Describe two-dimensional sections of three-dimensional figures

Learning goal: Identify the two-dimensional figure formed where a plane intersects a solid. Describe how the plane's position and direction affect the section, especially in right rectangular prisms and right rectangular pyramids.

A plane is a flat surface extending in two dimensions. A plane section is the figure shared by the plane and a solid at their intersection. Imagine a perfectly thin flat cut and focus on the shape exposed at the cut, not on the entire remaining solid. A face already on the outside and a section through the interior can have the same shape without being the same location.

A right rectangular prism has rectangular bases and lateral edges perpendicular to those bases. A plane parallel to its base and passing through the interior produces a rectangle congruent to the base. If the base is four by three units, every such parallel interior section is four by three. Moving this plane up or down changes its location but not its dimensions because the prism maintains the same cross-sectional shape along its height.

A vertical plane parallel to a side face also produces a rectangle, with dimensions corresponding to the prism's height and the appropriate base direction. A different vertical plane can run diagonally across the base; its section is still a rectangle in a right prism, but its width follows that diagonal route through the base. Do not assume every rectangular section has exactly the dimensions of an outside face.

Other planes can produce different polygons. A plane cutting off a corner near one vertex of a rectangular prism can intersect three incident edges and form a triangle. The number and placement of intersected faces matter. A prism's name does not imply that every possible slice must reproduce its base, and a drawing of one slice is not a list of all possible sections. Tilted planes can also cut a rectangular prism in a parallelogram, a trapezoid, a pentagon or a hexagon, because a plane section of a prism has one side for each face the plane crosses, and a rectangular prism has six faces.

A right rectangular pyramid has a rectangular base and an apex positioned directly above its center. A plane parallel to the base and between the base and apex produces a smaller rectangle similar to the base. As the plane approaches the apex, the section shrinks. A plane through the apex that is parallel to the base meets the solid only at that single point, not in a positive-area rectangle. A plane perpendicular to the base and parallel to one base edge that crosses the interior without passing through the apex cuts a trapezoid. Other perpendicular planes can cut other shapes: one that slices off a single base corner cuts a triangle.

A plane containing the pyramid's apex and a line through its base can produce a triangular section. For example, a plane through the apex and a diagonal of the rectangular base cuts a triangle whose vertices are the apex and the diagonal's two endpoints. This section is different from the smaller rectangle formed by a parallel plane. Describing the cutting plane is essential to identifying the shape.

The illustration uses perspective to show the solids. A rectangle in a three-dimensional view may appear as a slanted parallelogram on the page because depth is projected into the flat picture. The geometric section is still a rectangle in its own plane. The companion flat outline identifies the actual two-dimensional shape without requiring the learner to measure perspective angles as if they were true angles.

Separate a section from a shadow. A shadow projects light onto another surface and can change with lighting direction even when the solid stays fixed. A plane section instead comes from points where the plane passes through the solid. A silhouette viewed from the side is another different representation. Each can be useful, but they answer different geometric questions.

To investigate without cutting a real object, construct a wireframe or use a digital solid with a movable plane. Mark the intersections on the edges and connect them along the relevant faces. A paper or modeling-material demonstration can also help when guided by an adult, but the mathematical section is idealized as flat and without thickness. Record the intended plane before interpreting a shape created by an uneven physical cut.

Use precise language in a claim: “A plane parallel to the base inside this right rectangular prism makes a congruent rectangle” is more informative than “prisms make rectangles.” The first includes the condition that supports the result. A good counterexample, such as a corner triangle, explains why dropping that condition would make the general statement false.

Worked comparison: Both a prism and a pyramid have four-by-three rectangular bases. Halfway up the prism, a base-parallel section remains four by three. Halfway up a right pyramid, the parallel section has half the base's linear dimensions, two by 1.5, because the edges converge linearly toward the apex. The two solids share a base but respond differently to moving a parallel plane.

Worked apex section: Draw the pyramid's apex and two opposite base vertices. A plane containing those three noncollinear points intersects the pyramid in the triangle connecting them. The triangle's interior is the section; the rest of the base rectangle is not part of that same cut.

Worked representation error: A student measures a slanted corner in the perspective drawing and concludes the base-parallel section is not rectangular. The page projection does not preserve every spatial angle. Use the plane's geometric relationship and the flat section outline to identify the actual shape.

Investigation: Use a physical model, a wireframe with marked intersection points or a digital geometry tool to examine three planes in a prism and two in a pyramid. For each, sketch the solid and cutting plane, then draw the section separately as a flat figure. Include a base-parallel slice and an apex-containing pyramid slice. Predict before changing the plane, then compare the observation with the prediction.

Independent performance: Create a five-panel section report stating the solid, plane orientation, resulting polygon and a reason. Include a counterexample to the claim that every rectangular-prism section is a rectangle. Explain why a perspective view, a shadow and a plane section should not be treated as interchangeable. Have a partner identify your intended plane from the description before viewing the answer outline.