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.

Cylinder: A horizontal cut will always yield a circle. Vertical cuts, depending on orientation, can result in rectangles or ellipses.

Sphere: Any planar cut through the center of a sphere will always produce a circle.

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.

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.

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.

Examples

Example 1:
If a cylinder with a base diameter of \(8 \text{ cm}\) is sliced vertically through its center, what shape is revealed?

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).

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?

Solution:
The cross section will be a rectangle measuring \(5 \text{ cm} \times 4 \text{ cm}\).

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?

Solution:
The diameter is \(2 \times 7 \text{ cm} = 14 \text{ cm}\).

Example 4:
If you slice a cube diagonally from one corner to the opposite corner on the same face, what shape will you get?

Solution:
The cross section will be a triangle.

Example 5:
If a square-based pyramid is sectioned parallel to its base about midway up, what is the resulting cross section?

Solution:
The cross section will be a square, but smaller than the base of the pyramid.

Example 6:
If a cone is sliced vertically, passing through its apex and base, what shape is revealed?

Solution:
The cross section will be a sector of a circle, also described as a ‘curved triangle’.

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:

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.