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11.02 Cross sections of solids

Lesson

Concept summary

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 plane during the cross-section.

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.

Notice that 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.

Worked examples

Example 1

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

A pentagonal pyramid.

Solution

Slice the shape parallel to the base.

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

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

A pentagon.

Example 2

What shape could possibly give the following cross-section?

A pentagon with two adjacent right interior angles.

Approach

A two dimensional shape has depth of zero. To construct a solid with the cross-section, we can increase the depth to greater than zero.

Solution

Making the depth of the cross-section greater than zero.

A prism with pentagonal bases and resembles a house. Each pentagonal base has two right interior angles.

Reflection

There are multiple ways to construct a shape from its cross-section.

Outcomes

MA.912.GR.4.1

Identify the shapes of two-dimensional cross-sections of three-dimensional figures.

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