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Maths Project on Volume and Surface Area of Cube and Cuboid

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Volume and Surface Area of Cube and Cuboid

Objective

To explore the changes in surface areas and volumes of cuboids with respect to each other.

Description

Case 1:

Took the cuboids having equal volumes and following dimensions:-

1) Length = 20cm, Breadth = 10cm, Height = 5cm.

2) Length = 10cm, Breadth = 10cm, Height = 10cm.

3) Length = 100cm, Breadth = 5cm, Height = 2cm.

Calculation:-

= 20 x 10 x 5 = 1000 cubic cm. Surface area of cuboid = 2(lb + bh + hl)

= 2(20 x 10 + 10 x 5 + 5 x 12) = 2(200 + 50 + 60) = 620 square cm.

2. Volume of cuboid =lbh

= 10 x 10 x 10 = 1000 cubic cm. Surface area of cuboid = 2(lb + bh + hl)

= 2(10 x 10 + 10 x 10 + 10 x 10) = 2(100 + 100 + 100) = 600 square cm.

= 100 x 5 x 2

= 1000 cubic cm. Surface area of cuboid = 2(lb + bh + hl)

= 2(100 x 5 + 5 x 2 + 2 x 100) = 2(500 + 10 + 200) = 1420 square cm.

Observation:-

All these cuboids have volume =1000 cubic cm that is volumes are equal. The surface areas are not equal.

The surface of cuboid which is a cube is minimum.

Case 2:

Took the cuboids having equal volumes and following dimensions:-

1) Length = 14cm, Breadth = 6cm, Height = 5.4cm.

2) Length = 8cm, Breadth = 8cm, Height = 8cm.

3) Length = 16cm, Breadth = 6.4cm, Height = 4cm.

= 14 x 6 x 5.4 = 453.6 cubic cm.

Surface area of cuboid = 2(lb + bh + hl)

= 2(14 x 6 + 6 x 5.4 + 5.4 x 14) = 2(84 + 32.4 + 75.6) = 384 square cm.

2) Volume of cuboid =lbh

= 8 x 8 x 8 = 512 cubic cm.

Surface area of cuboid = 2(lb + bh + hl)

= 2(8 x 8 + 8 x 8 + 8 x 8) = 2(64 + 64 + 64) = 384 square cm.

= 16 x 6.4 x 4 = 409.6 cubic cm. Surface area of cuboid = 2(lb + bh + hl)

= 2(16 x 6.4 + 6.4 x 4+ 4 x 16) = 2(102.4 + 25.6+ 64) = 384 square cm.

Observation:

All these cuboids have surface area = 384 square cm that is surface areas are equal. The volumes are not equal.

The cuboid which a cube has largest volume.

Final Conclusion:

1) Of all the cuboids with equal volumes, the cube has the minimum surface area.

2) Of all the cuboids with equal surface areas, the cube has the maximum volume.

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