3-D figures · Ages 12–13
Cross-sections
Prisms: same as the base; pyramids: smaller near the top; sides = faces cut.
How to do it
Method
Prisms: same as the base; pyramids: smaller near the top; sides = faces cut
A cylinder cut through its axis: a rectangle, the height by the diameter
Another way
Count the faces cut
Through 3 of a cube's faces: a triangle
In full
Cut a solid with a plane and the flat shape of the cut is a cross-section. Prisms and cylinders cut parallel to the base give a copy of the base; cut at right angles to the base, a prism gives a rectangle, and a cylinder cut through its axis gives a rectangle as long as the cylinder is high and as wide as its diameter. Pyramids and cones cut parallel to the base give the base's shape, smaller nearer the apex; cut upright through the apex, a triangle. A sphere gives a circle every way, biggest through the centre. To tell how many sides the cut has, count the faces the plane passes through: through 3 faces, a triangle. A plane can cross at most 6 faces of a cube, so a cube's cross-section can be a triangle, a quadrilateral, a pentagon or a hexagon, but never a circle.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
The solid is cut parallel to its base; the shaded part is the cross-section. It is the same shape as the base and the same size.
True or false?
Show the answer and working
Answer: True
Prisms and cylinders (the cube, the cuboid, the cylinder) cut parallel to the base give an exact copy of it; a cone's or pyramid's cut has the base's shape but shrinks towards the apex.
Example 2
A cylinder has base radius 2 cm and height 7 cm. Cut through its axis, the cross-section is a rectangle: cm high, cm wide, with area cm².
Fill in each blank
Show the answer and working
Answer: 7, 4, 28
It is as high as the cylinder, 7 cm, and as wide as the base's diameter, 2 × 2 = 4 cm: area 7 × 4 = 28 cm².
Example 3
A sphere has radius 13 cm. A plane cuts it, and the cross-section is a circle. At most, how many cm across is that circle?
Show the answer and working
Answer: 26 cm
The cut through the centre is the biggest: its radius is the sphere's, 13 cm, so it is 2 × 13 = 26 cm across.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.A plane cuts the solid in the picture; the shaded part is the cross-section. What shape is it?
- A
- B
- C
- D
Choose the right picture
Answer:
2.The solid is cut parallel to its base; the shaded part is the cross-section. It is the same shape as the base and the same size.
True or false?
Answer:
3.An upright plane cuts the solid in the picture; the shaded part is the cross-section. What shape is it?
- A
- B
- C
- D
Choose the right picture
Answer:
4.A plane cuts the solid in the picture; the shaded part is the cross-section. What shape is it?
- A
- B
- C
- D
Choose the right picture
Answer:
5.A sphere has radius 15 cm. A plane cuts it, and the cross-section is a circle. At most, how many cm across is that circle?
Answer:
All the answers
- 1. D
- 2. False
- 3. C
- 4. D
- 5. 30 cm
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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