7.2: Cross Sections and Scale | Geometry Honors

How it works

A cross section is the intersection of a solid with a plane. Its shape depends on the direction of the cut. For similar solids, lengths, surface areas, and volumes scale by the first, second, and third powers of the same factor.

L\mapsto kL,\quad A\mapsto k^2A,\quad V\mapsto k^3V

Worked examples

Example 1

Similar solids have length factor $\displaystyle 3$ . Give the surface-area and volume factors.

Surface area scales by $\displaystyle 9$ and volume by $\displaystyle 27$ .

Example 2

A plane parallel to a cylinder base cuts the cylinder. What is the section?

It is a circular disk congruent to the base, provided the cut meets the cylinder between its bases.

Problems

Problem 7.2.1

A $\displaystyle 3$ by $\displaystyle 4$ by $\displaystyle 5$ rectangular prism is cut parallel to its $\displaystyle 3$ by $\displaystyle 4$ base halfway up. Give the cross-section area.

Perspective rectangular prism labeled base dimensions 3 and 4 and perpendicular height 5. — The sketch is a perspective schematic; labels give the dimensions.

Show worked solution

The section is a $\displaystyle 3$ by $\displaystyle 4$ rectangle, so its area is $\displaystyle 12$ square units.

Perspective rectangular prism labeled base dimensions 3 and 4 and perpendicular height 5. The solution adds labeled relationships. — A horizontal section has area 12. Volume = 60 and closed-box surface area = 94.

Problem 7.2.2

Similar solids have volume ratio $\displaystyle 64:27$ . Find the length ratio.

Show worked solution

Take cube roots to get $\displaystyle 4:3$ .

Problem 7.2.3

A sphere of radius $\displaystyle 5$ is cut by a plane $\displaystyle 3$ units from its center. Find the cross-section radius.

Show worked solution

A right triangle gives $\displaystyle r=\sqrt{25-9}=4$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 7.2.1

Problem 7.2.2

Problem 7.2.3

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