Combined solids: a circus tent
A circus tent is a cylinder with a cone on top. Change the radius, the height of the wall and the height of the cone, and find how much air it holds and how much canvas it needs.
Core concept
A combined solid's volume is the sum of its parts' volumes, and its surface is the sum of the parts' outer surfaces, leaving out the faces where they join
Topic: Area and volume
Many real objects are made of two simple solids joined together. A circus tent is a cylinder of radius r and height H with a cone of the same radius and height h on top. Its volume: How much space something takes up, measured in millilitres or cubic centimetres. All key words is the volume of the cylinder plus the volume of the cone: πr²H + ⅓πr²h. Its canvas covers only the outside: the curved wall of the cylinder, 2πrH, and the curved roof of the cone, πrl, where the slant height l = √(r² + h²). The circle where the cone sits on the cylinder is inside, so it is not counted, and the floor needs no canvas. In the same way a water tank made of a cylinder with a hemisphere on top has volume πr²H + ⅔πr³, since a sphere's volume is 4/3 πr³ and its surface 4πr².
What you will learn
- Find the volume of a solid made of a cylinder and a cone.
- Find the curved surface area of the combined solid, leaving out hidden faces.
- Use Pythagoras to find the slant height of a cone.
Before you start
- Know the volume and curved surface area of a cylinder and a cone.
Materials
- Card, a compass, glue and scissors to make a model tent
Equipment
- None
Procedure
- Record a tent with r = 3 m, H = 3 m and h = 4 m.
- Make the wall 5 m high and record.
- Record r = 6 m, H = 3 m, h = 8 m.
- Record r = 5 m, H = 4 m, h = 12 m.
How this simulation is simplified
- The tent has no floor, and the cone sits exactly on top of the cylinder.