Question

Describe the shape resulting from a vertical, horizontal, and angled cross section for each figure

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Answer 1

In mathematics, a cross section is the shape obtained when a solid figure is cut by a plane. The shape resulting from a vertical, horizontal, and angled cross section depends on the figure being considered. For example, a vertical cross section of a cube is a square, a vertical cross section of a cone can be a point, and a vertical cross section of a wood block reveals the annual rings.

Step-by-step explanation:

In mathematics, a cross section is the shape or profile that is obtained when a solid figure is cut by a plane. Let's consider different figures and describe the shape obtained from a vertical, horizontal, and angled cross section of each:

1. Cube: A vertical cross section of a cube would be a square, a horizontal cross section would also be a square, and an angled cross section would be a rhombus.
2. Conic section: The shape resulting from a vertical, horizontal, and angled cross section of a conic section depends on the type of curve. For example, a vertical cross section of a cone can result in a point, a horizontal cross section can result in a circle, and an angled cross section can result in an ellipse.
3. Wood block: A vertical cross section of a wood block would show the annual rings and parallel lines, a horizontal cross section would show the same pattern, and an angled cross section would still reveal the parallel lines.

Answer 2

Problem 22

Answers:

• vertical: rectangle
• horizontal: rectangle
• angled: parallelogram

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Step-by-step explanation:

The vertical and horizontal cross sections are fairly straight forward. They are simply mirror images of the outward showing faces. The angled cross sections are a bit more complicated and there's a lengthy proof involved, but long story short, the angled cutting plane divides the 3D solid such that we have 2 sets of lines that have the same slope (if we consider a 2D view), which leads to 2 sets of parallel sides.

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Problem 23

Answers:

• vertical: either a triangle or quadrilateral
• horizontal: triangle
• angled: either a triangle or quadrilateral

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Step-by-step explanation:

The horizontal cross section is always a triangle because the bottom base face is a triangle. The other two types of cross sections are either triangles or quadrilaterals depending on where the cutting plane is situated. For vertical cross sections that go through the apex point, we get a triangle. For vertical cross sections that do not go through the apex, then we get a quadrilateral. Sometimes a trapezoid is possible here, but not always. It's better to just consider it a quadrilateral to be the most general. A similar situation happens with the angled cuts as well.

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Problem 24

Answers:

• vertical: triangle, but only if plane is crossing through apex
• horizontal: circle
• angled: ellipse or parabola

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Step-by-step explanation:

Imagine you shined a flashlight onto the cone such that the flashlight is perfectly level and flat. It would cast a shadow that is a triangle. This is one way to think of a cross section. If you vertically slice. The horizontal cross sections are always circles due to the circular base of the cone. The angled cross sections are either ellipses or parabolas. For more information, look in your math textbook about conic sections (just ignore the second cone however).

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Problem 25

Answers:

• vertical: rectangle
• horizontal: circle
• angled: ellipse

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Step-by-step explanation:

The horizontal cross sections are circles for similar reasoning as the cone horizontal cross section. However, this time the vertical cross sections are rectangles. The widest possible rectangle is the result of the vertical cutting plane passing through the center of the circular base. Angled cross sections are ellipses. Though some portions of the ellipse may be cut off depending on what the actual angle is.