The sum of length, breadth and height of a cuboid is 20 cm. If the length of the diagonal is 12 cm, then find the total surface area of cuboid.
256 cm2
Finding Total Surface Area of a Cuboid
The question asks us to find the total surface area of a cuboid given the sum of its length, breadth, and height, and the length of its diagonal.
Let the length, breadth, and height of the cuboid be \(l\), \(b\), and \(h\) respectively.
Given Information
- Sum of length, breadth, and height: \(l + b + h = 20\) cm
- Length of the diagonal of the cuboid: \(d = 12\) cm
Formulas for a Cuboid
The key formulas needed for this problem are:
- Length of the diagonal, \(d = \sqrt{l^2 + b^2 + h^2}\)
- Total surface area, \(TSA = 2(lb + bh + hl)\)
Step-by-Step Solution
We are given \(l + b + h = 20\). Let's square both sides of this equation:
\((l + b + h)^2 = 20^2\)
\((l + b + h)^2 = 400\)
Now, we expand the left side using the algebraic identity \((x + y + z)^2 = x^2 + y^2 + z^2 + 2(xy + yz + zx)\):
\(l^2 + b^2 + h^2 + 2(lb + bh + hl) = 400\)
We also know the formula for the diagonal: \(d = \sqrt{l^2 + b^2 + h^2}\). Squaring this gives:
\(d^2 = l^2 + b^2 + h^2\)
We are given \(d = 12\) cm, so \(d^2 = 12^2 = 144\).
Thus, we have \(l^2 + b^2 + h^2 = 144\).
Now substitute this value into the expanded equation:
\(144 + 2(lb + bh + hl) = 400\)
Notice that the term \(2(lb + bh + hl)\) is the formula for the total surface area (TSA) of the cuboid.
So, the equation becomes:
\(144 + TSA = 400\)
To find the TSA, subtract 144 from both sides:
\(TSA = 400 - 144\)
\(TSA = 256\)
The total surface area of the cuboid is 256 cm\(^2\).
Summary of Calculation
| Step | Equation/Formula | Value |
|---|---|---|
| 1 | Sum of dimensions \((l+b+h)\) | 20 cm |
| 2 | Diagonal \(d\) | 12 cm |
| 3 | Square of sum \((l+b+h)^2\) | \(20^2 = 400\) |
| 4 | Diagonal squared \((d^2 = l^2+b^2+h^2)\) | \(12^2 = 144\) |
| 5 | Expanded square of sum | \(l^2+b^2+h^2 + 2(lb+bh+hl) = 400\) |
| 6 | Substitute values | \(144 + TSA = 400\) |
| 7 | Calculate TSA | \(TSA = 400 - 144 = 256\) cm\(^2\) |
The calculated total surface area matches one of the given options.
Revision Table: Cuboid Formulas
| Property | Formula (l, b, h are dimensions) |
|---|---|
| Volume | \(V = l \times b \times h\) |
| Total Surface Area (TSA) | \(TSA = 2(lb + bh + hl)\) |
| Lateral Surface Area (LSA) | \(LSA = 2h(l+b)\) |
| Length of Diagonal | \(d = \sqrt{l^2 + b^2 + h^2}\) |
Additional Information: Understanding Cuboids
A cuboid is a three-dimensional solid shape bounded by six rectangular faces. It is also known as a rectangular prism. Here are some key properties:
- It has 6 faces, 12 edges, and 8 vertices.
- Opposite faces are identical rectangles.
- The angle between any two adjacent faces is 90 degrees.
- A cube is a special type of cuboid where all edges are of equal length (\(l=b=h\)).
Understanding the relationship between the sum of dimensions, diagonal, and surface area is a common application of algebraic identities in geometry problems. The identity \((l+b+h)^2 = l^2+b^2+h^2 + 2(lb+bh+hl)\) is particularly useful in such cases, directly linking the sum of dimensions, the square of the diagonal, and the total surface area.
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