The sum of the length and breadth of a cuboid is 16 cm. If the height of the cuboid is 1/4 of the sum of its length and breadth, then what is the lateral surface area of the cuboid?
128 cm2
Calculating the Lateral Surface Area of a Cuboid
Let the length, breadth, and height of the cuboid be $l$, $b$, and $h$ respectively.
Understanding the Given Information
We are given the following information about the cuboid:
- The sum of the length and breadth is 16 cm. This can be written as:
- $l + b = 16 \text{ cm}$
- The height of the cuboid is $\frac{1}{4}$ of the sum of its length and breadth. This can be written as:
- $h = \frac{1}{4} \times (l + b)$
Finding the Height of the Cuboid
We can substitute the given sum of length and breadth into the equation for the height:
$h = \frac{1}{4} \times (l + b)$
Since $l + b = 16 \text{ cm}$, we have:
$h = \frac{1}{4} \times 16 \text{ cm}$
Calculating the height:
$h = 4 \text{ cm}$
Calculating the Lateral Surface Area of the 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 formula for the lateral surface area of a cuboid is:
$\text{LSA} = 2 \times \text{height} \times (\text{length} + \text{breadth})$
In terms of $l$, $b$, and $h$:
$\text{LSA} = 2h(l + b)$
We have already found that $h = 4 \text{ cm}$ and we are given that $l + b = 16 \text{ cm}$. Now, we substitute these values into the formula:
$\text{LSA} = 2 \times 4 \text{ cm} \times 16 \text{ cm}$
$\text{LSA} = 8 \text{ cm} \times 16 \text{ cm}$
$\text{LSA} = 128 \text{ cm}^2$
Therefore, the lateral surface area of the cuboid is 128 cm$^2$.
Summary of Calculations
| Parameter | Value/Relation | Calculated Value |
|---|---|---|
| Sum of Length and Breadth ($l+b$) | Given | 16 cm |
| Height ($h$) | $\frac{1}{4} \times (l+b)$ | $\frac{1}{4} \times 16 \text{ cm} = 4 \text{ cm}$ |
| Lateral Surface Area (LSA) | $2h(l+b)$ | $2 \times 4 \text{ cm} \times 16 \text{ cm} = 128 \text{ cm}^2$ |
Revision Table: Cuboid Formulas
| Concept | Formula | Description |
|---|---|---|
| Volume | $V = l \times b \times h$ | Space occupied by the cuboid |
| Total Surface Area (TSA) | $\text{TSA} = 2(lb + bh + hl)$ | Sum of the areas of all six faces |
| Lateral Surface Area (LSA) | $\text{LSA} = 2h(l + b)$ | Sum of the areas of the four side faces |
| Diagonal | $d = \sqrt{l^2 + b^2 + h^2}$ | Length of the longest segment connecting opposite vertices |
Additional Information: Understanding Surface Area
Surface area is a measure of the total area that the surface of a three-dimensional object occupies. For a cuboid, there are two main types of surface area often calculated: total surface area and lateral surface area.
- Total Surface Area: This includes the area of all six faces (top, bottom, front, back, left, right).
- Lateral Surface Area: This includes only the area of the four vertical faces (front, back, left, right). It is often referred to as the area of the walls if the base is the floor and the top is the ceiling.
In this problem, we specifically calculated the lateral surface area using the given dimensions.
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:
