What is a section and how do we find it?
When a flat plane cuts a solid, the cut face is called a section. For a solid with flat faces it is a polygon. Each side of the polygon is the line where the plane meets one face. So the plane gives one side per face, and a section can never have more sides than the solid has faces. A cube has 6 faces, so its section has at most 6 sides. A tetrahedron has 4 faces, so at most 4.
Three points that are not in one line fix exactly one plane. That is the usual way a problem gives the cutting plane. Two other ways: a line and a point outside it, or two crossing lines.
Rules that make the drawing easy
Rule 1. You may join two points only if they lie on the same face. Then the segment between them lies on that face and is part of the section.
Rule 2. If two faces are parallel, the cutting plane meets them in parallel lines. So once you have one side on a face, draw a parallel to it on the opposite face. This is often the fastest step for a cube or a box.
Rule 3. A line that lies in the cutting plane and a line that lies in a face plane meet at a point that is on both. Use this to find new points.
The trace method
The trace of the cutting plane is the straight line where it meets the plane of the base (the ground plane of the solid). Steps:
- Take two points of the section that lie on one face. Join them and extend the line until it meets the base plane. This meeting point is on the trace.
- Do the same with two points on another face. You get a second point of the trace.
- Join the two points. This is the trace line.
- See where the trace crosses the edges of the base. These crossings are new points of the section.
- Join points that share a face, going round. Shade the polygon.
Why does it work? Two flat planes meet in a straight line, and two points are enough to draw it.
Internal projection method
Sometimes the lines you need would meet far outside the page. Then use internal projection. Take a point of the section inside the solid. Project it onto the base: straight down for a prism or box (parallel to the side edges), or along the line from the top vertex for a pyramid. Now you have a flat drawing on the base. The line through two points and the line through their two projections meet on the trace. Use that meeting point just like X in the 3D scene above.
Both methods need the same idea: a point common to the cutting plane and to the base plane lies on the trace.
Special sections worth remembering
- Cube, plane parallel to a face: a square equal to the face.
- Cube, plane through two opposite edges: a rectangle a by a√2.
- Cube, plane through the midpoints of the three edges at one corner: an equilateral triangle with area √3a²/8.
- Cube, plane through the midpoints of six edges: a regular hexagon with side a/√2 and area 3√3a²/4.
- Pyramid, plane parallel to the base: a smaller copy of the base. If it is at fraction k of the height from the top, area is k² times the base.
- Regular tetrahedron, plane through the midpoints of four edges (two pairs of opposite edges): a square with side a/2.
Key formulas and definitions
- Sides of a section ≤ number of faces of the solid
- 3 points not in a line → exactly 1 plane
- Parallel faces → parallel section sides
- Plane ∥ base of pyramid at fraction k of height from apex: area = k² × base area
- Cube edge a: hexagon section area = 3√3a²/4; corner-midpoint triangle area = √3a²/8
Worked examples
1. A plane passes through the midpoints of the three edges that meet at one corner of a cube of edge 4 cm. Find the section and its area.
The plane meets 3 faces, so the section is a triangle. Each side is the diagonal of a small square of side 2 cm, so each side is 2√2 cm. It is equilateral. Area = (√3/4) × (2√2)² = (√3/4) × 8 = 2√3 ≈ 3.46 cm².
2. A plane goes through the top front edge of a cube and the opposite bottom back edge. Name the shape and give its area for edge 3 cm.
The plane cuts the front and back faces along the two edges, and the top and bottom faces along parallel lines. It is a rectangle. One side is the edge, 3 cm; the other is a face diagonal, 3√2 cm. Area = 3 × 3√2 = 9√2 ≈ 12.73 cm².
3. P and Q lie on the front face of a cube and R lies on its right face. How do we find a point of the trace using P and Q?
P and Q are on one face, so join them and extend line PQ until it hits the base plane at X. X lies on the cutting plane (it is on line PQ) and on the base plane, so X is on the trace. Repeat with S (where PQ meets the right edge) and R to get a second trace point Y. Join X and Y.
Common mistakes
- Joining two points that are on different faces. The segment would go through the inside of the solid, not along the section.
- Forgetting to extend lines. If two lines do not meet inside the solid, extend them to the base plane to find the trace.
- Drawing more sides than faces. A section has at most one side per face.
- Drawing the trace through only one point. You always need two points to fix a line.