Cross-Sections of Three-Dimensional Figures

A cross section is the flat shape left at the intersection of a solid and a plane. Imagine cutting a loaf of bread and looking at the exposed face: that face is a cross section. To predict the shape, picture the cutting plane's direction and identify which parts of the solid's surface it reaches. Merely naming the solid is not enough; the same solid can produce different cross sections from different slices.

Read the plane's orientation

A slice parallel to a base often creates a cross section congruent to or similar to the base, depending on the solid. A slice perpendicular to a base produces a different profile. A diagonal slice can create still another polygon or curved region. In a diagram, use the plane marker, shaded cut, or written description to determine the angle of the slice before naming the shape.

Also distinguish the cross section from the solid's overall surface area, a face, or a net. A face is one of the flat sides already bounding a solid. A cross section is created by the cut through the interior. The slice may run through the whole object and reveal a shape that is not one of its outside faces.

Cross sections of prisms

A prism has two parallel, congruent bases joined by parallelogram-shaped lateral faces. A slice parallel to the bases makes a cross section congruent to a base. Thus, a right rectangular prism sliced parallel to its rectangular base gives a rectangle, and a triangular prism sliced parallel to its triangular bases gives a triangle.

A slice perpendicular to the bases of a right prism can form a rectangle. An oblique slice may create a different polygon, and its exact shape depends on which edges or faces the plane crosses. For an irregular or oblique prism, do not automatically assume a perpendicular cut looks like a perfect rectangle; inspect the solid and plane specified. When a problem uses a familiar right prism, the parallel-versus-perpendicular distinction is usually the main clue.

Cross sections of pyramids

A pyramid has one polygonal base and triangular side faces meeting at an apex. A slice parallel to the base creates a smaller, similar polygon. For a square pyramid, a parallel slice is a smaller square; for a triangular pyramid, it is a smaller triangle. The cross section gets smaller as the plane moves toward the apex and larger as it moves toward the base.

A vertical slice through the apex of a square pyramid can produce a triangle. A slice that does not pass through the apex may create a different polygon, such as a trapezoid, depending on the plane and which side faces it intersects. Trace the plane across the solid rather than assuming every pyramid cross section is triangular.

Cross sections of cylinders and cones

A right circular cylinder sliced parallel to its circular bases has a circular cross section. A vertical plane through its central axis creates a rectangle. A slanted plane can create an ellipse. The cross section changes as the plane tilts, even though the original solid remains the same cylinder.

A cone sliced parallel to its circular base produces a smaller circle. A plane passing through the cone's apex and the center of its base creates a triangle. A slanted plane can produce an oval or other conic-shaped section. On elementary diagrams, focus on whether the cut is parallel to the base or passes through the vertex; these cues determine the intended answer.

Cross sections of spheres

A plane slicing through a sphere produces a circle. A plane through the sphere's center creates its largest circular cross section, called a great circle. A plane that misses the center creates a smaller circle. If a plane is tangent to a sphere, it touches at only one point, so there is no two-dimensional circular cross section.

A sphere has no flat faces or edges, but its planar cross sections are still flat shapes. Do not confuse the curved spherical surface with the circular region exposed by a cut. If the cut is off-center, the circle's radius is shorter than the sphere's radius.

Use the diagram as a tracing problem

For a drawing-based question, mark where the plane enters and exits the solid. Follow the boundary around the slice: every time the plane meets a flat face, the cross section may gain a straight side; meeting a curved surface may create a curved boundary. Then compare the result with the base, an existing face, or a common shape such as a circle, rectangle, triangle, or trapezoid.

For example, a vertical plane through the center axis of a right circular cylinder intersects the curved wall along two vertical lines and the top and bottom bases along two parallel line segments. Together these boundaries form a rectangle. A horizontal plane intersects the curved wall in a circle and lies parallel to the two circular bases. The cut direction changes the intersection shape.

Cross-section area and volume are different

A cross section is a shape; its area is a two-dimensional measurement. Volume measures the three-dimensional space inside the entire solid. A problem may ask for the cross section's shape only, its area, or how that area changes at different heights. Do not use a volume formula when the question asks for the slice itself. If dimensions are supplied and area is requested, first identify the cross-sectional shape, then apply its area formula.

Common mistakes and a dependable method

The biggest mistake is naming a shape based only on the solid. A cylinder can yield a circle, rectangle, or ellipse; a pyramid's slice depends on whether it is parallel to the base or passes through the apex. Another error is treating every parallel slice as equal in size: that is true for a prism or cylinder, while a parallel section of a pyramid or cone changes size.

1. Identify the solid and its base, apex, axis, or curved surface.
2. Read whether the plane is parallel, perpendicular, or angled relative to those features.
3. Trace the boundary where the plane intersects the solid.
4. Name the resulting two-dimensional shape; distinguish its shape from its area or volume.
5. Use a formula only if the question asks for a measurement, and keep units appropriate to that measurement.

Exam takeaway

A cross section is the two-dimensional intersection made by a plane slicing a solid. Identify both the solid and the plane's direction. Parallel slices of prisms and cylinders match their bases; parallel slices of cones and pyramids are smaller similar shapes. Trace the cut for any other orientation.