Prism | Shape, Examples, Types, and Surface Area

Prisms are basic 3D shapes that have two flat ends and rectangular side faces. What sets them apart is their consistent shape along their length, which can be different types of polygons, like triangles, squares, or rectangles. Prisms are essential in geometry, helping us understand volume, surface area, and shapes. In everyday life, we see prisms in things like buildings and optical devices like cameras and glasses. They’re practical and important in many ways.

In this article, we will learn about all the details related to Prism including the shape of prism, cross-section of prism, examples of prism, various types of Prism, difference between a pyramid and prism, etc.

What is Prism in Geometry?

A prism is a part of the polyhedron family and has identical polygons at the top and bottom. The other faces of a prism are called lateral faces and they have the same shape all along their length, but it doesn’t have curved faces. Prisms are named based on their cross-sections. For example, a hexagonal prism looks like a metallic nut and a rectangular prism is similar to a fish tank. Many everyday objects represent different types of prisms.

Prism Definition

A prism is a 3D shape with flat, identical faces at its two ends. The other faces are flat as well. The name of a prism depends on the shape of its base. There are different types of prisms named after their base shapes.

Prism Shape

A prism has a solid shape consisting of two identical ends (such as triangles, squares, rectangles, etc.), having flat faces or surfaces and maintaining a uniform cross-section across its length. The cross-section resembles a triangle, thus it is named a triangular prism. The shape of the prism is devoid of curves. Consequently, a prism can have square, rectangular, pentagonal, and other polygonal shapes but cannot take on a circular form.

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.

Prisms Examples

There are various examples of Prisms, some of those are:

• Rectangular Prism
  • Cross-section: Rectangle
  • All angles are right angles.
  • Example: A book or a rectangular box.
• Triangular Prism
  • Cross-section: Triangle
  • Example: A tent or a Toblerone chocolate box.
• Pentagonal Prism
  • Cross-section: Pentagon
  • Example: A pencil or a tall decorative candle.
• Hexagonal Prism
  • Cross-section: Hexagon
  • Example: A bolt or a piece of honeycomb.
• Octagonal Prism
  • Cross-section: Octagon
  • Example: A column in architecture or a certain type of spice container.
• Cylindrical Prism (Cylinder)
  • Cross-section: Circle
  • Example: A can of soda or a roll of paper towels.
• Triangular Prism (With an Isosceles Triangle Base)
  • Cross-section: Isosceles Triangle
  • Example: Certain architectural columns or some prism-shaped crystals.

Types of Prism

Based on various criteria prism can be classified into various categories. These criteria are:

• Type of Polygon as Its Base
• Shape of the Base
• Alignment of Center of the Base

Let’s discuss each criteria of classification of Prism in more detail.

Prism Based on the Type of Polygon Base

Based on the Type of polygon base, prism can be classified as:

• Regular Prism
• Irregular Prism

Regular Prism

A regular prism 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

In contrast, an irregular prism 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 the Shape of the Base

Prisms are named based on the shape of their cross-sections, which means the shape you get when you cut them. The diffferent types of Prism based on the shape of base are:

Triangular Prism

• Bases: Two triangular faces, parallel and congruent.
• Sides: Three rectangular faces that connect 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).

People Also View:

• Rectangular Prism
• Pentagonaal Prism
• Hexagonal Prism

Prism Based on Alignment of Base

Prism can also be have more types based on the alignment of the base. examples of prism based on alignment are:

• Right Prism
• Oblique Prism

Let’s understand them seperately.

Right Prism

A right prism is a solid shape with flat ends that align perfectly and creates rectangular bases. Its side faces are also rectangular, tha 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.

Difference Between Right Prism and Oblique Prism

Key difference between both right and oblique prism are listed in the following table:

Aspect	Right Prism	Oblique Prism
Definition	Faces and joining edges are perpendicular to base faces.	Faces and joining edges are not perpendicular to base faces.
Side Face Shape	Rectangles	Parallelograms
Surface Area Formula	Surface area = [Base length x height] + 2[prism length x side length] + [prism length x base length]	Surface area = [Base length x height] + 2[prism length x side length] + [prism length x base length]
Volume Formula	Volume = ½ [base length x height x prism length]	Volume = ½ [base length x height x prism length]

What is a Pyramid?

A pyramid is a three-dimensional geometric solid characterized by:

• Base: A single polygonal base which can be any shape (triangle, square, pentagon, etc.).
• Faces: Triangular lateral faces that meet at a common point above the base, known as the apex.

Types of Pyramids

Pyramids are classified based on the shape of their base:

• Triangular Pyramid (Tetrahedron): A pyramid with a triangular base.
• Square Pyramid: A pyramid with a square base.
• Pentagonal Pyramid: A pyramid with a pentagonal base, and so forth.

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. This fundamental variation in base structure distinguishes prisms as having two parallel bases and pyramids with just one.

Difference Between Prism and Pyramid
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

Formula for Prism

A prism has mainly two formulas, one is the surface area of the prism and another one is volume of prism. The surface area formula computes the combined area of all its faces, providing valuable information for material requirements. Whereas, the volume formula calculates the amount of space the prism occupies, aiding in capacity and content assessments. These formulas are essential tools in geometry for various real-world applications.

Read More: Prism Formulas

Surface Area of Prism

There are two kinds of areas regarding prisms:

• Lateral Surface Area
• Total Surface Area

Let’s discuss these in detail.

Lateral Surface Area of Prism

The 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)

OR

Total Surface Area = 2 × Base Area + (Base Perimeter × Height)

Read More: Surface Area of Prism

Formula for 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. Here, V is the volume, B is the base area, and H is the height of the prism. 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 formula 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)

People Also View:

• Volume of Square Prism
• Volume of Triangular Prism
• Volume of Rectangular Prism

Solved 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:

according to the given information:

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:

according to the given information:

side length = 5 units

height = 10 units

The 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

Practice Questions on What is a Prism

Q1. Find the volume of a hexagonal prism with a side length of 4 units and a height of 9 units.

Q2. Calculate the total surface area of a pentagonal prism with a base side length of 6 centimeters and a height of 10 centimeters.

Q3. If the base area of a triangular prism is 36 square inches and its height is 5 inches, what is its volume?

Q4. Find the total surface area of a trapezoidal prism with bases measuring 5 and 8 centimeters, and a height of 10 centimeters.

FAQs on What is a 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 the Volume of a Prism Calculated?

The 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.

What is the Surface Area Formula for a Prism?

The surface area (SA) of a prism is given by: 2 × Base Area + Perimeter of Base × Height

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 the Altitude of a Prism?

The 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 the Lateral Surface Area of a Prism Calculated?

The 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.