The length of a cuboid is 10 cm. If the breath of the cuboid is half of its lengths and height of the cuboid is twice of its length, then what is the lateral surface area of the cuboid?
600 cm2
Calculating Lateral Surface Area of a Cuboid
The question asks us to find the lateral surface area of a cuboid given its dimensions. We are provided with the length of the cuboid, and the breadth and height are described in terms of the length.
Understanding Cuboid Dimensions
Let the length of the cuboid be \(l\), the breadth be \(b\), and the height be \(h\).
- Given: Length \(l = 10\) cm.
- Breadth is half of the length: \(b = \frac{l}{2} = \frac{10}{2} = 5\) cm.
- Height is twice of the length: \(h = 2 \times l = 2 \times 10 = 20\) cm.
So, the dimensions of the cuboid are: length = 10 cm, breadth = 5 cm, and height = 20 cm.
Lateral Surface Area Formula for a Cuboid
The lateral surface area (LSA) of a cuboid is the sum of the areas of its four side faces (excluding the top and bottom faces). The four side faces are two rectangles of dimensions \(l \times h\) and two rectangles of dimensions \(b \times h\).
The formula for the lateral surface area of a cuboid is:
\[ \text{LSA} = 2 \times (\text{length} + \text{breadth}) \times \text{height} \]
\[ \text{LSA} = 2 \times (l + b) \times h \]
Step-by-Step Calculation of Lateral Surface Area
Now, we substitute the values of \(l\), \(b\), and \(h\) into the formula:
\[ \text{LSA} = 2 \times (10 \text{ cm} + 5 \text{ cm}) \times 20 \text{ cm} \]
First, calculate the sum of length and breadth:
\[ 10 \text{ cm} + 5 \text{ cm} = 15 \text{ cm} \]
Now substitute this value back into the formula:
\[ \text{LSA} = 2 \times (15 \text{ cm}) \times 20 \text{ cm} \]
Perform the multiplication:
\[ \text{LSA} = 30 \text{ cm} \times 20 \text{ cm} \]
\[ \text{LSA} = 600 \text{ cm}^2 \]
Result
The lateral surface area of the cuboid is 600 cm\(^2\).
Revision Table: Cuboid Formulas
| Measurement | Formula |
|---|---|
| Length | \(l\) |
| Breadth | \(b\) |
| Height | \(h\) |
| Volume | \(l \times b \times h\) |
| Lateral Surface Area (LSA) | \(2 \times (l + b) \times h\) |
| Total Surface Area (TSA) | \(2 \times (lb + bh + hl)\) |
| Diagonal of Cuboid | \(\sqrt{l^2 + b^2 + h^2}\) |
Additional Information on Surface Area
The surface area of a 3D shape like a cuboid is the total area of all its faces. We discussed the lateral surface area, which includes only the side faces. The total surface area (TSA) includes all six faces: the four side faces, the top face, and the bottom face.
- Lateral Faces: The four vertical rectangles around the sides. Their combined area is \(2(lh) + 2(bh) = 2h(l+b)\), which is the LSA formula.
- Top and Bottom Faces: These are two identical rectangles, each with dimensions \(l \times b\). Their combined area is \(lb + lb = 2lb\).
- Total Surface Area: LSA + Area of Top and Bottom Faces = \(2h(l+b) + 2lb = 2(hl + hb + lb)\). This matches the TSA formula.
Understanding the difference between lateral and total surface area is important in geometry problems.
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:
