Maths Project on Volume and Surface Area of Cube and Cuboid
Maths Project on 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.
Case 1: Cuboids with Equal Volumes
Dimensions taken:
- Length = 20 cm, Breadth = 10 cm, Height = 5 cm
- Length = 10 cm, Breadth = 10 cm, Height = 10 cm
- Length = 100 cm, Breadth = 5 cm, Height = 2 cm
Calculations
1. Volume of cuboid = l × b × h = 20 × 10 × 5 = 1000 cubic cm
Surface area of cuboid = 2(lb + bh + hl) = 2(20 × 10 + 10 × 5 + 5 × 20) = 2(200 + 50 + 100) = 700 square cm
2. Volume of cuboid = l × b × h = 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 = l × b × h = 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 three cuboids have volume = 1000 cubic cm, that is, the volumes are equal. The surface areas are not equal. The surface area of the cuboid which is a cube (10 × 10 × 10) is minimum.
Case 2: Cuboids with Equal Surface Areas
Dimensions taken:
- Length = 14 cm, Breadth = 6 cm, Height = 5.4 cm
- Length = 8 cm, Breadth = 8 cm, Height = 8 cm
- Length = 16 cm, Breadth = 6.4 cm, Height = 4 cm
Calculations
1. Volume of cuboid = l × b × h = 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 = l × b × h = 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
3. Volume of cuboid = l × b × h = 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 three cuboids have surface area = 384 square cm, that is, the surface areas are equal. The volumes are not equal. The cuboid which is a cube (8 × 8 × 8) has the largest volume.
Final Conclusion
- Of all the cuboids with equal volumes, the cube has the minimum surface area.
- Of all the cuboids with equal surface areas, the cube has the maximum volume.