The length, breadth and height of a cuboid are 4 cm, 6 cm and 10 cm respectively. Find the total surface area.
248 cm2
Calculating Total Surface Area of a Cuboid
The problem asks us to find the total surface area of a cuboid given its dimensions: length, breadth, and height.
Understanding Cuboid Dimensions
A cuboid is a three-dimensional shape with six rectangular faces. The dimensions given are:
- Length (l) = 4 cm
- Breadth (b) = 6 cm
- Height (h) = 10 cm
Formula for Total Surface Area of a Cuboid
The total surface area of a cuboid is the sum of the areas of all six faces. Since opposite faces are identical, the formula is derived by adding the areas of the three unique pairs of faces:
Area of front and back faces = $2 \times (length \times height) = 2lh$
Area of top and bottom faces = $2 \times (length \times breadth) = 2lb$
Area of left and right faces = $2 \times (breadth \times height) = 2bh$
Total Surface Area (TSA) = Area of front/back + Area of top/bottom + Area of left/right
The formula for the total surface area (TSA) of a cuboid is:
TSA = $2(lb + bh + hl)$
Step-by-Step Calculation of Total Surface Area
Now, let's substitute the given dimensions into the formula:
- Length (l) = 4 cm
- Breadth (b) = 6 cm
- Height (h) = 10 cm
TSA = $2 \times ((4 \text{ cm} \times 6 \text{ cm}) + (6 \text{ cm} \times 10 \text{ cm}) + (10 \text{ cm} \times 4 \text{ cm}))$
First, calculate the area of each pair of faces:
- $l \times b = 4 \text{ cm} \times 6 \text{ cm} = 24 \text{ cm}^2$
- $b \times h = 6 \text{ cm} \times 10 \text{ cm} = 60 \text{ cm}^2$
- $h \times l = 10 \text{ cm} \times 4 \text{ cm} = 40 \text{ cm}^2$
Next, sum these areas:
$lb + bh + hl = 24 \text{ cm}^2 + 60 \text{ cm}^2 + 40 \text{ cm}^2 = 124 \text{ cm}^2$
Finally, multiply the sum by 2 to get the total surface area:
TSA = $2 \times 124 \text{ cm}^2 = 248 \text{ cm}^2$
Final Answer Confirmation
The total surface area of the cuboid with length 4 cm, breadth 6 cm, and height 10 cm is 248 cm2.
Revision Table: Cuboid Properties
| Property | Formula | Description |
|---|---|---|
| Volume | $V = l \times b \times h$ | Space occupied by the cuboid |
| Lateral Surface Area | $LSA = 2h(l + b)$ | Area of the four side faces (excluding top and bottom) |
| Total Surface Area | $TSA = 2(lb + bh + hl)$ | Area of all six faces |
| Diagonal of Cuboid | $d = \sqrt{l^2 + b^2 + h^2}$ | Length of the longest diagonal inside the cuboid |
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, it's like unfolding the box and finding the total area of the resulting 2D shape (net). Calculating surface area is important in various real-world applications, such as determining the amount of paint needed to cover a wall, the material required to wrap a gift, or the amount of heat that can be radiated or absorbed by an object.
The units for surface area are always square units (e.g., cm2, m2, in2) because it represents an area.
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:
