Prism: Shape, Examples, Types, and Surface Area
Last Updated :
13 Dec, 2024
A prism is a three-dimensional shape with two identical, parallel polygonal bases and rectangular lateral faces connecting the corresponding sides of the bases. Prisms are named after the shape of their base; for example, a hexagonal prism has hexagonal bases, and a rectangular prism has rectangular bases. Prisms do not have curved faces.
In real life, we encounter prisms in various objects like buildings, optical devices, and containers. They are commonly used in construction, architecture, and design due to their simple and practical structure.
PrismTypes of Prisms
Prisms can be classified based on three criteria:
- Type of Polygon as Its Base
- Shape of the Base
- Alignment of Center of Base
Prism Based on Base of Polygon
Based on the type of polygon base, the prism can be classified as:
- Regular Prism
- Irregular Prism
Regular Prism: It is characterized by a base that takes the form of a regular polygon, which means all its sides and angles are equal. This results in a prism with uniform and symmetric properties. The faces and edges are organized in a structured and predictable manner, making calculations and geometric analysis more straightforward.
Irregular Prism: It features a base in the shape of an irregular polygon, where the sides and angles are not equal. This leads to a prism with non-uniform and asymmetric characteristics. The faces and edges exhibit a less predictable arrangement, making geometric calculations and analysis more complex due to the lack of symmetry.
Prisms Based on Shape of Base
Prisms are named based on the shape of their cross-sections, which means the shape you get when you cut them. The different types of Prism based on the shape of the base are:
Triangular Prism
- Bases: Two triangular faces, parallel and congruent.
- Sides: Three rectangular faces that connect the corresponding sides of the triangular bases.
- Properties: Volume can be calculated by the formula V = Area of base x height. The surface area includes the areas of two triangular bases and three rectangular sides.
Square Prism
- Bases: Two square faces, parallel and congruent.
- Sides: Four rectangular faces connecting the corresponding sides of the square bases.
- Properties: Like other prisms, the volume is the area of the base times the height. The surface area calculation includes the areas of two square bases and four rectangular sides.
Rectangular Prism
- Bases: Two rectangular bases, parallel and congruent.
- Sides: Four rectangular faces that may differ in dimensions, connecting the sides of the bases.
- Properties: The volume is calculated as length x width x height. Surface area calculations must consider all six faces, which are rectangular.
Pentagonal Prism
- Bases: Two congruent, parallel pentagons.
- Sides: Five rectangular sides connecting the corresponding sides of the pentagon bases.
- Properties: The volume formula remains the same as for other prisms. Calculating the surface area involves adding the areas of the pentagonal bases to the areas of the five rectangular sides.
Hexagonal Prism
- Bases: Two congruent, parallel hexagons.
- Sides: Six rectangular sides connecting the corresponding sides of the hexagonal bases.
- Properties: Its volume is the product of the base area and height. Surface area calculations include the areas of the two hexagonal bases and the six connecting rectangular faces.
Octagonal Prism
- Bases: Two congruent, parallel octagons.
- Sides: Eight rectangular sides that connect the corresponding sides of the octagon bases.
- Properties: Volume is determined by the base area multiplied by the height, with the surface area including the octagonal bases and the eight rectangular sides.
Trapezoidal Prism
- Bases: Two congruent, parallel trapezoids.
- Sides: Four rectangular sides plus two additional parallelogram sides (unless the trapezoids are right trapezoids, in which case all sides are rectangular).
- Properties: The volume calculation involves the area of the trapezoidal base and the height. The surface area calculation needs to account for both trapezoidal bases and the sides (rectangular and potentially parallelogram).
Prism Based on Alignment of Base
Prism can also have more types based on the alignment of the base. Examples of prism based on alignment are:
Right Prism
A right prism is a solid shape with flat ends that align perfectly and create rectangular bases. Its side faces are also rectangular, that gives a consistent, upright structure. This geometric structure plays a crucial role in various mathematical concepts and practical applications.
Oblique Prism
An oblique prism seems slanted because its flat ends aren't perfectly aligned. The sides form parallelograms, creating an inclined shape. This occurs due to the prism's construction. It's this structure that causes the visual effect of tilting when observed from certain angles.
Cross Section of a Prism
A cross-section forms when a 3D object is sliced by a plane along its axis. In simpler terms, you can think of it as cutting a 3D object with a flat plane to create a different shape.
If a plane parallel to its base intersects a prism, the resulting cross-section will match the shape of the base. For instance, when a plane cuts through a square pyramid in the same direction as its base, the cross-section will also be a square. This means the shape after the cut is the same as the starting shape.
A prism has mainly two formulas, one is the surface area of the prism and another one is the volume of a prism. Let's learn them in detail.
Surface Area of Prism
There are two kinds of areas regarding prisms:
- Lateral Surface Area
- Total Surface Area
Lateral Surface Area of Prism
Lateral Area of a prism is the sum of the areas of all its side faces. On the other hand, the Total Surface Area of a prism is the sum of its lateral area and the area of its bottom and top faces.
To find the lateral surface area of a prism can be calculated using the formula:
Lateral Surface Area = Base Parameter × Height
Total Surface Area of Prism
For the total surface area of a prism, there are two methods to calculate: by adding two times the base area to the lateral surface area, or by adding two times the base area to the product of base perimeter and height.
Total Surface Area = 2 × (Base Area )+ Lateral Surface Area
FThe formulaformulasfor Surface Area of Various Prisms
There are seven types of prisms we have discussed earlier, and each type has different base shapes. Therefore, the formulas for finding the surface area of the prism vary depending on the specific type of prism.
Shape | Base | Lateral Surface Area Formula | Total Surface Area Formula |
|---|
Triangular Prism | Triangular | Perimeter of Base × Height (Ph) | 2 × Base Area + Perimeter of Base × Height (2Ab + Ph) |
|---|
Square Prism | Square | 4 × Base Side Length × Height (4aH) | 2 × (Base Area) + 4 × Base Side Length × Height (2a² + 4aH) |
|---|
Rectangular Prism | Rectangular | 2 × (Base Perimeter × Height) (2(l + w)H) | 2 × (Base Area) + 2 × (Base Perimeter × Height) (2lw + 2(l + w)H) |
|---|
Pentagonal Prism | Pentagonal | Base Perimeter × Height (Ph) | 2 × Base Area + Base Perimeter × Height (2Ab + Ph) |
|---|
Hexagonal Prism | Hexagonal | Base Perimeter × Height (Ph) | 2 × Base Area + Base Perimeter × Height (2Ab + Ph) |
|---|
Octagonal Prism | Octagonal | Base Perimeter × Height (Ph) | 2 × Base Area + Base Perimeter × Height (2Ab + Ph) |
|---|
Trapezoidal Prism | Trapezoidal | Base Perimeter × Height (Ph) | 2 × Base Area + Base Perimeter × Height (2Ab + Ph) |
|---|
Volume of Prism
Volume is how much space a prism takes up. To find the volume of a prism, simply multiply the base area by its height. The volume of a prism is represented as V = B × H. The base area is measured in square units (units²), and the height is in linear units (units), so the unit of volume is given as units³.
Volume (V) = Area of Base (A) × Height (H)
Formula for Volume of Various Prisms
Various formulas for calculating the volume of different prisms are:
Shape | Base | Volume of Prism |
|---|
Triangular Prism | Triangle | Volume = ½ × Base × Height × Length (½bhL) |
|---|
Square Prism | Square | Volume = Base Area × Height (a²h) |
|---|
Rectangular Prism | Rectangle | Volume = Base Area × Height (lwh) |
|---|
Pentagonal Prism | Pentagon | Volume = 5/2 × Base × Height (5/2abh) |
|---|
Hexagonal Prism | Hexagon | Volume = 3 × Base × Height (3abh) |
|---|
Octagonal Prism | Octagon | Volume = 2 × (1 + √2) × Base × Height (2(1+√2)a²h) |
|---|
Trapezoidal Prism | Trapezoidal | Volume = ½ × (Sum of Bases) × Height × Height (½(a + b)h²) |
|---|
Difference Between Prism and Pyramid
A prism and a pyramid are distinct three-dimensional geometric shapes. The key difference lies in their base configuration. A prism has two identical parallel bases, which are typically polygons, and its sides are rectangular.
In contrast, a pyramid has a single base, often a polygon, and triangular sides that converge at a common point called the apex.
Characterstics | Pyramid | Prism |
|---|
Base Shape | Single polygon (usually triangular) | Two congruent polygons (usually rectangular) |
|---|
Bases | One triangular base and three triangular faces | Two parallel and congruent bases with rectangular or polygonal side faces |
|---|
Edges | Varies depending on the base shape | Consistent number of edges (equal on both bases) |
|---|
Vertices | Four or more vertices depending on the base | Six or more vertices, depending on the base |
|---|
Volume | V = (1/3) × Base Area × Height | V = Base Area × Height |
|---|
Example | The Great Pyramid of Giza | Rectangular Prism, Triangular Prism |
|---|
Visual Shape | Pointed top, triangular sides | Rectangular or polygonal sides, parallel bases |
|---|
Article Related to Prism:
Examples on Prism
Example 1: Find the volume of a rectangular prism with a length of 8 units, a width of 4 units, and a height of 6 units.
Solution:
Given,
- L = 8 units
- B = 4 units
- H = 6 units
We know that,
Volume of Rectangular Prism = L × B × H
putting the values in formula, we get:
V = 8 × 4 × 6
Volume = 192 units3
Example 2: Calculate the total surface area of a square prism with a side length of 5 units and a height of 10 units.
Solution:
Given,
- Side Length = 5 units
- Height = 10 units
Lateral surface area (LSA) formula for a square prism is:
LSA= 4 × side length × height
Putting the values in formula, we get:
LSA = 4 × 5 × 10
LSA = 200 units2
Now,
Total Surface Area = 2 × base area + LSA
TSA = 2 × (side length)2 + 200 units2
= 2 × 52 + 200 units2
= 2 × 25 + 200 units2
= 50 + 200 units2
∴ TSA = 250 units2
FAQs on Prism
Define Prism in Geometry.
A prism is a three-dimensional solid with two identical, parallel bases and rectangular sides connecting corresponding vertices of the bases.
How is Volume of a Prism Calculated?
Volume (V) of a prism is calculated using the formula: V = Base Area × Height, where the base area is the area of one of the bases.
Surface Area (SA) of a prism is given by the formula SA: 2 × Base Area + Perimeter of Base × Height
Can a Prism have any Shape for its Bases?
Yes, as long as the bases are identical and parallel. They can be any polygon - square, rectangle, triangle, etc.
How is a Prism different From a Pyramid?
A prism has two identical bases and rectangular sides, while a pyramid has one base and triangular sides that meet at a common vertex.
What are Some Examples of Prisms in Everyday life?
Examples include rectangular prisms (like boxes), triangular prisms (like certain roof shapes), and more complex prisms in architecture.
How do you Classify Prisms based on the Shape of their Bases?
Prisms can be classified by the shape of their bases, such as rectangular prisms, triangular prisms, pentagonal prisms, etc.
What is Altitude of a Prism?
Altitude (or height) of a prism is the perpendicular distance between the two bases.
Can a Prism have a Slant Height?
Some prisms, like oblique prisms, may have slant heights. The slant height is the distance between the vertices of the bases along the lateral faces.
How is Lateral Surface Area of a Prism Calculated?
Lateral area of a prism is the sum of the areas of its lateral faces and is calculated using the formula: LA = Perimeter of Base × Height.
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