Prisms: the volume and canvas of a tent
A trekking group is choosing a tent. Change the width and height of its triangular end and its length, and find how much air it holds (volume) and how much canvas it needs (surface area).
Core concept
A prism has the same cross-section all along it, so its volume is the area of its end × its length; its surface is the two ends plus the rectangles round it
Topic: Solid figures
A prism is a solid with the same shape all along its length, like a ridge tent, a Toblerone box or a house roof. A tent is a triangular prism: its end is a triangle with base b and height h, and it is l long. The volume: How much space something takes up, measured in millilitres or cubic centimetres. All key words is the area: How much flat space a shape covers, measured in square units like cm². All key words of the end × the length: V = ½bh × l. The canvas covers the two triangular ends and the two sloping sides. Each sloping side is a rectangle l long and as wide as the slant side of the triangle, s = √((b/2)² + h²) by Pythagoras. So the canvas (without a floor) is 2 × ½bh + 2 × s × l. A cylinder is a prism with a circle for its end: V = πr²h; a sphere has V = 4/3 πr³. Another lab pours water between them.
What you will learn
- Find the volume of a prism as the area of the cross-section × the length.
- Find the surface area of a triangular prism, using Pythagoras for the slant side.
- Explain why tents with the same volume can need different amounts of canvas.
Before you start
- Know the area of a triangle and Pythagoras' theorem.
Materials
- A sheet of card, a ruler and scissors to make a model tent
Equipment
- None
Procedure
- Record a tent with b = 6 m, h = 4 m and l = 5 m.
- Make it 8 m long and record.
- Record a tent with b = 8 m, h = 3 m and l = 5 m, and compare it with the first.
- Record a small tent with b = 4 m, h = 2 m and l = 4 m.
How this simulation is simplified
- The tent has no floor, and its end is an isosceles triangle.