Cross Section
In short: Cross section names the flat shape exposed when an imaginary plane slices a solid, and the shape you see depends on the tilt of the cut and the solid it passes through.
7th GradeGeometry7.G.A.3
What is Cross Section?
Slice a loaf of bread and the face of the cut is a cross section. In geometry the knife is an imaginary plane, and the exposed face is a two-dimensional figure: a rectangle, a circle, a triangle or something else.
One solid yields different faces for different cuts. A box sliced parallel to one of its faces shows a rectangle identical to that face. A cone sliced parallel to its base shows a circle, while a cone sliced straight down through its tip shows a triangle. The angle of the plane decides.
How to spot it
- A statement mentions a plane that slices a prism, pyramid, cylinder, cone or sphere, and wants the cut shape named.
- The words slice, cut, parallel to the base or perpendicular to the base are in the wording.
- A picture of a solid contains one shaded flat region inside it.
- The task asks for the size of the exposed face rather than the outside of the whole solid.
The method
- Sketch the solid and mark where the plane goes: parallel to a face, perpendicular to it, or tilted.
- Find which edges or curved surfaces the plane crosses and name the outline those crossings trace.
- Read the dimensions of that outline from the measurements of the solid.
- Apply the area formula for the outline, such as length times width for a rectangle.
Cross Section: solved example
Problem
A shipping carton is 8 cm by 5 cm by 12 cm. Picture a plane running parallel to its 8-by-12 face. What shape does the plane expose and what is its area? Then repeat for a plane parallel to a 5-by-12 face.
Solution
Parallel to the 8-by-12 face, the exposed shape is a rectangle 8 cm by 12 cm, so its area is \(8 \times 12 = 96\) square centimeters.
Parallel to the 5-by-12 face, the rectangle is 5 cm by 12 cm and \(5 \times 12 = 60\). The cuts are rectangles of 96 cm² and 60 cm².
Try it
Question 1
What shape appears when a plane passes through a basketball?
Show the answer
A circle, large if the plane goes through the center and smaller if it is off to one side.
Question 2
A soup can of radius 4 inches is cut parallel to its lids. What is the area of the exposed face?
Show the answer
It is a circle of radius 4: \(\pi \times 4^2 = 16\pi \approx 50.27\) square inches.
Cross Section vs. Net
A cross section shows up when you cut into a solid and look at one inner face. A net shows up when you unfold the outer skin of a solid flat. The first reveals a single shape from the middle; the second keeps every outside face and lays them next to each other.
Keep going
Lessons
Worksheets and quizzes
- Cross-Sections of Three-Dimensional Figures – Worksheet
- Cross-Sections of Three-Dimensional Figures – Quiz
- Cross Sections and Solids of Revolution – Worksheet
- Cross Sections and Solids of Revolution – Quiz
Related words
Whole dictionary: A–Z · 7th Grade · Geometry