- Let’s analyze cross sections by slicing three-dimensional solids.
2.1: Slice This
Imagine slicing a cylinder with a straight cut. The flat surface you sliced along is called a cross section. Try to sketch all the possible kinds of cross sections of a cylinder.
2.2: Slice That
The triangle is a cross section formed when the plane slices through the cube.
- Sketch predictions of all the kinds of cross sections that could be created as the plane moves through the cube.
- The 3 red points control the movement of the plane. Click on them to move them up and down or side to side. You will see one of these movement arrows appear. Sketch any new cross sections you find after slicing.
Delete the cube and build another solid by following the directions in its Tooltip. Make predictions about the the kinds of cross sections that could be created if the plane moves through the solid. Move your plane to confirm.
2.3: Stack ‘Em Up
Each question shows several parallel cross-sectional slabs of the same three-dimensional solid. Name each solid.
3D-printers stack layers of material to make a three-dimensional shape. Computer software slices a digital model of an object into layers, and the printer stacks those layers one on top of another to replicate the digital model in the real world.
- Draw 3 different horizontal cross sections from the object in the image.
- The layers can be printed in different thicknesses. How would the thickness of the layers affect the final appearance of the object?
- Suppose we printed a rectangular prism. How would the thickness of the layers affect the final appearance of the prism?
In earlier grades, you learned some vocabulary terms about solid geometry: A sphere is the set of points in three-dimensional space the same distance from some center. A prism has two congruent faces (or sides) that are called bases. The bases are connected by parallelograms. A cylinder is like a prism except the bases are circles. A pyramid has one base. The remaining faces are triangles that all meet at a single vertex. A cone is like a pyramid except the base is a circle.
We often analyze cross sections of solids. A cross section is the intersection of a solid with a plane, or a two-dimensional figure that extends forever in all directions. For example, some cheese is sold in cylindrical blocks. If you stand the cheese on end and slice vertically, you will get a rectangle, as shown. This rectangle is a cross section of the cylinder.
Here are 3 more examples of cross sections created by intersecting a plane and a cylinder.
If you wanted to serve your cylindrical cheese at a party, you might cut it into several pieces, like this. The pieces are thin cylinders. They are like cross sections, but they are three-dimensional. All the cuts were made parallel to one another. By looking at the slices, or by stacking them up, you could figure out that the original shape of the cheese was a cylinder.
What if another cheese plate contained slices whose radii got bigger to a maximum size and then got smaller again? The cheese was probably in the shape of a sphere. A sphere has circular cross sections. The size of the circular cross sections increases as you get closer to the center of the sphere, then decreases past the center.
- 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.
- cross section
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.