1. Introduction
A chef takes a knife and slices a carrot in half. What shape results? Depending on the direction of the cut, the resulting shape may resemble a circle (for a widthwise cut), a triangle (for a lengthwise cut from the tip of the carrot to its base), or an oval (for a diagonal cut). The circle, triangle, and oval are each a type of plane figure, a two-dimensional shape on a plane. Recall that a plane is a flat, two-dimensional figure without depth that has at least three non- collinear points and extends infinitely in all directions.
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

2. Introduction, continued
When planes intersect with solid figures, or three- dimensional objects that have length, width, and height (or depth), plane figures are created. In this example, the carrot can be thought of as the solid figure, the knife as the intersecting plane, and the resulting shape as the plane figure.
Plane figures have many applications in real-world problems. Technical drawings of solid figures, such as the parts of a redesigned spacecraft, often rely on back, bottom, front, and rear views of a figure that exists in three dimensions or in three-dimensional coordinate systems.
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

3. Introduction, continued
These applications can be as complex as the real-world circumstances in which they are found, but many rely on relatively basic properties of plane and solid geometry for their description and use in calculations.
Similarly, two-dimensional figures can be used to generate solid figures with transformations in three- dimensional space. Solids can be generated by the horizontal or vertical movement of a plane figure through a third dimension, or by the rotation of a plane figure about an axis of rotation or line in a third dimension.
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

4. Key Concepts
In its basic form, a cross section is the plane figure formed by the intersection of a plane and a solid figure, where the plane is at a right angle to the surface of the solid figure. The cross section can also be formed by an axis of rotation of the solid figure, given that the axis of rotation is perpendicular to the intersecting plane.
An axis of rotation is a line about which a plane figure can be rotated in three-dimensional space to create a solid figure, such as a diameter or a symmetry line. For example, a line that connects the centers of the circular bases of a cylinder is an axis of rotation.
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

5. Key Concepts, continued
The shape of a cross section depends on the solid figure and the location of the cross section.
A polyhedron is a three-dimensional object that has faces made of polygons. Such polygonal plane figures include squares, rectangles, and trapezoids. Regular polyhedra are solids that have faces that are all congruent regular polygons. (Recall that congruent means having the same shape, size, lines, and angles; the symbol for representing congruency between figures is ≅.)
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

6. Key Concepts, continued
The cross sections of regular polyhedra are similar figures. Similar figures are two figures that are the same shape but not necessarily the same size. In other words, corresponding sides have equal ratios. The symbol for representing similarity between figures is ∼. One example of a regular polyhedron is a triangular prism with bases consisting of equilateral triangles.
Cross sections of some polyhedra such as pyramids form similar figures.
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

7. Key Concepts, continued
An infinite number of plane figures can be formed by the intersection of a plane with a solid figure, but the cross section is a special case due to its perpendicularity with the surface of the solid.
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

8. Key Concepts, continued
A solid figure can be generated by translating a plane figure along one of its two dimensions. In three dimensions, translation refers to the horizontal or vertical movement of a plane figure in a direction that is not in the plane of the figure, such that a solid figure is produced. Imagine a pop-up cylindrical clothes hamper—it can be compressed into a flat circle for storage, but when you need to use it you lift up the top part to “open” it up into its cylindrical form.
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

9. Key Concepts, continued
A solid figure can also be generated by the rotation of a plane figure in three dimensions. A rotation is a transformation in which a plane figure is moved about one of its sides, a fixed point, or a line that isn’t in the plane of the figure, such that a solid figure is produced. Similarly, imagine a toothpick with a small paper flag attached. Twirling the toothpick quickly back and forth between your fingers causes the paper flag to form the illusion of a cylinder.
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

10. Common Errors/Misconceptions
thinking that a cross section can be formed by any intersection of a plane and a solid figure
labeling cross sections formed by planes and solids as having linear or cubic units rather than square units
thinking that the axis of rotation of a plane figure is a single point instead of a line
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

11. Guided Practice Example 1
Compare the areas of two rectangular cross sections of an air-conditioning duct along a length of the duct that shrinks in width from 20 inches to 12 inches and in height from 8 inches to 6 inches. Find the difference of the areas, and determine the percentage that each cross section is of the other. Then, name the solid figure from which the two rectangular cross sections are formed.
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

12. Guided Practice: Example 1, continued
Calculate the area of the larger cross section.
The larger cross section is a rectangle, so use the formula for the area of a rectangle, area = length • width. In this case, the “length” is actually the height, so use the formula area = height • width, or A = hw.
The larger cross section has a height of 8 inches and a width of 20 inches. Substitute these measurements into the formula A = hw and then solve to determine the area of the larger cross section.
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

13. Guided Practice: Example 1, continued
A = hw Rectangular area formula
A = (8)(20) Substitute 8 for h and 20 for w.
A = Simplify.
Recall that area is expressed in square units.
Thus, the area of the larger cross section is 160 square inches.
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

14. Guided Practice: Example 1, continued
Calculate the area of the smaller cross section.
The smaller cross section is also a rectangle, so use the formula A = hw. However, this cross section has a height of 6 inches and a width of 12 inches. Substitute these measurements into the formula A = hw and then solve to determine the area of the smaller cross section.
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

15. Guided Practice: Example 1, continued
A = hw Rectangular area formula
A = (6)(12) Substitute 6 for h and 12 for w.
A = Simplify.
The area of the smaller cross section is 72 square inches.
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

16. Guided Practice: Example 1, continued
Find the difference of the cross section areas.
Subtract the area of the smaller cross section from the area of the larger cross section.
160 – 72 = 88
The difference of the areas is 88 square inches.
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

17. Guided Practice: Example 1, continued
Calculate the percentage that the smaller area is of the larger area.
Divide the smaller area by the larger area and then multiply by 100 to find the percentage.
Percentage formula
Substitute 72 for the smaller area and 160 for the larger area.
= Simplify.
The smaller area is 45% of the size of the larger area.
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

18. Guided Practice: Example 1, continued
Calculate the percentage that the larger area is of the smaller area.
Divide the larger area by the smaller area and then multiply by 100 to find the percentage.
Percentage formula
Substitute 160 for the larger area and 72 for the smaller area.
≈ Simplify.
The larger area is about 222% of the size of the smaller area.
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

19. ✔ Guided Practice: Example 1, continued
Determine the solid figure from which the cross sections are formed.
The areas of the rectangular cross sections are unequal, and the rectangular cross sections are not similar. Given this information, the only solid figure from which these cross sections could have been formed is like a pyramid with a rectangular base, except that instead of coming to a point at the top, it comes to a line like the ridge of a roof. ✔
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

20. Guided Practice: Example 1, continued
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

21. Guided Practice Example 2
The diagram shows the two cross sections, I and II, of a solid figure. What is the geometric solid from which the cross sections are formed?
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

22. Guided Practice: Example 2, continued
Name the plane figure represented by cross section I.
Cross section I is a trapezoid because it has one pair of parallel sides.
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

23. Guided Practice: Example 2, continued
Name the plane figure represented by cross section II.
Cross section II is a rectangle because it has four right angles and two pairs of congruent and parallel sides.
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

24. Guided Practice: Example 2, continued
Describe how the cross sections are oriented with respect to each other and why.
The cross sections represent two of the three dimensions of the solid. By definition, a cross section intersects a solid figure at a right angle to the surface of the solid figure. Therefore, since both cross sections must intersect the figure at a right angle and there are two different cross sections, the cross sections are perpendicular to each other.
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

25. Guided Practice: Example 2, continued
Name the solid figure implied by the cross sections.
The two cross sections are the only ones for this solid figure. Therefore, the trapezoid is the base of a prism. The rectangle is a cross section of the prism that is parallel to the faces of the prism formed by the parallel sides of the trapezoid, because it has a width that is less than the longer parallel side of the trapezoid and greater than the shorter parallel side of the trapezoid. The diagram that follows shows the solid and its cross sections.
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

26. Guided Practice: Example 2, continued✔
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects

27. Guided Practice: Example 2, continued
6.8.1: Two-Dimensional Cross Sections of Three-Dimensional Objects