Solids, Nets & Cross-Sections
A net is what you get when you unfold a three-dimensional solid and lay every face flat. A cross-section is the opposite move — the flat shape revealed when you slice straight through a solid. Both are ways of seeing a 3D object through 2D pieces.
Nets tell you what faces a solid has and how they connect, which is exactly what you need to find surface area. Cross-sections show up in modeling and design, and on tests as 'what shape is the slice.'
Reading a net
Count the faces and their shapes. A net of two congruent triangles and three rectangles folds into a triangular prism — the triangles become the two ends, the rectangles wrap around as the sides.
The rule of thumb: the repeated flat shapes are the bases, and the rectangles between them are the lateral faces. A net of six squares is a cube; a net of one square and four triangles is a square pyramid.
Slicing for a cross-section
The cross-section depends on the angle of the cut. Slice a cylinder straight across (parallel to the base) and you get a circle; slice it straight down and you get a rectangle.
Slice a cube parallel to a face and you get a square, but tilt the cut and you can get a rectangle, a triangle, or even a hexagon. The cut plane's orientation is everything.
The figure shows the horizontal slice of a cylinder — a cut parallel to the base reveals a circle.
Worked examples
Example 1: matching a net to a solid
A net has two congruent triangles and three rectangles. Which solid is it?
Answer: A triangular prism
Example 2: a horizontal slice of a cone
A cone is sliced parallel to its base. What is the cross-section?
Answer: A circle
Example 3: a vertical slice of a cylinder
A cylinder is sliced straight down through its center, perpendicular to the base. What is the cross-section?
Answer: A rectangle
Try one yourself
Common questions
How do I know how many faces a solid's net has?
Count the faces of the solid: a rectangular prism has 6, a triangular prism has 5, a square pyramid has 5. The net has exactly one flat piece per face.
Can one solid have more than one net?
Yes. A cube has 11 different nets — many ways to unfold the same six squares. They all fold back into the same cube.
What is the most surprising cube cross-section?
A hexagon. Slice a cube at the right diagonal angle through six faces and the cross-section is a regular hexagon — not something most people expect from a box of squares.
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