bottles of Coca Cola did they buy to fill that pool?
Coca-Cola Pool
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Michael McKinnon of Gaines Street Pies explains how when making pizza the volume is conserved but the surface area changes.
Volume and Surface Area of Pizza Dough
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Carbon can take many forms, including foam! Learn more about how geometry and the Monte Carlo Method is important in understanding it.
Carbon Foam and Geometry