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10. Modeling in 2 & 3 Dimensions
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Geometry

10.02 Cross sections of solids

Lesson
Practice

Cross sections of solids

A solid is a term used when talking about a three-dimensional object.

By slicing through a solid, we produce a two-dimensional shape called a cross section. The figure created from a cross section depends on the orientation or angle of the intersecting plane.

Cross section

The two-dimensional shape made by slicing through a three-dimensional figure

Apex

The point (vertex) furthest from the base of an object

Exploration

  1. Draw at least two solid figures that would produce circular cross sections.
  2. Draw at least two solid figures that would produce triangular cross sections.
  3. Draw a solid figure that would produce a non-rectangular polygon cross section.
  4. What additional information would be useful in deciding what three-dimensional solid to draw from the cross section shape?

A solid may form many different shapes by taking different cross sections. In particular, knowing a cross section of a solid isn't enough information to uniquely determine the original solid. Many different solids can produce identical cross sections.

Cross sections are formed when taking a slice of a figure horizontally, vertically, or diagonally. The shapes that form from cross sections may be more or less obvious, like when we take a vertical slice of a cube versus a diagonal slice.

A cube showing a square cross section.

By taking a slice of a cube that is parallel to one of its sides, we form a cross section that is a square which is congruent to the sides of the cube.

A cube showing a triangle cross section with its vertices passing through the midpoints of three adjacent edges of the cube.

By taking a slice of a cube that passes through the midpoint of three edges, as if we were cutting off a corner of the cube, we form a cross section that is an equilateral triangle in shape.

Examples

Example 1

Find a cross section parallel to the base and identify the shape formed by the cross section.

A pentagonal pyramid.
Worked Solution
Create a strategy

Slice the shape parallel to the base.

A pentagonal pyramid cut by a plane parallel to its base.
Apply the idea

The cross section is the intersection of the shape and the plane. The cross section is a pentagon.

Example 2

Consider the following cross section sliced from a solid figure:

Draw two different figures that the cross section could have come from.

Worked Solution
Create a strategy

We can slice a solid horizontally, vertically, or diagonally in order to get a cross section. We can think of a few triangular-shaped solids that could produce the cross section shown.

Apply the idea
Two-dimensional representation of a three-dimensional pyramid with a triangular base. The pyramid is rendered within a square frame, which appears to be a window or a picture frame. The pyramid itself has a solid fill of light blue with a slightly darker outline, indicating its three-dimensional form through shading and perspective.

A rectangular pyramid cut vertically from its apex could produce the triangular cross section.

A triangular prism standing upright. The prism has a light blue fill with a darker blue outline enhancing its geometric shape. One of its vertical rectangular faces is bisected by a line, creating a mirrored effect and suggesting depth, while the base is represented with a dashed line, indicating the bottom edge that is not directly visible.

A cone cut vertically could produce the triangular cross section.

Reflect and check

A cube, like the image shown for the equilateral triangle in the lesson could be cut at a steeper angle to produce an isosceles triangle.

A two-dimensional representation of a three-dimensional geometric figure. Inside a transparent cube with a light grey outline, there is a solid, light blue tetrahedron (a pyramid with a triangular base). The tetrahedron is centrally placed and shares one of its edges with the cube, illustrating a spatial relationship between the two shapes. The cube's edges are drawn in full lines, while the tetrahedron is highlighted with a slightly darker outline.

Example 3

Consider the cylinder shown below:

A circular cylinder.

Identify three differently shaped cross sections, including at least one that comes from a diagonal slice.

Worked Solution
Create a strategy

Imagine slicing a plane through the cylinder vertically, horizontally, and diagonally at different points to produce differently-shaped cross sections.

Apply the idea

Three shapes that could be formed are a rectangle, an ellipse, and a circle.

A sequence of three two-dimensional illustrations representing three-dimensional geometric shapes intersecting with planes. Speak to your teacher for more information
Reflect and check

We could also slice a cylinder from one side down to its base, creating a semicircle.

A two-dimensional representation of a three-dimensional cylinder. It is light blue with a darker outline for definition. Inside the cylinder, a curved dashed line creates an elliptical shape at the base, suggesting a circular cut or the cylinder's cross-section.

Example 4

A 3D printer uses computer assistance to stack layers of material that make a three-dimensional shape. The printer creates an object out of several layers to create a physical model of a computer image. Shown below are the layers of a model that a 3D printer has created. What solid figure is created by the printer?

A series of seven circles in a horizontal row, increasing in size from left to right. Speak to your teacher for more information.
Worked Solution
Create a strategy

If a 3D printer is stacking the layers, we can imagine the cross sections being cut horizontally to the printing surface.

Apply the idea

The printer is printing a cone.

Idea summary

A single three-dimensional solid may have cross sections of different shapes, depending on how the solid is sliced (vertical, horizontal or diagonal).

Outcomes

G.DF.1

The student will create models and solve problems, including those in context, involving surface area and volume of rectangular and triangular prisms, cylinders, cones, pyramids, and spheres.

G.DF.1a

Identify the shape of a two-dimensional cross section of a three-dimensional figure.

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