Lesson 12-1: Describe Cross Sections of Three-Dimensional Figures

Property

A cross section is the 2D shape formed when a flat plane slices through a 3D solid figure. The resulting 2D shape depends entirely on whether the slice is horizontal, vertical, or angled.

Examples

• Cylinder: A horizontal slice parallel to the bases produces a circle. A vertical slice straight down produces a rectangle.
• Square Pyramid: A horizontal slice parallel to the base produces a square. A vertical slice directly through the apex produces a triangle.
• Cube: A slice parallel to a face produces a square, but an angled slice through a cube can produce a rectangle, a triangle, or even a hexagon!

Explanation

Imagine playing a game where you slice through flying fruit with a sword! When your flat sword slices straight through a 3D object, the flat interior surface exposed by the cut is the cross section. A common misconception is that slicing a shape always produces its base shape. But remember: changing the angle of your slice can create entirely different shapes!

Property

To find the area of composite figures, you must first be a master of the basic 2D shape formulas:

• Rectangle / Parallelogram:
  $$A = bh$$
  (base x height)
• Triangle:
  $$A = \frac{1}{2}bh$$
  (half of base x height)
• Trapezoid:
  $$A = \frac{1}{2}(b_1 + b_2)h$$
  (half of the sum of the bases x height)
• Circle:
  $$A = \pi r^2$$
  (pi x radius squared)

Examples

• Triangle: A triangle with a base of 8 cm and a height of 5 cm has an area of 1/2 x 8 x 5 = 20 square cm.
• Trapezoid: A trapezoid with bases of 6 m and 10 m, and a height of 4 m has an area of 1/2 x (6 + 10) x 4 = 32 square m.
• Circle: A circle with a radius of 7 inches has an area of pi x 7 squared = 49pi square inches.

Explanation

Before tackling a giant composite figure, remember that every complex shape is just a combination of simple shapes. Think of these formulas as your geometric toolbox. As long as you know how to find the area of a basic rectangle, triangle, or circle, you can solve any complex shape puzzle by breaking it down!

Property

When a pyramid or cone is sliced by a plane parallel to its base, the resulting cross section is similar to the base. The dimensions of the cross section can be found using the linear scale factor, which is the ratio of the distance from the vertex to the total height:

Property

When a three-dimensional figure (such as a cone or pyramid) is sliced parallel to its base, the resulting cross section is similar to the base. If the scale factor between the corresponding linear dimensions of the two similar cross sections is $k$, then the ratio of their areas is $k^2$.