What will be the length of longest diagonal of the cuboid having length 13 cm width 11 cm and height 20 cm?
26.27 cm
Calculating the Longest Diagonal of a Cuboid
A cuboid is a three-dimensional solid shape bounded by six rectangular faces. It is also known as a rectangular prism. The longest diagonal of a cuboid is a line segment connecting two opposite vertices that do not lie on the same face.
Formula for the Longest Diagonal
The length of the longest diagonal (\(d\)) of a cuboid with length (\(l\)), width (\(w\)), and height (\(h\)) is given by the formula derived from extending the Pythagorean theorem to three dimensions:
\[ d = \sqrt{l^2 + w^2 + h^2} \]
Applying the Formula with Given Dimensions
In this problem, we are given the dimensions of the cuboid:
- Length (\(l\)) = 13 cm
- Width (\(w\)) = 11 cm
- Height (\(h\)) = 20 cm
Now, we substitute these values into the formula:
\[ d = \sqrt{(13 \, \text{cm})^2 + (11 \, \text{cm})^2 + (20 \, \text{cm})^2} \]
Calculate the squares of each dimension:
- \(13^2 = 169\)
- \(11^2 = 121\)
- \(20^2 = 400\)
Sum the squared values:
\[ 13^2 + 11^2 + 20^2 = 169 + 121 + 400 = 690 \]
Now, take the square root of the sum:
\[ d = \sqrt{690} \, \text{cm} \]
Calculating the square root of 690 gives an approximate value:
\[ \sqrt{690} \approx 26.2679 \, \text{cm} \]
Rounding this value to two decimal places, we get:
\[ d \approx 26.27 \, \text{cm} \]
Conclusion
The length of the longest diagonal of the cuboid is approximately 26.27 cm.
Comparing this result with the given options:
- Option 1: 23.45 cm
- Option 2: 26.27 cm
- Option 3: 34.65 cm
- Option 4: 22.65 cm
The calculated length matches Option 2.
Revision Table: Cuboid Formulas
| Concept | Formula | Description |
|---|---|---|
| Volume (V) | \(V = l \times w \times h\) | Space occupied by the cuboid |
| Surface Area (SA) | \(SA = 2(lw + lh + wh)\) | Total area of all faces |
| Longest Diagonal (d) | \(d = \sqrt{l^2 + w^2 + h^2}\) | Length of the longest distance between two vertices |
Additional Information: Pythagorean Theorem in 3D
The formula for the longest diagonal of a cuboid is a direct application of the Pythagorean theorem in three dimensions. Consider a cuboid with vertices at (0,0,0) and (l,w,h). The distance between these two points is the length of the longest diagonal. Using the distance formula in 3D (which is derived from the Pythagorean theorem), the distance is \(\sqrt{(l-0)^2 + (w-0)^2 + (h-0)^2}\), which simplifies to \(\sqrt{l^2 + w^2 + h^2}\). This shows how the familiar theorem extends from 2D right triangles to 3D rectangular solids.
Important Questions from Solid Figures
The area of the floor of a cubical room is 192 m 2. The length of the longest rod that can be kept in that room is :
A solid metallic rectangular block of dimensions 112 cm × 44 cm × 25 cm is melted and recast into a cylinder of radius 35 cm. The curved surface area (in cm 2) of the cylinder is: (Take π = 22/7)
If the volume of a cube is 175616 cm 3, what is its side?
The volume of a right circular cone is 1232 cm 3. If the height of the cone is 24 cm, then what will be the radius of its base?
A right triangle contains the right angle between the sides 5 cm and 7 cm. A cone is generated by revolving about the side 5 cm. The volume of this cone is:
