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
A cylindrical tube, open at both ends, is made of a metal sheet which is 0.5 cm thick. Its outer radius is 4 cm and length is 2 m. How much metal (in cm 3) has been used in making the tube?
The volume of a right circular cone is 308 cm 3 and the radius of its base is 7 cm. What is the curved surface area (in cm 2) of the cone? (Take π = \(\frac{22}{7} \) )
The slant height and radius of a right circular cone are in the ratio 29 ∶ 20. If its volume is 4838.4 π cm 3, then its radius is:
Six cubes, each of edge 2 cm, are joined end to end. What is the total surface area of the resulting cuboid in cm 2?
A solid cube of side 8 cm is dropped into a rectangular container of length 16 cm, breadth 8 cm and height 15 cm which is partly filled with water. If the cube is completely submerged, then the rise of water level (in cm) is:
