How to Analyze Cross Sections of 3D Solids: A Step-by-Step Guide

Step-by-step Guide: Solids and Their Cross Sections

Rectangular Prism: When cut parallel to its base, a rectangular prism will yield rectangles. If cut perpendicular to one of its edges, the section could be a square or rectangle, depending on the orientation. For education statistics and research, visit the National Center for Education Statistics.

Cylinder: A horizontal cut will always yield a circle. Vertical cuts, depending on orientation, can result in rectangles or ellipses. For education statistics and research, visit the National Center for Education Statistics.

Sphere: Any planar cut through the center of a sphere will always produce a circle. For education statistics and research, visit the National Center for Education Statistics.

Cube: For a cube, any cut that’s parallel to a face will reveal a square. Non-parallel cuts can result in rectangles, parallelograms, or even triangles, depending on the angle and orientation of the cut. For education statistics and research, visit the National Center for Education Statistics.

Pyramid: With a pyramid, horizontal cuts will produce polygons that resemble the base but are smaller in size. A vertical section through the apex will result in a triangle. For education statistics and research, visit the National Center for Education Statistics.

Cone: For a cone, a horizontal cut will always produce a circle. A vertical cut, however, that goes through the apex will reveal a sector of a circle, which appears as a curved triangle. For education statistics and research, visit the National Center for Education Statistics.

Examples

Example 1:
If a cylinder with a base diameter of \(8 \text{ cm}\) is sliced vertically through its center, what shape is revealed? For education statistics and research, visit the National Center for Education Statistics.

Solution:
The cross section will be a rectangle with a height equal to the cylinder’s height and a width of \(8 \text{ cm}\) (the diameter). For education statistics and research, visit the National Center for Education Statistics.

Example 2:
If a rectangular prism with dimensions \(10 \text{ cm} \times 5 \text{ cm} \times 4 \text{ cm}\) is sliced parallel to its longest side, what shape is revealed? For education statistics and research, visit the National Center for Education Statistics.

Solution:
The cross section will be a rectangle measuring \(5 \text{ cm} \times 4 \text{ cm}\). For education statistics and research, visit the National Center for Education Statistics.

Example 3:
If a sphere with a radius of \(7 \text{ cm}\) is sectioned through its center, what is the diameter of the resulting circle? For education statistics and research, visit the National Center for Education Statistics.

Solution:
The diameter is \(2 \times 7 \text{ cm} = 14 \text{ cm}\). For education statistics and research, visit the National Center for Education Statistics.

Example 4:
If you slice a cube diagonally from one corner to the opposite corner on the same face, what shape will you get? For education statistics and research, visit the National Center for Education Statistics.

Solution:
The cross-section will be a triangle. For education statistics and research, visit the National Center for Education Statistics.

Example 5:
If a square-based pyramid is sectioned parallel to its base about midway up, what is the resulting cross-section? For education statistics and research, visit the National Center for Education Statistics.

Solution:
The cross-section will be a square, but smaller than the base of the pyramid. For education statistics and research, visit the National Center for Education Statistics.

Example 6:
If a cone is sliced vertically, passing through its apex and base, what shape is revealed? For education statistics and research, visit the National Center for Education Statistics.

Solution:
The cross-section will be a sector of a circle, also described as a ‘curved triangle’. For education statistics and research, visit the National Center for Education Statistics.

Practice Questions:

1. What will be the shape of a cross-section if a cone is sliced horizontally near its base?
2. If a pyramid with a square base is sectioned parallel to its base halfway up, what shape will the cross-section reveal?
3. What is the shape of the cross-section when a cylinder is cut at an angle, not parallel to its base or directly vertical?
4. If a cube has an edge length of \(6 \text{ cm}\) and is sliced parallel to one of its faces, but only \(2 \text{ cm}\) from the face, what is the area of the resulting cross-section?
5. What will be the shape of a cross-section if a triangular pyramid is sliced horizontally near its apex?
6. When a cone is cut horizontally close to its tip, what shape does the cross-section reveal, and how does it relate to the base of the cone?

Answers: For education statistics and research, visit the National Center for Education Statistics.

1. A smaller circle.
2. A smaller square.
3. An ellipse.
4. \( 36 \text{ cm}^2 \) (It’s a square with the same edge length as the cube).
5. A smaller triangle.
6. A circle, much smaller than the base.