A cross-section is the 2D shape you see when you cut through a 3D solid. The shape of the cross-section depends on the solid and the angle of the cut. Prisms always produce the same cross-section as their end face; other solids — pyramids, cones, spheres — produce different shapes at different heights.
What is a cross-section?
Imagine slicing through a loaf of bread. The face of each slice is a cross-section of the loaf. More precisely, a cross-section is the intersection of a flat plane with a solid shape.
A perpendicular cross-section cuts at right angles to the axis of the solid — for a cylinder standing upright, a horizontal cut is perpendicular. Any other angle of cut is an oblique cross-section.
At GCSE you mainly deal with perpendicular cross-sections, but exam questions may show you a cross-section diagram and ask you to identify the shape or find its area.
What cross-sections do prisms produce?
A prism is defined as a solid with a uniform cross-section — the same shape all the way through, parallel to the two end faces.
| Prism type | Perpendicular cross-section |
|---|---|
| Cuboid (rectangular prism) | Rectangle |
| Cube | Square |
| Triangular prism | Triangle |
| Cylinder | Circle |
| Hexagonal prism | Regular hexagon |
Because the cross-section is uniform, the volume of any prism = cross-sectional area × length (or height). This is why recognising the cross-section is the first step in a prism volume question.
Key exam link: if you see a cross-section area given in a problem, and the solid is a prism or cylinder, the volume is simply area × length — no other formula needed.
What cross-sections do pyramids and cones produce?
Unlike prisms, pyramids and cones do not have a uniform cross-section. Slicing parallel to the base always produces the same shape as the base, but scaled down.
| Solid | Perpendicular cut parallel to base | Cross-section shape |
|---|---|---|
| Square-based pyramid | At any height | Smaller square (similar to base) |
| Cone | At any height | Circle (smaller than base) |
| Triangular pyramid | At any height | Smaller triangle |
Scaling rule: if you cut a pyramid or cone at height h from the base, where the full solid has height H, the linear scale factor of the cross-section compared to the base is (H − h) / H. Areas scale by the square of this factor.
Example: A cone of height 12 cm and base radius 6 cm is cut parallel to its base at 4 cm above the base. What is the radius of the circular cross-section?
Distance from apex = 12 − 4 = 8 cm. Linear scale factor = 8/12 = 2/3.
Radius = 6 × (2/3) = 4 cm.
What cross-sections does a sphere produce?
Every cross-section of a sphere is a circle. The largest possible cross-section passes through the centre (a great circle) and has the same radius as the sphere. Cuts further from the centre produce smaller circles.
If a sphere of radius r is cut at a distance d from the centre (where d < r):
Cross-section radius = √(r² − d²)
Example: A sphere of radius 5 cm is sliced at 3 cm from the centre.
Cross-section radius = √(25 − 9) = √16 = 4 cm.
Oblique cross-sections
An oblique cut (not perpendicular to the axis) changes the shape of the cross-section:
- A cylinder cut obliquely produces an ellipse, not a circle.
- A cone cut at certain angles produces a parabola or hyperbola (beyond GCSE, but good to be aware of).
- A cuboid cut obliquely produces a rectangle, parallelogram, or other polygon depending on the angle.
At GCSE, questions involving oblique cross-sections will normally show you the shape and ask you to recognise it — you are unlikely to be asked to calculate its area.
Why does the cross-section matter in exam questions?
The cross-section is directly tested in three types of exam question:
- "What shape is the cross-section?" — identify from a diagram or description.
- Volume of a prism: Volume = cross-section area × length. You must find the cross-section area first.
- Comparing solids: a question may ask whether two solids with the same cross-section but different lengths can have the same volume.
Worked example: A triangular prism has a cross-section that is a right-angled triangle with legs 5 cm and 8 cm. The prism is 12 cm long. Find its volume.
Cross-section area = ½ × 5 × 8 = 20 cm²
Volume = 20 × 12 = 240 cm³
Frequently asked questions
Is a face the same as a cross-section?
Not exactly. A face is one of the flat surfaces bounding a solid. A cross-section is a cut through the interior. For a prism, the end face is the same shape as the perpendicular cross-section. For a pyramid, the base is a face, and a cross-section parallel to it is a similar (smaller) shape — not the base itself.
How do I know if a solid is a prism?
A solid is a prism if it has two identical, parallel end faces (the cross-sectional faces) and its sides are all rectangles or parallelograms. Cylinders are sometimes called "circular prisms" because they satisfy the uniform cross-section definition. Pyramids, cones, and spheres are not prisms.
Can a cross-section be curved?
A cross-section is always flat (a 2D plane shape) — it is the result of a flat cut. However, the boundary of the cross-section can include curved edges. For example, a horizontal slice through a cylinder produces a flat circular disc with a curved boundary.
What is a frustum and how does it relate to cross-sections?
A frustum is the portion of a cone or pyramid left after slicing off the top with a cut parallel to the base. The frustum has two parallel circular (or polygonal) cross-sections — the original base and the new top created by the cut. Frustum volume problems use both cross-section areas.
Explore cross-sections and 3D geometry with Professor Pi's guided questions at aitutors.me.