Key takeaway: A cross section is the two-dimensional shape revealed when a plane slices through a three-dimensional solid.
Grade 77.G.A.3Geometry and Measurement
Definition of Cross Section
A cross section is the flat face exposed when a thin wire slices through a block of cheese. Cut a carrot straight across and you see a circle; cut a loaf of bread and you see the outline of a slice.
What you see depends on both the solid and the tilt of the cut. A plane running parallel to the bottom of a prism exposes a twin of that bottom. A plane running straight down exposes a rectangle. Slanted cuts can reveal ovals or trapezoids. A cross section is always a plane figure, so its area uses two-dimensional formulas.
Formula and notation
The shape has no formula of its own. Its size comes from familiar plane formulas. For a cut parallel to the base of a right prism:
\[A_{\text{cut}} = A_{\text{base}}\]
For a vertical cut of height \(h\) and width \(w\), the area is \(w \cdot h\).
Key facts
- Slicing a cylinder parallel to its base always produces a circle.
- Slicing a cone through its tip, from top to bottom, produces a triangle.
- A sphere gives a circle for every plane that cuts it, large or small.
- A different angle on the same solid can reveal a different shape.
Cross Section example
A block of tofu measures 9 cm long, 5 cm wide and 7 cm tall. Chef Dalia makes one slice level with the 9-by-5 face and another slice level with the 5-by-7 face. Name each cross section and find its area.
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The level slice copies the 9-by-5 face, a rectangle: \(9 \cdot 5 = 45\) square centimeters.
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The upright slice copies the 5-by-7 face, another rectangle: \(5 \cdot 7 = 35\) square centimeters.
Both slices are rectangles, with areas 45 cm\(^2\) and 35 cm\(^2\).
Common mistake
Watch out
Pablo says that slicing a cone parallel to its base gives a triangle, because a cone looks like a triangle from the side.
Why it is wrong
A parallel cut across a cone gives a circle, smaller the nearer the cut is to the tip. Only a vertical cut through the tip gives a triangle. Picture the cut face, not the outline of the solid.
Related terms and lessons
- Grade 7 · Cross Sections of 3D FiguresLessonWorksheetQuiz
- Geometry · Cross Sections, Solids of Rotation and Geometric ModelingLessonWorksheetQuiz
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