Cross-sections of solids
A cross-section is the shape you get when a solid is cut by a plane. The key is to identify both where the cut passes and what shape that cut makes.
1) Picture the solid and the cut
- Sketch the solid if possible.
- Mark the plane of the cut.
- Ask whether the slice is parallel to a base, passes through the top, or cuts several faces.
2) Determine the 2D shape of the slice
- A cut parallel to a face usually makes a shape similar to that face.
- In prisms and cylinders, parallel cuts often keep the same general shape as the base.
- In pyramids and cones, parallel cuts make a smaller, similar figure.
- If the plane is slanted, trace where it intersects each face and connect those intersection points in order.
3) Use given measurements carefully
- Look for lengths, radii, heights, or coordinates that help locate the slice.
- If the figure is similar to a base shape, compare corresponding side lengths using a scale factor.
- Simplify any numerical expression at the end.
4) Check your answer
- Does the slice match the geometry of the solid?
- Are the vertices or sides connected in the correct order?
- If similarity is involved, do the proportions make sense?
A good final answer should name the cross-section and show the simplified result clearly.