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Cuboid Investigation

Cuboid Investigation
Volume: Total space inside the cuboid
Formula: 𝑉 = π‘™π‘’π‘›π‘”π‘‘β„Ž × π‘€π‘–π‘‘π‘‘β„Ž × β„Žπ‘’π‘–π‘”β„Žπ‘‘
Units: Usually π‘π‘š3 (β€œcubic centimetres”, β€œcentimetres cubed”, β€œcc”, β€œml”)
Surface area: The total area (flat space) of all 6 faces of the cuboid
Formula: 𝑆𝐴 = 2 × (π‘™π‘’π‘›π‘”π‘‘β„Ž × π‘€π‘–π‘‘π‘‘β„Ž + π‘™π‘’π‘›π‘”π‘‘β„Ž × β„Žπ‘’π‘–π‘”β„Žπ‘‘ + π‘€π‘–π‘‘π‘‘β„Ž × β„Žπ‘’π‘–π‘”β„Žπ‘‘)
Units: Usually π‘π‘š2 (β€œsquare centimetres”, β€œcentimetres squared”)
Investigate the different surface areas of cuboids with a volume of πŸ—πŸŽπŸŽπ’„π’ŽπŸ‘ .
What is the largest surface area you can find? What is the smallest?
ο‚· How do I begin?
Think of any three numbers that multiply together to make 900. These can be the
dimensions of your cuboid. Draw a diagram and work out the total surface area.
ο‚· What next?
Once you have three numbers that work, you can create a new set of three by scaling
one number up and another down. Eg, if you have 50π‘π‘š, 9π‘π‘š and 2π‘π‘š, then 25π‘π‘š,
18π‘π‘š and 2π‘π‘š will also work (halving one number and doubling another in this case).
Eg:
Dimensions:
Surface area:
πΏπ‘’π‘›π‘”π‘‘β„Ž = 2π‘π‘š
π‘Šπ‘–π‘‘π‘‘β„Ž = 50π‘π‘š
2 × (2 × 50 + 2 × 9 + 50 × 9) = πŸπŸπŸ‘πŸ”π’„π’ŽπŸ
Conclusions:
The cuboid with the largest surface area was: ____________________
It has a total surface area of: ____________________
The cuboid with the smallest surface area was: ____________________
It has a total surface area of: ____________________
What do you notice?
π»π‘’π‘–π‘”β„Žπ‘‘ = 9π‘π‘š
Cuboid Investigation SOLUTIONS
Volume: Total space inside the cuboid
Formula: 𝑉 = π‘™π‘’π‘›π‘”π‘‘β„Ž × π‘€π‘–π‘‘π‘‘β„Ž × β„Žπ‘’π‘–π‘”β„Žπ‘‘
Units: Usually π‘π‘š3 (β€œcubic centimetres”, β€œcentimetres cubed”, β€œcc”, β€œml”)
Surface area: The total area (flat space) of all 6 faces of the cuboid
Formula: 𝑆𝐴 = 2 × (π‘™π‘’π‘›π‘”π‘‘β„Ž × π‘€π‘–π‘‘π‘‘β„Ž + π‘™π‘’π‘›π‘”π‘‘β„Ž × β„Žπ‘’π‘–π‘”β„Žπ‘‘ + π‘€π‘–π‘‘π‘‘β„Ž × β„Žπ‘’π‘–π‘”β„Žπ‘‘)
Units: Usually π‘π‘š2 (β€œsquare centimetres”, β€œcentimetres squared”)
Investigate the different surface areas of cuboids with a volume of πŸ—πŸŽπŸŽπ’„π’ŽπŸ‘ .
What is the largest surface area you can find? What is the smallest?
ο‚· How do I begin?
Think of any three numbers that multiply together to make 900. These can be the
dimensions of your cuboid. Draw a diagram and work out the total surface area.
ο‚· What next?
Once you have three numbers that work, you can create a new set of three by scaling
one number up and another down. Eg, if you have 50π‘π‘š, 9π‘π‘š and 2π‘π‘š, then 25π‘π‘š,
18π‘π‘š and 2π‘π‘š will also work (halving one number and doubling another in this case).
Eg:
Dimensions:
Surface area:
πΏπ‘’π‘›π‘”π‘‘β„Ž = 2π‘π‘š
π‘Šπ‘–π‘‘π‘‘β„Ž = 50π‘π‘š
π»π‘’π‘–π‘”β„Žπ‘‘ = 9π‘π‘š
2 × (2 × 50 + 2 × 9 + 50 × 9) = πŸπŸπŸ‘πŸ”π’„π’ŽπŸ
Conclusions:
The cuboid with the largest surface area was: 1π‘π‘š × 1π‘π‘š × 900π‘π‘š
It has a total surface area of: 3602π‘π‘š2
(note: no limit on surface area if not restricted to whole numbers)
The cuboid with the smallest surface area was: 9π‘π‘š × 10π‘π‘š × 10π‘π‘š
It has a total surface area of: 560π‘π‘š2
(note: the cube 9.65 … π‘π‘š × 9.65 … π‘π‘š × 9.65 … π‘π‘š × gives the overall min: 559.3 … π‘π‘š2 )
What do you notice?
The more similar the lengths, the smaller the surface area.
The closer the cuboid gets to a cube, the smaller the surface area.
Cuboid Investigation SOLUTIONS Continued
Full list of possible surface areas for each integer factor triple:
Length Width Height
9
6
5
5
5
6
4
5
4
5
3
3
3
4
3
3
2
3
2
2
2
3
2
2
2
2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
10
10
12
10
9
6
15
6
9
5
15
12
10
5
6
5
18
4
15
10
9
3
6
5
3
2
30
25
20
18
15
12
10
9
6
5
4
3
2
1
10
15
15
18
20
25
15
30
25
36
20
25
30
45
50
60
25
75
30
45
50
100
75
90
150
225
30
36
45
50
60
75
90
100
150
180
225
300
450
900
SA
560
600
630
640
650
672
690
720
722
770
810
822
840
850
936
990
1072
1074
1080
1120
1136
1218
1224
1280
1512
1808
1920
1922
1930
1936
1950
1974
2000
2018
2112
2170
2258
2406
2704
3602

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