What is a section?
Imagine a very thin, flat knife. We call it a plane. When the plane cuts a solid, the flat face it makes is called a section or cross-section.
- The section is always a flat shape (a polygon for solids with flat faces).
- Each side of the section lies on one face of the solid.
- So: number of sides = number of faces the plane cuts.
To draw a section, find where the plane meets each edge. Join these points in order, face by face.
Sections of a cube
A cube has 6 faces, so a section can have 3, 4, 5 or 6 sides.
- Parallel to a face: a square equal to the face. Area = a².
- Through two opposite edges (diagonal plane): a rectangle a × a√2. Area = a²√2.
- Cutting off a corner: a triangle. If the plane cuts the 3 edges at equal distances, it is equilateral.
- Through the centre, at right angles to a long (space) diagonal: a regular hexagon with side a√2⁄2. Area = (3√3⁄4)a².
- Other tilted planes give a trapezium or a pentagon.
A cube section can never be a regular pentagon or a shape with 7 sides.
Sections of a pyramid, prism, cylinder and cone
Pyramid, plane parallel to the base: the section is a smaller copy of the base. If the cut is at a fraction k of the height measured from the apex, every length is k times the base length, and the area is k² times the base area.
Example: a square pyramid with base 4 cm cut halfway: side 2 cm, area 4 cm² (one quarter of 16 cm²).
Plane through the apex: a triangle. The diagonal section of a regular square pyramid passes through the apex and two opposite base corners.
Prism: a cut parallel to the base gives the same shape as the base. Cylinder: parallel to the base gives a circle; along the axis gives a rectangle; slanted gives an ellipse. Cone: circle, triangle, ellipse, or open curves (parabola, hyperbola).
Using perpendicular lines to find section areas
A line is perpendicular to a plane if it is at right angles to two crossing lines in that plane; then it is at right angles to every line in the plane. The height of a pyramid is such a line.
Three perpendiculars theorem: let PO be perpendicular to a plane and let a line ℓ lie in the plane. If the foot O’s line OA is perpendicular to ℓ, then the slant line PA is also perpendicular to ℓ (and the other way round). We use it to find the height of a triangular section or the slant height of a pyramid face, then area = ½ × base × height.
Sections and cuts in technical drawing
Engineers show hidden insides with imagined cutting planes.
- Section: shows only the shape lying in the cutting plane. A removed section is drawn outside the view; a superimposed section is drawn on top of the view with thin lines.
- Cut (sectional view): shows the cut face and everything visible behind the plane.
- Hatching: the cut face is filled with thin parallel lines, usually at 45°. Different materials use different patterns (metal, wood, glass, plastic).
- Simple cut: one cutting plane. Complex cut: two or more planes (stepped or turned). Full cut: through the whole part. Local cut: a small area, bounded by a wavy line.
- Half view + half cut: for symmetric parts, draw half as an outside view and half as a cut, split by the centre line.
Key formulas and definitions
- Sides of a section = number of faces the plane cuts
- Cube, parallel to a face: area = a²
- Cube, diagonal plane: rectangle a × a√2, area = a²√2
- Cube, regular hexagon: side = a√2⁄2, area = (3√3⁄4)a²
- Cube, corner cut at distance x from the corner: equilateral triangle, side x√2, area = (√3⁄2)x²
- Pyramid cut parallel to base at k of the height from apex: area = k² × base area
Worked examples
1. A cube has edge 6 cm. Find the area of the section through two opposite edges.
The section is a rectangle 6 cm by the face diagonal 6√2 ≈ 8.49 cm. Area = 6 × 6√2 = 36√2 ≈ 50.9 cm².
2. A plane cuts a 4 cm cube through the midpoints of six edges, making a regular hexagon. Find its side and area.
Side = distance between midpoints of two edges at a corner = √(2² + 2²) = 2√2 ≈ 2.83 cm. Area = (3√3⁄2) × side² = (3√3⁄2) × 8 = 12√3 ≈ 20.8 cm².
3. A plane cuts a corner of a cube, meeting the three edges 3 cm from the corner. Find the section.
Each side joins two points 3 cm along perpendicular edges: √(3² + 3²) = 3√2 cm. All sides equal, so it is equilateral. Area = (√3⁄4)(3√2)² = (√3⁄4) × 18 ≈ 7.79 cm².
4. A square pyramid has base 10 cm and height 12 cm. A plane parallel to the base is 3 cm below the apex. Find the section area.
k = 3⁄12 = 1⁄4. Side = 10 × 1⁄4 = 2.5 cm. Area = 2.5² = 6.25 cm² (= 100 × 1⁄16).
5. A regular square pyramid has base edge 6 cm and height 4 cm. Find the area of its diagonal section (through the apex and two opposite base corners).
The section is a triangle with base = base diagonal 6√2 ≈ 8.49 cm and height = the pyramid height 4 cm (the height lies in this plane). Area = ½ × 6√2 × 4 = 12√2 ≈ 16.97 cm².
6. Same pyramid (base 6 cm, height 4 cm). Find the area of the section through the apex and the midpoints of two opposite base edges.
This triangle has base 6 cm (it joins the two midpoints, passing through the centre) and height 4 cm. Area = ½ × 6 × 4 = 12 cm². Each slant side is the face’s slant height √(3² + 4²) = 5 cm, found with the three perpendiculars idea.
Common mistakes
- Thinking a cube can give a 7- or 8-sided section. It has only 6 faces, so at most 6 sides.
- Using the edge instead of the face diagonal for the diagonal section: the long side is a√2, not a.
- For a pyramid cut, scaling the area by k instead of k². Lengths scale by k, areas by k².
- In drawing, mixing up a section (only the cut face) and a cut (cut face plus what is behind), or hatching areas that are not cut.