Cross-Sections of 3D Figures Math Example 5

Follow the full solution, then compare it with the other examples linked below.

Example 5

hard
A rectangular prism (box) has dimensions 44 cm ×6\times 6 cm ×8\times 8 cm. A plane cuts through the prism diagonally, connecting midpoints of the four longest edges (the edges of length 8 cm). Describe and find the area of the resulting cross-section.

Solution

  1. 1
    Step 1: The four edges of length 8 cm are the four vertical edges of the box. Their midpoints are at height 4 cm. The cutting plane passes through these four midpoints.
  2. 2
    Step 2: The four midpoints form a rectangle. The distances between adjacent midpoints are the widths of the base: 4 cm and 6 cm. So the cross-section is a rectangle.
  3. 3
    Step 3: The cross-section is a horizontal rectangle (since all four midpoints are at the same height) with dimensions 44 cm ×6\times 6 cm.
  4. 4
    Step 4: Area of cross-section =4×6=24= 4 \times 6 = 24 cm².

Answer

The cross-section is a 44 cm ×6\times 6 cm rectangle with area 2424 cm².
When the cutting plane is parallel to the base and passes through the midpoints of the vertical edges, it produces a rectangle congruent to the base cross-section. The midpoints all lie at the same height, so the cross-section is horizontal and has the same width and length as the prism's base.

About Cross-Sections of 3D Figures

A cross-section is the flat, two-dimensional shape revealed when a plane cuts through a three-dimensional solid. For example, slicing a cylinder parallel to its base gives a circle, while slicing it at an angle gives an ellipse.

Learn more about Cross-Sections of 3D Figures →

More Cross-Sections of 3D Figures Examples

Example 1 easy

A horizontal plane cuts through the middle of a right circular cone (parallel to the base). What 2D

Example 2 medium

Identify and describe the cross-sections formed when a plane cuts a cube in the following ways: (a)

Example 3 hard

A regular hexagonal prism is cut by a plane perpendicular to its bases that passes through two oppos

Example 4 easy

A plane cuts through a sphere. What shape is the cross-section, and when is it the largest possible

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