Question

The sum of the length, breadth and height of a cuboid is 38 cm and the lengths of its diagonal is 22 cm. Then find the total surface area of the cuboid.

Answer 1

Hint:- Formula for diagonal of the cuboid is \[d = \sqrt {{a^2} + {b^2} + {c^2}} \] where d is the diagonal and a, b and are the length, breadth and height of the cuboid. And the total surface area of the cuboid is 2(ab + bc + ca). So, let us use the identity of \[{\left( {a + b + c} \right)^2}\] to find the total surface area of the cuboid.

Complete step-by-step solution -
Now as we can see from the above figure that diagonal of the cuboid ABCDHGFE is the line joining the points A and D.
And as we know that a, b, c are the length, breadth and height of any cuboid then its diagonal length must be equal to \[\sqrt {{a^2} + {b^2} + {c^2}} \].
So, let the length of be AB = DC = EF = HG = l cm
Breadth of the cuboid will be AE = DH = BF = CG = b cm
And, the height of the cuboid will be equal to AD = EH = BC = FG = h cm.
And it is given that the sum of the length, breadth and height of the cuboid is 38 cm.
So, l + b + h = 38 cm (1)
And, \[\sqrt {{l^2} + {b^2} + {h^2}} \] = 22 cm (2)
On squaring both sides of the above equation. We get,
\[{l^2} + {b^2} + {h^2} = 484\] (3)
Now as we know that the formula for total surface area is 2(ab + bc + ca). So, the total surface area of the above cuboid will be 2(lb + bh + hl).
So, to find the value of the total surface area of the cuboid we must use the identity of \[{\left( {a + b + c} \right)^2}\]. As we know that \[{\left( {a + b + c} \right)^2} = {a^2} + {b^2} + {c^2} + 2\left( {ab + bc + ca} \right)\].
So, \[{\left( {l + b + h} \right)^2} = {l^2} + {b^2} + {h^2} + 2\left( {lb + bh + hl} \right)\] (4)
Now putting the value of \[l + b + h\] and \[{l^2} + {b^2} + {h^2}\] in equation 4. We get,
\[{\left( {38} \right)^2} = 484 + 2\left( {lb + bh + hl} \right)\]
Now solving the above equation. We get,
\[1444 = 484 + 2\left( {lb + bh + hl} \right)\]
Subtracting 484 to both sides of the above equation we get,
\[960 = 2\left( {lb + bh + hl} \right)\]
As RHS of the above equation is the formula for calculating the total surface area of a cuboid. So, LHS must be equal to the total surface area of the cuboid.
Hence, the total surface area of the cuboid is 960\[c{m^2}\].
Note:- The main diagonal of the cuboid is one which cuts the cuboid through the centre of it. And length if the diagonal is \[\sqrt {{l^2} + {b^2} + {h^2}} \] if l, b and h are the dimensions of the cuboid. So, we have to use the formula of calculate the value of \[{\left( {l + b + h} \right)^2}\] which is \[{\left( {l + b + h} \right)^2} = {l^2} + {b^2} + {h^2} + 2\left( {lb + bh + hl} \right)\]. So, from this identity we can easily get the total surface area of the cuboid value of \[l + b + h\] is given in the question and \[{l^2} + {b^2} + {h^2}\] is equal to the square of the diagonal which is also given.