A prism is a three-dimensional solid which has identical faces at both ends. The other faces are flats. A prism is named after its base. Therefore, there are different types of prisms named on the basis of the shape of their bases. Let us explore more about the world of prisms!
|1.||Definition of Prism|
|2.||Types of Prisms|
|4.||FAQ's on Prisms|
Definition of a Prism
A prism is an important member of the polyhedron family that has congruent polygons at the base and top. The other faces of a prism are called lateral faces. It means that a prism does not have a curved face. A prism has the same cross-section all along its length. The prisms are named depending upon their cross-sections. A metallic nut is the best example to represent hexagonal prism, fish aquarium represents rectangular prism and many more such live appliances in our surroundings represent the prisms.
Types of Prisms
Before reading about various types of prisms let us understand on what basis types of prisms can be obtained. Prisms can be classified on the following basis:
Prisms Based on the Type of Polygon, of the Base
A prism is classified on the basis of the type of polygon base it has. There are two types of prisms in this category named as:
- Regular Prism: If the base of the prism is in the shape of a regular polygon, the prism is a regular prism.
- Irregular Prism: If the base of the prism is in the shape of an irregular polygon, the prism is an irregular prism.
Prisms Based on the Alignment of the Identical Bases.
There are two different prisms based on the alignment of the bases named:
- Right Prism: A right prism has two flat ends that are perfectly aligned with all the side faces in the shape of rectangles.
- Oblique Prism: An oblique prism appears to be tilted and the two flat ends are not aligned and the side faces are parallelograms.
Prisms Based on the Shape of the Bases
A prism is named on the basis of the shape obtained by the cross-section of the prism. They are further classified as:
- Triangular Prism: A prism whose bases are triangle in shape is considered a triangular prism.
- Square Prism: A prism whose bases are square in shape is considered a square prism
- Rectangular prism: A prism whose bases are rectangle in shape is considered a rectangular prism (a rectangular prism is cuboidal in shape).
- Pentagonal Prism: A prism whose bases are pentagon in shape is considered a pentagonal prism.
- Hexagonal Prisms: A prism whose bases are hexagon in shape is considered a hexagonal prism.
- Octagonal Prism: A prism whose bases are octagon in shape is considered an octagonal prism.
- Trapezoidal Prism: A prism whose bases are trapezoid in shape is considered a trapezoidal prism.
There are two basic formulas we read in geometry about all the respective 3-dimensional shapes. The two formulas are the area of the shape and volume of the shape. Let us learn these two in the case of prisms.
Surface Area of Prisms
There are two types of areas we read about, first, the lateral surface area of the prism, and second, the total surface area of the prism. Let us learn in detail.
The lateral area of a prism is the sum of the areas of all its lateral faces whereas the total surface area of a prism is the sum of its lateral area and area of its bases.
The lateral surface area of prism = base perimeter × height
The total surface area of a prism = Lateral surface area of prism + area of the two bases = (2 × Base Area) + Lateral surface area or (2 × Base Area) + (Base perimeter × height).
There are seven types of prisms we read above based on the shape of the bases. The bases of different types of prisms are different so are the formulas to determine the surface area of the prism. See the table below to understand this concept behind the surface area of various prisms:
|Shape||Base||Surface Area of Prisms = (2 × Base Area) + (Base perimeter × height)|
|Triangular Prism||Triangular||Surface area of triangular prism = bh + (s1 + s2 + b)H|
|Square Prism||Square||Surface area of square prism = 2a2 + 4ah|
|Rectangular Prism||Rectangular||Surface area of rectangular prism = 2(lb + bh + lh)|
|Trapezoidal Prism||Trapezoidal||Surface area of trapezoidal prism = h (b + d) + l (a + b + c + d)|
|Pentagonal Prism||Pentagonal||Surface area of pentagonal prism = 5ab + 5bh|
|Hexagonal Prism||Hexagonal||Surface area of hexagonal prism = 6b(a + h)
Surface area of regular hexagonal prism = 6ah + 3√3a2
|Octagonal Prism||Octagonal||Surface area of octagonal prism = 4a2 (1 + √2) + 8aH|
Volume of Prisms
The volume of a prism is defined as the total amount of space or vacuum a prism occupies. The volume of a prism is the product of the area of the base and height of the prism. Thus, the volume of a prism is represented as V = B × H where "V" is the volume, "B" is the base area, and "H" height of the prism. The base area is given in (units2) and the height of the prism is given in (units). Thus, the unit of volume of the prism is given as V = (units2) × (units) = units3. Look at the table below to understand the formula behind the volume of various prisms:
|Shape||Base||Volume of Prisms = Base area × height|
|Triangular Prism||Triangular||Volume of triangular prism = Area of triangle × height of the prism = ½ × b × h × l|
|Square Prism||Square||Volume of square prism = Area of square × height of the prism = a2 h|
|Rectangular Prism||Rectangular||Volume of rectangular prism = Area of rectangle × height of the prism = lwh|
|Trapezoidal Prism||Trapezoidal||Volume of trapezoidal prism = Area of trapezoid × height of the prism = ½ × (a + b) × h × h|
|Pentagonal Prism||Pentagonal||Volume of pentagonal prism = Area of pentagon × height of the prism = (5/2) × a × b × h|
|Hexagonal Prism||Hexagonal||Volume of hexagonal prism = Area of hexagon × height of the prism = 3abh|
|Octagonal Prism||Octagonal||Volume of octagonal prism = Area of octagon × height of the prism = 2a2(1+√2) × h|
Example 1: Help Eva find the volume of a prism whose base area is 35in2 and height is 6in.
Given, base area = 35 in2 and height = 6 in
We know that,
The volume of a prism (V) = Base area × Height
Thus, V = 35 × 6 = 210 in3
Therefore, the volume of the triangular prism is 210 in3
Example 2: What will be the surface area of the prism if the length, breadth, and height of the prism are 8 units, 6 units, and 4 units respectively?
Solution: Given information is length = 8 units, breadth = 6 units, and height = 4 units. With this information, it is clearly observable that the prism is a rectangular prism.
Surface area of a prism = (2 × Base Area) + (Base perimeter × height)
Base area = length × breadth = 48 units2
Base perimeter = 2 (l + b) = 28 units
Height = 4 units
Substituting the values of the base area, base perimeter, and height in the surface area formula we get,
Surface area of prism = (2 × 48) + (28 × 4) = 208 units2
Therefore, the surface area of the prism is 208 units2
FAQs on Prisms
What are Prisms in Geometry?
Prisms are an important member of the polyhedron family. A prism has two identical faces on which they are named upon. It is a 3-dimensional shape and can be categorized on the following basis:
- Prism based on the shape of the base
- Prism based on the alignment of the bases
- Prism based on the type of polygon base it has.
What Do You Understand by the Cross-Section of Prisms?
The cross-section of a prism is the shape obtained when you cut the prism through an axis. The cut is made in a straight line congruent to the bases in right angles. In other words, a prism has a cross-section exactly of the same shape and size throughout its length. For example, a triangular prism has a triangular cross-section end to end throughout its length.
How Can You Classify the Different Types of Prisms?
The classification of the prisms can be done on the following basis.
- Based on the shape of identical bases.
- Based on the type of polygon, at the base.
- Based on the alignment of bases.
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How is Right Prism different from Oblique Prism?
A right prism has two flat ends that are perfectly aligned whereas an oblique prism appears to be tilted and the two flat ends are not aligned. In a right prism, all the side faces are rectangles whereas the faces of the oblique prism are parallelograms.
What Is The Formula To Find the Surface Area of Prisms?
A prism is a three-dimensional solid with two identical bases. The total surface area of prisms can be found by adding 2 times the area of its base and its lateral surface area.
How Can You Calculate the Volume of the Prisms?
A prism is a polyhedron that has the same cross-section along all its length. The volume of the prisms can be calculated by multiplying its cross-sectional area by its total length. We can find the volume of the prism using the following steps:
- Step 1: Observe the pattern of the prism. Write down the given dimensions of the prism such as the edge (a), the height of the prism (h).
- Step 2: Substitute the dimensions in the general volume of prism formula: V = a2 h
- Step 3: The value of the volume of a prism is once obtained, then write the unit of the volume as units3.
How is Regular Prism Different From Irregular Prisms?
If the base of the prism is in the shape of a regular polygon, the prism is a regular prism. Whereas, if the base of the prism is in the shape of an irregular polygon, the prism is an irregular prism.