Planar sections of solids

What is a cross-section?

Cutting a solid with a plane produces a cross-section: the flat shape obtained at the point of intersection. Understanding the shape of this cross-section helps us to visualise a solid and to solve problems involving the calculation of partial areas or volumes.

Cross-section of a cube or a rectangular prism

If a cube is cut by a plane parallel to one of its faces, the cross-section is a square identical to that face. If the plane is parallel to an edge but not to a face, the cross-section may be a rectangle. An inclined plane may yield a triangle, a pentagon or even a hexagon, depending on its orientation.

Cross-section of a cylinder

• Plane parallel to the bases: the cross-section is a disc with the same radius as the bases.
• Plane parallel to the axis: the cross-section is a rectangle.

Cross-section of a pyramid or a cone

If a pyramid (or a cone) is cut by a plane parallel to the base, the cross-section is a reduced version of the base (a similar polygon for the pyramid, a smaller disc for the cone). This follows from Thales’ theorem generalised to three-dimensional space.

Example: a pyramid with a height of 12 cm is cut by a plane parallel to the base, 4 cm from the apex. The reduction ratio is 4/12 = 1/3. If the base has an area of 90 cm², the cross-section has an area of (1/3)² × 90 = 10 cm².

Cross-section of a sphere

Any cross-section of a sphere by a plane is a circle (or a point if the plane is tangent). If the plane passes through the centre, the result is a great circle with the same radius as the sphere.

Common pitfall

For cross-sections parallel to the base of a pyramid or a cone, the lengths are multiplied by the ratio k, but the areas are multiplied by k² (and the volumes by k³). Forgetting to square the ratio when calculating an area is a very common mistake.