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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.

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:- 1. Volume of cuboid =lbh = 20 × 10 × 5 = 1000 cubic cm. Surface area of cuboid = 2(lb + bh + hl) = 2(20 × 10 + 10 × 5 + 5 × 12) = 2(200 + 50 + 60) = 620 square cm. 2. Volume of cuboid =lbh = 10 × 10 × 10 = 1000 cubic cm. Surface area of cuboid = 2(lb + bh + hl) = 2(10 × 10 + 10 × 10 + 10 × 10) = 2(100 + 100 + 100) = 600 square cm. 3. Volume of cuboid =lbh = 100 × 5 × 2 = 1000 cubic cm. Surface area of cuboid = 2(lb + bh + hl) = 2(100 × 5 + 5 × 2 + 2 × 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. 1. Volume of cuboid =lbh = 14 × 6 × 5.4 = 453.6 cubic cm. Surface area of cuboid = 2(lb + bh + hl) = 2(14 × 6 + 6 × 5.4 + 5.4 × 14) = 2(84 + 32.4 + 75.6) = 384 square cm. 2) Volume of cuboid =lbh = 8 × 8 × 8 = 512 cubic cm. Surface area of cuboid = 2(lb + bh + hl) = 2(8 × 8 + 8 × 8 + 8 × 8) = 2(64 + 64 + 64) = 384 square cm. Volume of cuboid =lbh = 16 × 6.4 × 4 = 409.6 cubic cm. Surface area of cuboid = 2(lb + bh + hl) = 2(16 × 6.4 + 6.4 × 4+ 4 × 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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