Geometry - Rectangular Prism
- Rectangular Prism
- Types of Rectangular Prism
- How to Find the Volume of a Rectangular Prism?
- How to Find the Lateral Surface Area (LSA) of a Rectangular Prism?
- How to Find the Total Surface Area (TSA) of a Rectangular Prism?
- Real-life Examples
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Rectangular Prism
A `3D` shape of a rectangular object is known as a rectangular prism. The three dimensions are known as length, width, and height. It is also called a cuboid. Refer to the below image for a better understanding.
It has `6` faces and `12` edges. All these faces are rectangular. Refer to the net of the rectangular prism which is provided in the image below.
Types of Rectangular Prism
`1`. Right rectangular prism - In this type of rectangular prism all the angles are right angles.
`2`. Oblique rectangular prism - In this type of rectangular prism the bases are not not aligned one directly above the other.
How to Find the Volume of a Rectangular Prism?
We know a rectangular prism has six faces of rectangular shape. Therefore, the base's area is equal to the rectangle's area.
Area of the rectangle `= "length" × "width" = L × W`
And, rectangular prism’s height `= H`
The rectangular prism’s volume `= "Prism’s base area" × "height"`
`V= (L × W) × H`
How to Find the Lateral Surface Area (LSA) of a Rectangular Prism?
Lateral surface area refers to the combined areas of its side faces (excluding the bases). It is denoted by LSA. To determine the area of the side faces we can use a rectangular prism’s net configuration.
Lateral surface area `(LSA) =` Total area of its side faces
`LSA = W × H + W × H + L × H + L × H`
`LSA = 2(W × H) + 2(L × H)`
`LSA = 2WH + 2LH`
`LSA = 2 (W + L)H`
How to Find the Total Surface Area (TSA) of a Rectangular Prism?
The total surface area of a rectangular prism can be obtained by adding the lateral surface area and the area of its bases. It is denoted by the `TSA`. It can be calculated as,
`"Total surface area (TSA)" = "Lateral surface area" + 2("base area")`
`TSA = 2 (W + L)H + 2(L × W)`
`TSA = 2WH + 2LH + 2LW`
It can be written as `TSA = 2(WH+LH+2LW)`.
Real-life Examples of Rectangular Prism
Here are some real-life examples (images) of the rectangular prism.
Solved Examples
Example `1`. Determine the lateral surface area of a rectangular prism of length is `20` cm, width is `10` cm, and height is `15` cm.
Solution.
Length `(L) = 20` cm, width `(W) = 10` cm, and height `(H) = 15` cm
`LSA = 2 (W + L)H`
`LSA = 2 (10 + 20) 15`
`LSA = 2 × 30 × 15`
`LSA = 900` square cm
Therefore, the lateral surface area of the rectangular prism is `900` square centimeters.
Example `2`. Determine the volume of a rectangular prism with a length of `12` m, width of `5` m, and height of `8` m.
Solution.
Length `(L) = 12` m, width `(W) = 5` m, and height `(H) = 8` m
Volume `(V) = L × W × H`
`= 12 × 5 × 8`
`= 480`
Thus, the total surface area of the rectangular prism is `480` cubic meters.
Example `3`. Shelby wants to pack a box of a rectangular shape with a length of `25` m, width of `20` m, and height of `30` m with a gift paper. How much gift paper does she need to pack the rectangular box?
Solution.
Given, `L = 25` cm, `W = 20` m, and `H = 30` m.
Note: She needs to cover each side of the rectangular box, which means we need to determine the total surface area of the rectangular box.
`LSA = 2 × L × H + 2 × W × H + 2 × L × W`
Total surface area `(LSA) = 2 × 20 × 30 + 2 × 25 × 30 + 2 × 25 × 20`
Total surface area `(LSA) = 1200 +1500 + 1000`
Total surface area `(LSA) = 3700`
Thus, she needs `3700` square meters of gift paper.
Practice Problems
Q`1`. Determine the rectangular prism’s volume.
- `360` cubic centimeters
- `300` cubic centimeters
- `250` cubic centimeters
- `200` cubic centimeters
Answer: a
Q`2`. Determine a rectangular prism’s total surface area with dimensions: length of `19` inches, width of `10` inches, and height of `15` inches.
- `1200` square inches
- `2390` square inches
- `1250` square inches
- `1300` square inches
Answer: c
Q`3`. Determine the lateral surface area of the rectangular prism with dimensions: length of `80` cm, width of `60` cm, and height of `100` cm.
- `20000` square centimeters
- `25000` square centimeters
- `28000` square centimeters
- `30000` square centimeters
Answer: c
Q`4`. Which of the following is not true?
- The formula for determining the lateral surface area of a rectangular is `2 (W + L)H`.
- The total surface area is included in the lateral surface area.
- The lateral surface area is included in the total surface area.
- The formula to calculate the volume of the Rectangular prism is length multiplied by width multiplied by height.
Answer: b
Q`5`. Determine the total surface area of the provided rectangular prism.
- `2200` cubic centimeters
- `2000` cubic centimeters
- `3000` cubic centimeters
- `2100` cubic centimeters
Answer: a
Frequently Asked Questions
Q`1`. What is a rectangular prism?
Answer: A rectangular prism is a `3D` geometric figure with six rectangular faces.
Q`2`. How do you calculate the volume of a rectangular prism?
Answer: The volume (\(V\)) of a rectangular prism is calculated using the formula \(V = \text{length} \times \text{width} \times \text{height}\).
Q`3`. What is the difference between the surface area and lateral surface area of a rectangular prism?
Answer: Surface Area `(SA)`: It is the total area of all six faces of a rectangular prism.
\( SA = 2WH + 2LH + 2LW \)
Lateral Surface Area `(LSA)`: It only considers the areas of the four lateral (side) faces, excluding the top and bottom faces.
\( LSA = 2(W+L)H \)
Q`4`. Can a rectangular prism have equal length, width, and height?
Answer: Yes, a rectangular prism with equal length, width, and height is a cube. A cube is a special case of a rectangular prism.
Q`5`. How many edges does a rectangular prism have?
Answer: A rectangular prism has `12` edges.