Cross-Sections of 3D Figures

Imagine taking a straight knife and slicing perfectly through a block of cheese. When you pull the two pieces apart, the shape of the newly exposed flat surface is called a cross-section.

In geometry, a cross-section is the two-dimensional ($2\text{D}$) shape that is created when a three-dimensional ($3\text{D}$) figure is sliced by a flat plane. The shape of the cross-section depends entirely on the $3\text{D}$ object and the angle of the slice.

Slicing Common 3D Figures

Let's look at the different shapes you can create by slicing standard $3\text{D}$ figures in various ways:

1. The Cube

A cube is made up of six identical square faces.

• Horizontal or Vertical Slice (Parallel to a face): Slicing straight across or straight down gives you a square.
• Diagonal Slice (Through opposite edges): Slicing from a top edge to an opposite bottom edge gives you a rectangle.
• Corner Slice: Slicing off just a corner of the cube gives you a triangle.

2. The Cylinder

A cylinder has two circular bases and a curved surface (like a soup can).

• Horizontal Slice (Parallel to the base): Slicing straight across gives you a circle.
• Vertical Slice (Perpendicular to the base): Slicing straight down from the top base to the bottom base gives you a rectangle.
• Angled Slice: Slicing at an angle through the curved side gives you an ellipse (an oval shape).

3. The Square-Based Pyramid

A pyramid has a square base and four triangular faces that meet at a top point (the apex).

• Horizontal Slice (Parallel to the base): Slicing straight across gives you a smaller square.
• Vertical Slice (Through the apex): Slicing straight down from the top point to the base gives you a triangle.
• Angled Slice: Slicing at an angle through the sides can give you a trapezoid.

4. The Sphere

A sphere is perfectly round in every direction (like a basketball).

• Any Slice: No matter what angle you slice a sphere from, the cross-section will always be a circle. Slicing exactly through the center gives the largest possible circle.

Example Problems

Example 1: What shape results from slicing a cube horizontally? Answer: Because the slice is parallel to the flat square base of the cube, the resulting cross-section is a square.

Example 2: What shape results from slicing a cylinder at an angle? Answer: If you slice a cylinder at an angle (not parallel to the circular base and not straight down), the cut is stretched out. This creates an ellipse.

Example 3: What possible cross-sections can a pyramid have? Answer: Depending on how you slice a square-based pyramid, you can get a square (parallel to the base), a triangle (vertically through the top point), or a trapezoid (at an angle through the sides).