Geometry - Prism
- What is a Prism?
- Components of Prism
- Types of Prism
- Cross Section of a Prism
- Other types of Prisms
- Surface area of Prism
- Volume of the Prism
- Solved Examples
- Practice Problems
- Frequently Asked Questions
What is a Prism?
A prism is a three-dimensional shape consisting of edges and faces under two polygons called bases. The faces are joined by the vertical edges, forming lateral faces. The prism has two faces, top and bottom, and they are congruent and are called bases.
Based on the shape of the base, the prism is named. So, if a prism’s base is triangular, then it is called a triangular prism, a prism with a rectangular face is called a rectangular prism, etc.
Components of Prism
The components are described as follows:
- Bases: Bases are two congruent and parallel polygonal faces that are present at the end.
- Edges: Edges are the line segments that are made by joining the vertices of the prism on the outer periphery.
- Vertices: The corners and points of the prism are called vertices.
- Height: The distance between the bases, which is measured perpendicular to the bases, is called the height of the prism.
- Faces: Faces are the surfaces formed by the edges and vertices of the prism.
Types of Prism
- Rectangular Prism: A rectangular prism is also called a cuboid. The characteristics of a rectangular prism are that its bases and lateral faces are rectangular. A box is a rectangular prism.
- Triangular Prism: A triangular prism has triangular bases, but its lateral faces are rectangular. They are used in architectural buildings.
- Pentagonal Prism: The pentagonal prism has its base pentagon in shape. The lateral faces have a count of five rectangular faces.
- Hexagonal Prism: A hexagonal prism has six edges at its base. Therefore, it has six lateral rectangular faces. They are used in different geometrical practices.
- Octagonal Prism: An octagonal prism is composed of eight edges. Therefore it has eight rectangular faces. They are rarely used in practice although they exist.
Cross Section of a Prism
A cross-section of a prism is obtained by cutting the prism with a plane that is parallel to its base. Therefore, the cross-section of a prism is a two-dimensional shape. The cross-section can be made by the plane with different orientations. If the cutting plane of the prism is parallel to its base then its cross section will have the shape of its base. For example, a rectangular prism will have a rectangular cross-section, a triangular prism will have a triangular cross-section, etc, as shown below.
Other types of Prisms
- Right Triangular Prism: A triangular prism is a right triangular prism when the base of the prism is a right triangle. It has two parallel and congruent right triangular bases and three rectangular faces.
- Square Prism: A square prism is a prism that has square bases. Although these prisms are less common, they are useful. It has four rectangular faces.
- Regular Prism: A regular prism is a prism that has bases as a regular polygon, meaning all the sides and angles of the bases are equal. They can include regular hexagonal prism, etc.
- Irregular Prism: An irregular prism is a prism that has bases as an irregular polygon, meaning all the sides and angles of the bases are not equal. They can include rectangular prism, etc.
- Oblique Prism: An oblique prism is another type of prism in which the lateral faces are not perpendicular to the base. It also means that the lateral edges are not perpendicular to the base edges. A parallelepiped is an example of an oblique prism.
- Trapezoidal prism: A prism with a trapezoidal base is called a trapezoidal prism, whereas its lateral faces are parallelograms. It consists of six faces, eight vertices, and `12` edges.
Surface area of Prism
The surface area of the prism is calculated from the total surface area. The total surface includes the bases and lateral faces.
The surface area of any prism can be calculated using the following formula.
`"Surface area of the prism" = 2×("Area of the base of the prism")+("Perimeter of the base")× \text{height}`
Volume of the Prism
The volume of the prism can be calculated by the following formula
`"Volume of the prism" = "Area of the base" × "height of the prism"`
Solved Examples
Example `1`: A rectangular prism of `10` cm `×` `12` cm is extended to a height of `14` cm. Calculate the surface area of the prism.
Solution:
Here, L`=12` cm, B`=10` cm, and H`=14` cm.
`"Surface area of the prism" = 2×("Area of the base of the prism")+("Perimeter of the base")×"height"`
`"Surface area of the prism" = 2×(12×10)+(2×(12+10))×14`
`= 2×120+(2×22)×14`
`= 240+44×14`
`= 240+616`
`=856` `\text{cm}^2`
Example `2`: A prism has a triangular base. The length of the triangular base is `4` cm and the height of the triangular base is `7` cm. The length of the triangular prism is `10` cm. Find the volume of the prism.
Solution:
Here, the base of the triangle `= 4` cm, the height of the triangle `= 7` cm. The height of the prism `=` the length of the triangular prism `= 10` cm.
The area of the triangular base can be calculated by the formula
`"Area of the base" =1/2×"base"×"height of the triangle"`
`"Area of the base" =1/2×4×7=14 \text{cm}^2`
`"Volume of the prism"= "Area of the base" × "height of the prism"`
`"Volume of the prism" = 14×10 =140` `\text{cm}^3`
Example `3`: A pentagonal prism has an area of its base as `40` `\text{cm}^2`. The height of the prism is `15` cm and the base perimeter is `30` cm. Find the total surface area of the prism.
Solution:
Here is the area of the base `= 40` `\text{cm}^2`
The lateral surface area of the prism is calculated as follows:
`"Lateral surface area" = "Perimeter of the base" × "height"`
`"Lateral surface area" = 30×15 = 450` `\text{cm}^2`
`"Total surface area" = 2×("Base area") + "Lateral surface area"`
`"Total surface area" = 2×40+450=80+450=530 \text{cm}^2`
Practice Problems
Q`1`. A prism with a pentagonal base can be called by the name
- Rectangular Prism
- Triangular Prism
- Pentagonal Prism
- Cuboid
Answer: c
Q`2`. Calculate the area of the base of a hexagonal prism of length `20` cm if the volume of the prism is `360` `\text{cm}^3`.
- `16` `\text{cm}`
- `18` `\text{cm}`
- `20` `\text{cm}`
- `14` `\text{cm}`
Answer: b
Q`3`. The surface area of a rectangular prism with a length of `12` cm, width of `5` cm, and height of `9` cm is `" _____"`.
- `330` `\text{cm}^2`
- `526` `\text{cm}^2`
- `127` `\text{cm}^2`
- `426` `\text{cm}^2`
Answer: d
Frequently Asked Questions
Q`1`. Is there any difference between the right and oblique prism?
Answer: In the right prism, the two of its lateral faces are perpendicular to each other and the bases, while in the oblique prism, the faces are at an angle to the base.
Q`2`. Give some examples of prisms that we see in real life.
Answer: Here are some examples we see in daily life books, chocolate bars, cargo containers, camping tents, tissue boxes, Rubik's cubes, etc.
Q`3`. Is a rectangular prism a regular or irregular prism?
Answer: No, a rectangular prism is an irregular prism because all the sides of a rectangle are not of equal measure.
Q`4`. Name some types of regular prisms.
Answer: Equilateral triangular prism, square prism, and cube are some examples of regular prisms.