In this lesson students start working with cones, and learn that the volume of a cone is \(\frac13\) the volume of a cylinder with a congruent base and the same height. First, students learn a method for quickly sketching a cone, and the meaning of the radius and height of a cone. Then they watch a video (or if possible, a live demonstration) showing that it takes three cones of water to fill a cylinder with the same radius and height. At this point, it is taken as a mysterious and beautiful fact that the volume of a cone is one third the volume of the associated cylinder. A proof of this fact requires mathematics beyond grade level.
Students write the volume of a cone given a specific volume of a cylinder with the same base and height, and vice versa. Then they use the formula for the volume of a cylinder learned in previous lessons to write the general formula \(V= \frac13\pi r^2 h\) for the volume, \(V\), of a cone in terms of its height, \(h\), and radius, \(r\). Finally, students practice computing the volumes of some cones. There are opportunities for further practice in the next lesson.
- Calculate the volume of a cone and cylinder given the height and radius, and explain (orally) the solution method.
- Compare the volumes of a cone and a cylinder with the same base and height, and explain (orally and in writing) the relationship between the volumes.
Let’s explore cones and their volumes.
For the Which Has a Larger Volume activity, it is suggested that students have access to geometric solids.
During the From Cylinders to Cones activity, students will need to view a video. Alternatively, do a demonstration with a cone that could be filled with water and poured into a cylinder.
- I can find the volume of a cone in mathematical and real-world situations.
- I know the formula for the volume of a cone.
base (of a prism or pyramid)
The word base can also refer to a face of a polyhedron.
A prism has two identical bases that are parallel. A pyramid has one base.
A prism or pyramid is named for the shape of its base.
A cone is a three-dimensional figure like a pyramid, but the base is a circle.
A cross section is the new face you see when you slice through a three-dimensional figure.
For example, if you slice a rectangular pyramid parallel to the base, you get a smaller rectangle as the cross section.
A cylinder is a three-dimensional figure like a prism, but with bases that are circles.
A prism is a type of polyhedron that has two bases that are identical copies of each other. The bases are connected by rectangles or parallelograms.
Here are some drawings of prisms.
A pyramid is a type of polyhedron that has one base. All the other faces are triangles, and they all meet at a single vertex.
Here are some drawings of pyramids.
A sphere is a three-dimensional figure in which all cross-sections in every direction are circles.
The surface area of a polyhedron is the number of square units that covers all the faces of the polyhedron, without any gaps or overlaps.
For example, if the faces of a cube each have an area of 9 cm2, then the surface area of the cube is \(6 \boldcdot 9\), or 54 cm2.
Volume is the number of cubic units that fill a three-dimensional region, without any gaps or overlaps.
For example, the volume of this rectangular prism is 60 units3, because it is composed of 3 layers that are each 20 units3.
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