As they did with cylinders in a previous lesson, students in this lesson use the formula \(V=\frac13 \pi r^2 h\) to find the radius or height of a cone given its volume and the other dimension. Then they apply their understanding about the volumes of cylinders and cones to decide which popcorn container and price offers the best deal. Depending on the amount of guidance students are given, this last activity can be an opportunity to explain their reasoning and critique the reasoning of others (MP3).
- Calculate the value of one dimension of a cylinder, and explain (orally and in writing) the reasoning.
- Compare volumes of a cone and cylinder in context, and justify (orally) which volume is a better value for a given price.
- Create a table of dimensions of cylinders, and describe (orally) patterns that arise.
Let’s figure out the dimensions of cones.
- I can find missing information of about a cone if I know its volume and some other information.
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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