Experiment with the volume of two cylinders made from the same size paper. This interactive exercise focuses on using what you know about cylinders to make a prediction about their volume and then requires calculating the actual volume to see if your prediction was accurate.
Remember how to compute the volume of a cylinder or prism using the cross-sectional area and length (height) of the object? If the cross-sectional area is known and constant along the height, the volume calculation is easy. But, what if the cross-sectional area changes in a known manner along the line that is the height, like it does for a cone or pyramid? How could a single method in calculus be used to determine the volume of either of these types of solids?
This informational material will explain how to calculate the volume of special solid figures, like cones, by using cross-sections from the solid figure. The three-dimensional case of Cavalieri's Principle is introduced. There are corresponding videos available. Practice questions with a PDF answer key are provided.
Students will test their knowledge of calculating the volume of cones using cross-sections on a graph in this interactive.
How do you find the volume of a cone given its cross-section? Consider half the cross-section of the cone where the region is formed by the lines y = 0, x = 45 and the changing standard equation.
In this interactive, students will:
Use the Pythagorean theorem to unroll a cone and find its surface area with this interactive video from the School Yourself Geometry series.
Apply Cavalieri's principle to determine when two solids must definitely have the same volume with this interactive video from the School Yourself Geometry series.
How can you find the volume of prisms that aren't rectangular? Learn how with this interactive video from the School Yourself Geometry series.
Use the Pythagorean theorem and apply Cavalieri's principle to cones and cylinders to find the volume of a sphere with this interactive video from the School Yourself Geometry series.
Compare the volume of varied cylindrical glasses filled to different heights. This interactive exercise focuses on using what you know about cylinders to make a prediction about their volume and then requires calculating the actual volume to see if your prediction was accurate.
This resource is part of the Math at the Core: Middle School collection.
Use your problem-solving skills to find out if the pot will overflow when Dan adds meatballs to his pasta sauce. This interactive exercise focuses on using the volume equations for cylinders and spheres to figure out the multistep problem of how many meatballs it would take to fill the space left in the pot.
In school, you learn about shapes with sides and edges, but there are weird shapes out there (beyond our 3 dimensions) that defy our normal idea of geometry. QuanQuan and Jenny explain, knit, and 3D print their way through these strange shapes.
Students study the basic properties of two-dimensional and three-dimensional space, noting how ideas shift between the dimensions. They learn that general cylinders are the parent category for prisms, circular cylinders, right cylinders, and oblique cylinders, and study why the cross-section of a cylinder is congruent to its base. Next students study the explicit definition of a cone and learn what distinguishes pyramids from general cones, and see how dilations explain why a cross-section taken parallel to the base of a cone is similar to the base. Students revisit the scaling principle as it applies to volume and then learn Cavalieri’s principle, which describes the relationship between cross-sections of two solids and their respective volumes. This knowledge is all applied to derive the volume formula for cones, and then extended to derive the volume formula for spheres. Module 3 is a natural place to see geometric concepts in modeling situations. Modeling-based problems are found throughout Topic B and include the modeling of real-world objects, the application of density, the occurrence of physical constraints, and issues regarding cost and profit.
