A cross-section is the flat face you see when you slice through a solid. A prism has the same shape all the way along, so every slice parallel to its base is an exact copy of the base: slice a triangular prism and you get a triangle, slice a hexagonal prism and you get a hexagon. The size never changes either, so the area and the perimeter of every parallel slice match the base. A pyramid is different, because its slices shrink towards the tip. Slices that all match, all the way along, tell you the solid is a prism.
Explore this idea with maths tools
Maths you can move
Build 8 + 5
Tap two gold counters to fill the ten-frame.
8 in the frame + 5 waiting. The total stays the same.
Words to know
- cross-section
- the flat face made when a solid is sliced
- prism
- a solid with two matching parallel bases and the same cross-section all the way along
- base
- the end face that gives a prism its name
- parallel
- going in the same direction and never meeting
Worked examples
A hexagonal prism is sliced parallel to its base. How many sides does the cut face have?
- 1.A slice parallel to the base is a copy of the base.
- 2.The base is a hexagon, which has 6 sides.
- 3.So the cut face has 6 sides.
= 6
A prism has 7 faces altogether. It is sliced parallel to its two end faces. How many sides does each slice have?
- 1.Two faces are the ends. The rest are rectangles, one for each side of the base.
- 2.7 − 2 = 5 rectangles, so the base has 5 sides.
- 3.A parallel slice copies the base, so it has 5 sides.
= 5
A prism is 12 cm long. Its end face has an area of 36 cm². It is sliced parallel to the end, 5 cm from one end. What is the area of the cut face, in cm²?
- 1.A parallel slice of a prism is a copy of the end face.
- 2.The distance along does not matter: the area stays the same.
- 3.The cut face has an area of 36 cm².
= 36
A rectangular prism is 3 cm wide, 5 cm high and 9 cm long. What is the perimeter of a slice parallel to its 3 cm by 5 cm end?
- 1.The slice is a 3 by 5 rectangle. The 9 cm length is not part of it.
- 2.Perimeter = 2 × (3 + 5).
- 3.2 × 8 = 16.
= 16
For parents
Method: Slice parallel, copy the base
What to say: What shape is the end of the solid? If I slice it parallel to the end, what will the cut face look like? Does it get smaller as we move along?
Common mistakes:
- Describes the slice as a rectangle, the shape of the side faces, instead of the shape of the base.
- Names a base with one more (or one fewer) side than the real base.
- Thinks a slice is always triangular, like a slice of pizza or a wedge of cake.
- Counts the sides of a slice as the faces of the whole prism (the sides plus the two ends).
- Mishandles the two end faces: gives one more side than the base, or one fewer face than the whole prism has.
- Says the slice has four sides because prisms are usually pictured as boxes.
- Calls the solid a pyramid although its parallel slices stay the same size.
- Names the prism for the wrong base, using a neighbouring shape.
- Gives the name of the slice (a flat shape) instead of the name of the solid.
- Thinks the cut face gets smaller the further along the prism the slice is, taking the distance away from the end area.
- Scales the area by how far along the slice is, as if the prism were a pyramid.
- Takes half the end area because the slice is somewhere in the middle of the prism.
- Uses the length of the prism as one side of the cut face, instead of the two sides of the end face.
- Multiplies the two sides to find the area instead of adding to find the perimeter.
- Adds all three measurements of the prism, so a measurement that is not on the slice is included.