What is a cross-section?
When a flat plane (think of a very thin knife) cuts through a solid, the new flat face is the cross-section. It is a 2D shape.
The shape depends on two things: the solid and the angle of the cut.
- A cut parallel to the base of a prism or cylinder always gives a copy of the base.
- A cut parallel to the base of a pyramid or cone gives a smaller copy of the base; it shrinks as you go up.
- A sphere cut in any direction gives a circle. The biggest one passes through the centre (a great circle).
Cross-sections of a cube
A cube has 6 faces, so a plane can meet at most 6 of them. That means a cube's cross-section has 3, 4, 5 or 6 sides:
- Square: cut parallel to a face.
- Rectangle: cut straight down through two opposite edges (corner to corner). For a cube of side a, it is a × a√2.
- Triangle: cut off one corner. If the cut passes through three corners next to one corner, you get an equilateral triangle.
- Pentagon: a tilted cut that meets 5 faces.
- Regular hexagon: a cut through the centre at right angles to a long diagonal; it meets all 6 faces.
A cube can never give a circle or a shape with more than 6 sides.
Cross-sections of cylinders, cones and spheres
| Solid | Cut across (horizontal) | Cut through the axis (vertical) | Tilted cut |
|---|---|---|---|
| Cylinder | Circle (same size) | Rectangle | Ellipse |
| Cone | Circle (smaller higher up) | Isosceles triangle | Ellipse, parabola or hyperbola |
| Sphere | Circle | Circle (the biggest) | Circle |
For a sphere of radius R cut at distance h from the centre, the circle has radius r = √(R² − h²). For a cone of height H and base radius R, at height y above the base the radius is r = R(H − y)/H.
Solids of rotation
Spin a flat shape a full turn (360°) around a straight line, the axis. The space it sweeps out is a solid of rotation (also called a solid of revolution).
- Rectangle spun about one side → cylinder (radius = the other side).
- Right triangle spun about one leg → cone (the leg on the axis is the height).
- Semicircle spun about its diameter → sphere.
- Rectangle spun about a line away from it → hollow cylinder (pipe).
- Right trapezium spun about its side that is perpendicular to both parallel sides → frustum (a cone with its top cut off).
- A circle spun about a line outside it → torus (doughnut).
Every solid of rotation has circular cross-sections at right angles to its axis. That is why rotation and cross-sections go together.
Volume from cross-sections and Cavalieri's principle
Think of a solid as a stack of very thin slices. Volume = sum of (slice area × slice thickness). This is why:
- Prism or cylinder: every slice has the same area B, so V = B × h. Cylinder: V = πr²h.
- Pyramid or cone: slices shrink to a point, giving V = ⅓ B h. Cone: V = ⅓πr²h.
- Sphere: V = ⁴⁄₃πr³.
Cavalieri's principle: if two solids have the same height and their cross-sections at every height have equal area, the solids have equal volume. A straight stack of coins and a leaning stack of the same coins have the same volume. The same idea shows that an oblique (slanted) cylinder has V = πr²h too.
Try it at home
Cut a cucumber, a carrot or a block of soap in different directions and draw each cross-section. For a cube, use a cube of paneer or clay: can you find a triangle and a hexagon? Then tape a paper rectangle, triangle and semicircle to a pencil and spin each one quickly between your palms: you will see a cylinder, a cone and a sphere appear.
Key formulas and definitions
- Volume of prism or cylinder = base area × height; cylinder V = πr²h
- Volume of pyramid or cone = ⅓ × base area × height; cone V = ⅓πr²h
- Volume of sphere V = ⁴⁄₃πr³
- Hollow cylinder (pipe) V = π(R² − r²)h
- Sphere cut at distance h from centre: section radius r = √(R² − h²)
- Cube of side a cut corner to corner: rectangle a × a√2
- Cavalieri: equal cross-section areas at every height → equal volumes
Worked examples
1. A cube of side 4 cm is cut parallel to a face. What is the cross-section and its area?
The section is a square, 4 cm × 4 cm. Area = 16 cm².
2. The same cube is cut straight down through two opposite vertical edges. Find the area of the section.
The section is a rectangle. Its height is 4 cm and its width is the face diagonal 4√2 ≈ 5.66 cm. Area = 4 × 4√2 = 16√2 ≈ 22.6 cm².
3. A rectangle 3 cm wide and 5 cm tall is spun about its 5 cm side. Name the solid and find its volume.
It makes a cylinder with r = 3 cm and h = 5 cm. V = πr²h = π × 9 × 5 = 45π ≈ 141.4 cm³.
4. A right triangle with legs 6 cm (on the axis) and 4 cm is spun about the 6 cm leg. Find the volume.
It makes a cone with h = 6 cm and r = 4 cm. V = ⅓πr²h = ⅓ × π × 16 × 6 = 32π ≈ 100.5 cm³.
5. A sphere of radius 5 cm is cut by a plane 3 cm from its centre. Find the area of the cross-section.
r = √(R² − h²) = √(25 − 9) = √16 = 4 cm. Area = πr² = 16π ≈ 50.3 cm².
6. A rectangle 4 cm tall is placed with its inner side 1 cm from the axis and its outer side 3 cm from the axis, then spun. Find the volume of the pipe.
R = 3 cm, r = 1 cm, h = 4 cm. V = π(R² − r²)h = π(9 − 1) × 4 = 32π ≈ 100.5 cm³.
7. A cone has base radius 6 cm and height 9 cm. It is cut parallel to the base 3 cm above the base. Find the radius and area of the section.
r = R(H − y)/H = 6 × (9 − 3)/9 = 4 cm. Area = π × 4² = 16π ≈ 50.3 cm².
Common mistakes
- Thinking a cube can only give squares. Tilted cuts give rectangles, triangles, pentagons and hexagons.
- Spinning a right triangle about the wrong leg. The leg on the axis becomes the height; the other leg becomes the radius.
- Using the diameter instead of the radius in πr²h or ⁴⁄₃πr³.
- Thinking a slanted (oblique) cylinder has a different volume. By Cavalieri's principle, same base area and same height give the same volume.