Mathematics Lab Activity-19 Class X

Mathematics Lab Activities on Surface Area & Volume for class X students with complete observation tables strictly according to the CBSE syllabus.

Chapter - 12

surface area and volume

Activity - 19

Objective

To find the relationship among the volumes of a right circular cone, a hemisphere and a right circular cylinder of equal radii and equal heights.

Material Required

Cardboard, acrylic sheet, cutter, a hollow ball, adhesive, marker, sand or salt.

Procedure

1. Make a cone of radius "a" units and height "a" units by cutting a sector of a circle of suitable radius using acrylic sheet and place it on the cardboard [see Fig. 1]

Figure 1

2. Take a hollow ball of radius, say, "a" units and cut this ball into two halves [see Fig. 2].

Figure 2

3. Make a cylinder of radius "a" unit and height "a" unit, by cutting a rectangular sheet of a suitable size. Stick it on the cardboard [see Fig. 3].

Figure 3

4. Fill the cone with sand (or salt) and pour it twice into the hemisphere. The hemisphere is completely filled with sand.

Therefore, volume of cone = (1/2) volume of hemisphere.

5. Fill the cone with sand (or salt ) and pour it thrice into the cylinder. The cylinder is completely filled with sand.

Therefore, volume of cone = (1/3) volume of cylinder.

6. Volume of cone : Volume of hemisphere : Volume of cylinder = 1 : 2 : 3

Observations & calculations

Let radius of cone = "a" units

Height of cone = "a" units

Volume of cone $=\frac{1}{3}\pi a^{3}$ ..........(i)

Radius of cone = "a" units

Volume of cone $=\frac{2}{3}\pi a^{3}$ .......(ii)

Radius of culinder = "a" units

Height of cylinder = "a" units

Volume of cylinder $=\pi a^{2}\times a$ $=\pi a^{3}$ ..... (iii)

Compairing all the three volumes given in eqn. (i), (ii), (iii) we get

$\frac{1}{3}\pi a^{3}\;:\;\frac{2}{3}\pi a^{3}\;:\;\pi a^{3}$

Cancel πa³ in all the ratios we get

$\frac{1}{3}\::\:\frac{2}{3}\::\:1$

Multiply all ratios by 3 we get

1 : 2 : 3

Result

If cone, hemisphere and cylinder have same radius and height then their volumes are in the ratio of 1 : 2 : 3.

Applications

1. This relationship is useful in obtaining the formula for the volume of a cone and that of a hemisphere/sphere from the formula of volume of a cylinder.

2. This relationship among the volumes can be used in making packages of the same material in containers of different shapes such as cone, hemisphere, cylinder.

THANKS FOR YOUR VISIT

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