In grade 7, students described the two-dimensional figures that result from slicing three-dimensional figures. Here, these concepts are revisited with some added complexity. Students analyze cross sections, or the intersections between planes and solids, by slicing three-dimensional objects. Next, they identify three-dimensional solids given parallel cross-sectional slices. In addition, they revisit solid geometry vocabulary terms from earlier grades: sphere, prism, cylinder, cone, pyramid, and faces.
Spatial visualization in three dimensions is an important skill in mathematics. Understanding the relationship between solids and their parallel cross sections will be critical to understanding Cavalieri’s Principle in later lessons. Cavalieri’s Principle will be applied to the development of the formula for the volume of pyramids and cones. Students use spatial visualization to make sense of three-dimensional figures and their cross sections throughout the lesson (MP1).
- Generate multiple cross sections of three-dimensional figures.
- Identify the three-dimensional shape resulting from combining a set of cross sections.
- Let’s analyze cross sections by slicing three-dimensional solids.
Obtain several cylindrical food items to cut with a plastic knife.
Devices are required for the digital version of the activity Slice That. If using the paper and pencil version, prepare various solids from clay or play dough, such as cubes, spheres, cones, and cylinders. Each group of 3-4 students should have access to a three-dimensional solid to analyze.
Alternatively, you might consider getting food items from the grocery store with interesting cross sections or three-dimensional foam solids from a craft store, and plastic knives to slice the solids.
- I can identify the three-dimensional shape that generates a set of cross sections.
- I can visualize and draw multiple cross sections of a three-dimensional figure.
axis of rotation
A line about which a two-dimensional figure is rotated to produce a three-dimensional figure, called a solid of rotation. The dashed line is the axis of rotation for the solid of rotation formed by rotating the green triangle.
A cone is a three-dimensional figure with a circular base and a point not in the plane of the base called the apex. Each point on the base is connected to the apex by a line segment.
The figure formed by intersecting a solid with a plane.
A cylinder is a three-dimensional figure with two parallel, congruent, circular bases, formed by translating one base to the other. Each pair of corresponding points on the bases is connected by a line segment.
Any flat surface on a three-dimensional figure is a face.
A cube has 6 faces.
A prism is a solid figure composed of two parallel, congruent faces (called bases) connected by parallelograms. A prism is named for the shape of its bases. For example, if a prism’s bases are pentagons, it is called a “pentagonal prism.”
A pyramid is a solid figure that has one special face called the base. All of the other faces are triangles that meet at a single vertex called the apex. A pyramid is named for the shape of its base. For example, if a pyramid’s base is a hexagon, it is called a “hexagonal pyramid.”
solid of rotation
A three-dimensional figure formed by rotating a two-dimensional figure using a line called the axis of rotation.
The axis of rotation is the dashed line. The green triangle is rotated about the axis of rotation line to form a solid of rotation.
A sphere is a three-dimensional figure in which all cross-sections in every direction are circles.