Cuboid

Rectangular Parallelepiped or Cuboid
A closed box composed of three pairs of rectangular faces placed opposite each other and joined at right angles to each other, also known as a rectangular parallelepiped. The cuboid is also a right prism, a special case of the parallelepiped, and corresponds to what in everyday parlance is known as a (rectangular) "box".

Say the lengths of the sides be denoted a, b, and c. A cuboid with all sides equal (a = b = c) is called a cube, and a cuboid with integer edge lengths a > b > c and face diagonals is called an Euler brick. If the space diagonal is also an integer, the cuboid is called a perfect cuboid.
The volume of a cuboid is given by V = a b c
and the total surface area is S = s(ab + bc + ca)
The lengths of the face diagonals are
dab =(a2 + b2)
dac =(a2 + c2)
dbc =(b2 + c2)
and the length of the space diagonal is
dabc =(a2 + b2 + c2)

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Example: The sum of length, breadth and depth of a cuboid is 19 cm and the length of its diagonal is 11 cm. Find the surface area of the cuboid.
Solution: Let the length, breadth and height of the cube be l cm, b cm and h cm respectively. Then,
l + b + h = 19       ..........(i)
and,
Diagonal = 11 cm
(l2 + b2 + h2) = 11
l2 + b2 + h2 = 121 .............(ii)
Now, l + b + h = 19
(l + b + h)2 = 192
l2 + b2 + h2 + 2(lb + bh + lh) = 361
121 + 2(lb + bh + lh) = 361
2(lb + bh + lh) = 240
Hence, the surface area of the cuboid is 240 cm2.

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