Cylinders, cones and spheres: pouring into a drum
Fill a cone or a ball with water and pour it into a drum of the same size to discover the volume formulas.
Core concept
A cone holds a third of the cylinder with the same base and height; a sphere holds two thirds of the cylinder that just fits round it
Topic: Volume and surface area of solids
A drum is a cylinder: its 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 circular base times the height, V = πr²h. A cone with the same base and height looks like it might hold half as much, but if you fill it with water and pour it into the drum, it fills only a third: V = ⅓πr²h. A ball (sphere) of radius r fits exactly inside a drum of radius r and height 2r, and fills two thirds of it: V = ⁴⁄₃πr³. Surface areas follow too: the curved side of a cylinder unrolls into a rectangle 2πr by h, a cone's curved surface is πrl where l is the slant height, and a sphere's surface is 4πr².
What you will learn
- Find the volume of a cylinder, a cone and a sphere.
- Explain why a cone's volume is a third of the matching cylinder's.
- Find curved surface areas, using the slant height of a cone.
Before you start
- Area of a circle; Pythagoras' theorem.
Materials
- A paper cone and a paper cylinder of the same base and height, and rice or water
Equipment
- A measuring jug
Procedure
- Tap the chip to choose a cylinder, a cone or a sphere; tap − or + for the radius and the height.
- Watch how full the matching drum gets when the solid's water is poured in.
- Record the drum and the cone for r = 7 cm, h = 10 cm.
- Record a ball in a drum whose height is twice the radius, and a can that holds about 1 litre.
How this simulation is simplified
- For the sphere, the drum has the sphere's radius and the chosen height.