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All courses Math III · G-GMD.4 37 of 55
Identify cross-sections of 3D objects and solids generated by rotating 2D objects.

Identify the cross-section formed when a prism is sliced parallel to its base

Problem
If a rectangular prism is cut parallel to its base, what cross-section is formed?
Prompt diagram for M3-037-A01-V01: a three-dimensional prism, oriented slicing plane, and highlighted intersection without its classification. Open full size
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Hint

Identify the shape of the prism's base before naming the cross-section.

A plane parallel to a prism's base creates a cross-section congruent to that base.

Solution walkthrough

01

Identify the base shape

\[\text{base}~\text{shape}~=~\text{rectangle}\]

The solid is a rectangular prism, so its bases are rectangles. The base shape is the shape to track because the cut is parallel to the base.

02

Use the parallel-slice rule

\[\text{slice}~\text{parallel}~\text{to}~\text{base}~->~\text{same}~\text{shape}~\text{as}~\text{base}\]

A plane parallel to a prism's base keeps the same outline as that base throughout the prism.

03

Name the cross-section

\[\text{cross}-\text{section}~\text{shape}~=~\text{rectangle}\]

Since the base is a rectangle, the flat shape made by the slice is also a rectangle.

04

Relate it to the base

\[\text{cross}-\text{section}~\text{congruent}~\text{to}~\text{rectangular}~\text{base}\]

For a prism, this parallel cross-section is not just the same type of polygon; it is congruent to the base because it is a same-shape layer of the prism.

05

State the answer

\[\text{The}~\text{cross}-\text{section}~\text{is}~a~\text{rectangle}~\text{congruent}~\text{to}~\text{the}~\text{base}.\]

The requested cross-section is a 2D rectangle, and it can be described more completely as a rectangle congruent to the base.

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Another way

  1. Think of the rectangular prism as a stack of congruent rectangles. A slice parallel to the stack's base picks out another rectangle.

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Common mistake

A common mistake is to answer with the 3D solid, such as rectangular prism, instead of the flat 2D cross-section. The cut creates a rectangle, not another prism.

Answer diagram for M3-037-A01-V01: the same slice with the exact cross-section name and extracted polygon.