How to Calculate Volume and Surface Area of a Cuboid
What is Cuboid?
A cuboid is around us in day-to-day life. We see it in the form of bricks, shoeboxes, cuboid objects, etc. A cuboid is a three-dimensional figure which has six rectangular faces, twelve edges, and eight vertices. The cuboid shape is formed with a closed rectangular face.
Cuboid Examples
The boxes we use, the lunch boxes we take to school, the bricks we use to build a house, the pencil box, etc. are the known examples existing around us. These are some common examples that we have seen in our surroundings. The following pictures have some of the common examples.
Cuboid Examples
What is Surface Area?
The surface area is the area of a solid object that measures the surface that occupies the object. The surface area of a cuboid is the area of 6 rectangular faces. There is a formula to calculate the surface area of a cuboid.
Surface Area
Formula for the Surface Area of Cuboid:
Curved surface area : \[2h\left( {l + b} \right)\]
Total Surface area : \[2\left( {lb + bh + hl} \right)\]
Where l is length, b is the breadth and h stands for height.
Cuboid Examples: Surface Area
Example 1: The cuboid has its dimension given as length is 8 cm, width is 6 cm and height is 5 cm. Find the Total surface area of the given cuboid.
Given
Height(h) \[ = {\rm{ }}5{\rm{ }}cm\]
Width(w) \[ = {\rm{ }}6{\rm{ }}cm\]
Length(l) \[ = {\rm{ }}8{\rm{ }}cm\]
Total surface area \[ = {\rm{ }}2\left( {lb + bh + hl} \right)\]
\[ = {\rm{ }}2\left[ {{\rm{ }}\left( {8 \times 6} \right){\rm{ }} + {\rm{ }}\left( {6 \times 5} \right){\rm{ }} + {\rm{ }}\left( {5 \times 8} \right)} \right]\]
\[ = {\rm{ }}2{\rm{ }}\left[ {{\rm{ }}48{\rm{ }} + {\rm{ }}30{\rm{ }} + {\rm{ }}40{\rm{ }}} \right]\]
\[ = {\rm{ }}2 \times 118\]
\[ = 236\]
Hence, the total surface area is \[236c{m^2}.\]
Volume Of Cuboid
The volume of a cuboid is the product of length, breadth, and height in cubic units.
The formula of the volume of cuboid: length \[ \times \] breadth \[ \times \] height
Let’s understand it by doing some examples.
Cuboid Example: Volume Of The Cuboid
Example 2: Calculate the volume of a cuboid with a length of 8 cm, breadth of 25 cm, and height of 50 cm.
Given
Length \[ = \;8{\rm{ }}cm\]
Breadth \[ = {\rm{ }}25{\rm{ }}cm\]
Height \[ = {\rm{ }}50{\rm{ }}cm\]
The volume of cuboid: length \[ \times \] breadth \[ \times \] height
\[ = \,8\, \times \,25\, \times \,50\]
\[ = \,10000\,\,c{m^3}\]
Solved Questions:
1. Find the surface area of a cuboid whose length is 5 cm, breadth is 6 cm and height is 9 cm.
Given
Length \[ = {\rm{ }}5{\rm{ }}cm\]
Breadth \[ = \,6{\rm{ }}cm\]
Height \[{\rm{ = }}9{\rm{ }}cm\]
Surface are of cuboid \[ = {\rm{ }}2\left( {lb + bh + hl} \right)\]
\[ = {\rm{ }}2{\rm{ }}\left[ {\left( {5 \times 6} \right){\rm{ }} + {\rm{ }}\left( {6 \times 9} \right){\rm{ }} + {\rm{ }}\left( {9 \times 5} \right)} \right]\]
\[ = {\rm{ }}2{\rm{ }}\left[ {{\rm{ }}30 + 54 + 45{\rm{ }}} \right]\]
\[ = {\rm{ }}2 \times 129\]
\[\; = {\rm{ }}258{\rm{ }}c{m^2}.\]
2. Find the volume of the cuboid whose length is 8 cm, breadth is 7 cm and height is 4cm.
Given
Length \[{\rm{ = }}8{\rm{ }}cm\]
Breadth \[{\rm{ = }}7{\rm{ }}cm\]
Height \[{\rm{ = }}4{\rm{ }}cm\]
The volume of the cuboid is length × breadth × height
\[ = {\rm{ }}8 \times 7 \times 4\]
\[ = {\rm{ }}224{\rm{ }}c{m^3}.\]
3. Find the sum of the surface area and the volume of the cuboid whose length is 4 cm, breadth is 2 cm and height is 1 cm.
Given
Length \[ = {\rm{ }}4{\rm{ }}cm\]
Breath \[ = {\rm{ }}2{\rm{ }}cm\]
Height \[ = {\rm{ }}1{\rm{ }}cm\]
Surface area of cuboid \[ = {\rm{ }}2\left( {lb + bh + hl} \right)\]
\[ = {\rm{ }}2{\rm{ }}\left[ {{\rm{ }}\left( {4 \times 2} \right){\rm{ }} + {\rm{ }}\left( {2 \times 1} \right){\rm{ }} + {\rm{ }}\left( {1 \times 4} \right)} \right]\]
\[ = {\rm{ }}2{\rm{ }}\left[ {{\rm{ }}8{\rm{ }} + {\rm{ }}2{\rm{ }} + {\rm{ }}4{\rm{ }}} \right]\]
\[ = {\rm{ }}2 \times 14\]
\[ = {\rm{ }}28{\rm{ }}c{m^2}\].
The volume of the cuboid is length × breadth × height
\[ = {\rm{ }}4 \times 2 \times 1\]
\[ = {\rm{ }}8{\rm{ }}c{m^3}.\]
Sum of the surface area and the volume \[ = \,\,28\,\, + \,\,8\,\, = \,\,36\,cm\]
Summary
In this chapter, we have studied the cuboid. A cuboid is a three-dimensional solid object and the measure that the object occupies is the surface of the cuboid and the product of length, breadth, and height is the volume of the object.We also solved the various examples and solved questions to understand the surface area and volume of a cuboid.
FAQs on Volume and Surface Area of a Cuboid Explained
1. What is the volume of a cuboid and how is it calculated?
The volume of a cuboid represents the total three-dimensional space it occupies. It is calculated by multiplying its three fundamental dimensions: length (l), breadth (b), and height (h). The formula is: Volume (V) = length × breadth × height or V = l × b × h. The result is expressed in cubic units, such as cm³ or m³.
2. What is the difference between the Total Surface Area (TSA) and Lateral Surface Area (LSA) of a cuboid?
The key difference lies in which faces are included in the area calculation:
Total Surface Area (TSA) is the sum of the areas of all six rectangular faces of the cuboid (top, bottom, and four sides). The formula is TSA = 2(lb + bh + hl).
Lateral Surface Area (LSA) is the sum of the areas of only the four side faces, excluding the top and bottom faces. The formula is LSA = 2h(l + b).
3. What are the fundamental properties of a cuboid?
A cuboid is a three-dimensional shape with the following defining properties:
It has 6 faces, and all faces are rectangles.
It has 8 vertices (corners) where the edges meet.
It has 12 edges (the lines where the faces meet).
The opposite faces of a cuboid are always parallel and equal in area.
4. How does a cuboid differ from a cube in terms of its properties and formulas?
A cube is a special type of cuboid where all dimensions are equal (length = breadth = height). This leads to differences in their formulas:
Dimensions: A cuboid has different values for length, breadth, and height, while a cube has a single value 'a' for all its sides.
Volume: The volume of a cuboid is l × b × h. For a cube, this simplifies to a × a × a = a³.
Surface Area: The TSA of a cuboid is 2(lb + bh + hl). For a cube, this simplifies to 6a².
5. Why are all three dimensions (length, breadth, and height) essential for defining a unique cuboid?
While you can calculate a cuboid's volume using just its base area and height (Volume = Base Area × Height), this information is insufficient to define a unique cuboid. For example, a base area of 24 sq. cm could belong to a cuboid with dimensions 6x4, 8x3, or 12x2. Each of these would have a different Total Surface Area. Therefore, all three individual dimensions—length, breadth, and height—are essential to uniquely determine both the shape and all properties of the cuboid, including its surface area.
6. If you double all three dimensions of a cuboid, what is the effect on its surface area and volume?
Doubling the length, breadth, and height of a cuboid has a non-linear effect on its surface area and volume:
The surface area becomes four times larger. This is because each term in the TSA formula (lb, bh, hl) is multiplied by 4 (since 2l × 2b = 4lb).
The volume becomes eight times larger. This is because the volume is the product of the three dimensions (2l × 2b × 2h = 8 lwh).
7. What are some real-world examples of a cuboid?
Many everyday objects are cuboids because their shape is stable and easy to stack. Common examples include a book, a smartphone, a brick, a matchbox, a duster, and a rectangular room. Each of these objects has six flat rectangular faces, eight corners, and twelve straight edges, fitting the geometric definition of a cuboid.
8. For a real-world task like painting a room, is it better to use TSA or LSA?
Neither formula is typically used directly. When painting a room, you usually paint the four walls and the ceiling, but not the floor. Therefore, the total area to be painted is calculated as:
Area to be painted = Lateral Surface Area (for the four walls) + Area of the Ceiling.
The formula would be 2h(l + b) + lb. This shows the importance of applying the concepts of area rather than just memorising formulas.
