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CAT 2025 Lesson : Mensuration - Cuboid & Cube

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3.3 - D Figures

3.1 Cuboid

A cuboid is a figure with three dimensions, namely length, breadth and height. Each face of a cuboid is a rectangle and all corners or vertices of a cuboid are 90-degree angles. Ultimately, the shape of a cuboid is like a rectangular box with varying length, breadth and height. Note that breadth is also referred to as width.

In the adjacent figure,
l\bold{l} is the length.
b\bold{b} is the breadth.
h\bold{h} is the height

Volume of a cuboid = l×b×hl \times b \times h
Lateral surface area (LSA) = 2(l+b)h2(l + b)h
Total surface area (TSA) = 2(lb+bh+hl)2(lb + bh + hl)

Longest Diagonal =l2+b2+h2\sqrt {l^2 + b^2 + h^2}

Example 4

The ratio of the length and breadth of a rectangular fish tank is 3 : 2. When 48 litre of water is poured into the tank, the water level raises to 20 cm. Then the breadth of the fish tank is

Solution

A rectangular tank is a cuboid. Let the length and breadth of the tank in cm be 3x3x and 2x2x respectively.

Volume of the tank =l×b×hl \times b \times h = 48 litres = 48000 cm3cm^3

⇒ 3x×2x×20=480003x \times 2x \times 20 = 48000

⇒ 6x2=24006 x^2 = 2400

⇒ x2=400=20x^2 = \sqrt{400} = 20

Breadth of the tank = 2x2x = 40 cm

Answer: 40 cm

Example 5

From each of the six sides of a cuboid of dimensions 8m×6m×5m8m \times 6m \times 5m, a block with 1 metre thickness is sliced and removed. What is the volume of the portion removed from the cuboid?

Solution

Volume of the cuboid before removal = 8×6×58 \times 6 \times 5= 240 m3m^3

When 1m of thickness is removed from each side, each of length, breadth and height reduce by 1m on each of the two sides they connect. Therefore, reduction in each of the dimensions is 2 m.

Volume of the resultant cube = (8−2)×(6−2)×(5−2)=6×4×3=72 m3(8 - 2) \times (6 - 2) \times (5 - 2) = 6 \times 4 \times 3 = 72 \ m^3

Volume of the portion removed = 240 – 72 = 168 m3m^3

Answer: 168 m3m^3

3.2 Cube

A cuboid where the length, breadth and height are equal is a Cube. This six-sided figure is shaped like a square box with all sides being equal and all the faces having the same area.
In the adjacent figure,
a is the length of each edge

Volume of a cube = a3a^{3}

Lateral surface area (LSA) = 4a24a^{2}
Total surface area (TSA) = 6a26a^{2}
Longest Diagonal = 3a\sqrt{3}a


Example 6

What is the total surface area (in cm2^{2}) of the largest cube that can be fit into a sphere of radius 7 cm?

Solution

Let aa be the side of the square.
The cube's longest diagonal will be the sphere's diameter

∴\therefore 3a=2×7\sqrt{3}a = 2 \times 7
⇒ a=143a = \dfrac{14}{\sqrt{3}}

Total Surface Area of the cube = 6a2=6×(143)2=6a^{2} = 6 \times \left( \dfrac{14}{\sqrt{3}} \right)^{2} = 392 cm2^{2}

Answer: 392

Example 7

A large cube is cut into 1000 identical smaller cubes. What is the ratio of the surface area of the larger cube to that of the sum of surface areas of the 1000 smaller cubes?

Solution

As the ratio is asked, we can assume a value for the side of the larger cube, say 10 cm (instead of a variable). The question does not state Total Surface Area (6a2a^{2}) or Lateral Surface Area (4a2a^{2}). However, the ratio will be unchanged as the constant on both sides will be cancelled. Let's use Lateral Surface Area here.

Volume of the larger cube =103=1000= 10^{3} = 1000 cm3^{3}

As the larger cube is divided into 1000 smaller cubes,
Volume of a smaller cube =10001000=1\dfrac{1000}{1000} = 1cm3^{3}
∴\therefore side of a smaller cube = 13=1\sqrt[3]{1} = 1cm

LSA of larger cube = 4×1024 \times 10^{2} = 400
LSA of 1000 smaller cubes =1000×4×12=4000= 1000 \times 4 \times 1^{2} = 4000

Ratio of LSAs = 400 : 4000 = 1 : 10

Answer: 1 : 10

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