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.
Be sure to view the activity, Meatballs: Volumes of Spheres and Cylinders - Activity (found under Support Materials for Teachers), to use with the video.
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.
Volume is the measure of how much space there is within a three-dimensional object (one with length, width, and height). Watch the video for an explanation of the formula for volume.
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.
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.
Uncover the secret behind how a square-wheeled tricycle can work at the National Museum of Mathematics. This interactive exercise focuses on working with the radius of various circles to find the circumference and area as well as challenging you to find the distance a square wheel travels around the track.
In Module 5, Topic B, students apply their knowledge of volume from previous grade levels (5.MD.C.3, 5.MD.C.5) to the learning of the volume formulas for cones, cylinders, and spheres (8.G.C.9). First, students are reminded of what they already know about volume, that volume is always a positive number that describes the hollowed-out portion of a solid figure that can be filled with water. Next, students use what they learned about the area of circles (7.G.B.4) to determine the volume formulas of cones and cylinders. In each case, physical models will be used to explain the formulas, first with a cylinder seen as a stack of circular disks that provide the height of the cylinder. Students consider the total area of the disks in three dimensions understanding it as the volume of a cylinder. Next, students make predictions about the volume of a cone that has the same dimensions as a cylinder. A demonstration shows students that the volume of a cone is one-third the volume of a cylinder with the same dimension, a fact that will be proved in Module 7. Next, students compare the volume of a sphere to its circumscribing cylinder (i.e., the cylinder of dimensions that touches the sphere at points but does not cut off any part of it). Students learn that the formula for the volume of a sphere is two-thirds the volume of the cylinder that fits tightly around it. Students extend what they learned in Grade 7 (7.G.B.6) about how to solve real-world and mathematical problems related to volume from simple solids to include problems that require the formulas for cones, cylinders, and spheres.
In Module 7, Topic D, students learn that radical expressions naturally arise in geometry, such as the height of an isosceles triangle or the lateral length of a cone. The Pythagorean Theorem is applied to three-dimensional figures in Topic D as students learn some geometric applications of radicals and roots (8.G.B.7). In order for students to determine the volume of a cone or sphere, they must first apply the Pythagorean Theorem to determine the height of the cone or the radius of the sphere. Students learn that truncated cones are solids obtained by removing the top portion above a plane parallel to the base. Students know that to find the volume of a truncated cone they must access and apply their knowledge of similar figures learned in Module 3. Their work with truncated cones is an exploration of solids that is not formally assessed. In general, students solve real-world and mathematical problems in three dimensions in Topic D (8.G.C.9). For example, now that students can compute with cube roots and understand the concept of rate of change, students compute the average rate of change in the height of the water level when water is poured into a conical container at a constant rate. Students also use what they learned about the volume of cylinders, cones, and spheres to compare volumes of composite solids.