Cross-section of a cuboid through three points

What it shows

When a plane cuts a cuboid, the cross-section is a polygon whose sides are the segments where the plane meets the faces. Opposite faces of a cuboid lie in parallel planes, and a plane cuts two parallel planes along parallel lines, so sides of the section lying on opposite faces come in parallel pairs. This is the key to constructing the section through three points M, N and P on different edges step by step. Because a cuboid has six faces, the section can be a triangle, quadrilateral, pentagon or hexagon, but never a polygon with more sides.

How to use

Choose a Box type, then move M on AB, N on AD and P on CC' with the sliders or by dragging the points along their edges. Rotate the box to view the shaded section from several angles, and switch hidden edges on or off. Ask students to sketch the section on paper first, starting from the line MN on the base, then compare. Let them find positions that change the number of sides.

Parameters you can change

• Box type Cuboid 4 × 3 × 2.5, Cube with edge 3
• Position of M on AB (AM/AB) 0–1
• Position of N on AD (AN/AD) 0–1
• Position of P on CC' (CP/CC') 0–1
• Show hidden edges (dashed)

Questions to explore

1. Why are the sides of the section on two opposite faces of the box always parallel?
2. What is the greatest number of sides a cross-section of a cuboid can have, and why?
3. Which line of intersection should you draw first when constructing the section through M, N and P?