Relationship between the volumes of a right circular cone, hemisphere and a right circular cylinder of same radii and heights

 

Topic :

Relationship between the volumes of a right circular cone, hemisphere and a right circular cylinder of same radii and heights

Objective: 

To show that the volume of a right circular cone, a hemisphere and a right circular cylinder of the same base radii and heights are in the ratio

=  1:2:3.

Pre-requisite knowledge :

(i) Formation of a right circular cone of equal base radius and height.

(ii) Formation of a right circular cylinder of equal base radius and

Materials required :

(i) Flexible transparent plastic sheets of different colours

(ii) Plastic ball

(iii) Geometry box

(iv) Pair of scissors

(v) Tape

(vi) Sand.

To perform the activity :

Steps:

1. Cut the plastic ball into two hollow hemispheres and take one of them

as shown in fig. 31.1. 

2. Measure a diameter of the hemisphere and find its radius. Let the radius

of the hemisphere be r units.

3. Cut a circular piece of radius √2r units with centre O from a plastic sheet. Then cut this circular piece along two radii OA and OB so that the length of the arc of the sector OAB is 2nr units as shown

4. Fold the sector OAB to make radii OA and OB coincide. Fix the radii OA and OB together with tape so that a hollow right circular cone is formed of base radius r unit and of height r units as shown in fig. 

5. Take another piece of plastic sheet of length 27r units and breadth r units. 6. Roll the sheet to form a hollow right circular cylinder of base radius r units and of height r units.

7. Cut another circular piece of radius r units from a plastic sheet and fix it at the bottom of the cylinder to obtain a right circular cylindrical vessel as shown in fig. 

8. Fill the cone with sand and pour the sand into the hemisphere. Repeat this process twice.

9. Fill the cone with sand and pour the sand into the cylindrical vessel. Repeat this process three times.

Result:

We observe that: 

(i) After filling the cone with sand twice and pouring the sand into

hemisphere, the hemisphere is exactly filled with the sand.

(ii) After filling the cone with sand three times and pouring the sand into the cylindrical vessel, the vessel is exactly filled with the sand.. It follows that if right circular cone, a hemisphere and a right circular cylinder have equal radii and same heights, then volume of cone: volume of hemisphere : volume of cylinder

=      1: 2:3.

Remark:

Mathematically, we know that:

volume of a right circular cone of base radius r and height h= 1/3 πr²h

volume of hemisphere of radius r = 2/3πr³

 volume of a right circular cylinder of base radius r and height h = πr²h

 In particular, if r = h, then the volume relationship between the three objects

    = 1/3πr³ : 2/3πr³:πr³

   = 1 : 2 : 3