What is a Prism in Math?

A prism features a very solid shape that consists of two identical ends (like triangle, square, rectangle, etc.), the flat faces or the surfaces, and the uniform cross-section across its length. The cross-section looks like a triangle hence called triangular prisms. The shape of a prism doesn't have any curve. So, a prism can also have a square, a rectangular, a pentagonal, and many other polygon shapes but not a circular shape.


Cross Section of Prism

The cross-section is the point where the shape obtained by the intersection of an object by a plane along its axis. It is also said as cutting a three-dimensional object with a plane to get another shape.

If a prism is intersected by a plane, parallel to the base, then the shape of the cross-section will be the same as the base. For instance, a square pyramid is cut by a plane, parallel to the base, then the shape of the cross-section of the pyramid will also be a square.


Types of Prism

Depending upon the cross-sections, the prisms are named. It is of two types, namely;

  • Regular Prism

  • Irregular Prism

Regular Prism

If the base of a prism is in the shape of a regular polygon, it is called a regular prism.

Irregular Prism

If the base of a prism is within the shape of an irregular polygon, then the prism is called an irregular prism.


Prism Based on Shape of Bases

Based on the shape of the bases, it is further categorized into different types, namely;

  • Triangular prism (has triangular bases).

  • Square prism (has square bases).

  • Rectangular prism (has rectangular bases).

  • Pentagonal prism (has pentagonal bases).

  • Hexagonal prism (has hexagonal bases).

Right Prism and Oblique Prism

Apart from regular and irregular, the prism is often classified into two more types;

  • Right Prism

  • Oblique Prism

The difference between both the prism for triangular bases are given below;


Difference Between Right and Oblique Prism

Right Prism

Oblique Prism

If the faces and therefore the joining edges are perpendicular to the bottom faces, then it's referred to as a right prism

If the faces and the joining edges are not perpendicular to the bottom faces, then it is known as the oblique prism

In the right prism, the side faces are the rectangles

In an oblique prism, the side faces are the parallelograms

Surface area equals the sum of [Base length × height] + 2[prism length × side length]+[prism length × base length]

Surface area equals [Base length × height] + 2[prism length × side length]+[prism length × base length]

Volume equals ½ [base length × height × prism length]

Volume equals ½ [base length × height × prism length]


Properties of Prism

  1. A prism may be a sort of three-dimensional (3D) shape with flat sides.

  2. It has two ends that are an equivalent shape and size (and appear as if a 2D shape).

  3. It has an equivalent cross-section right along with the form from end to end; meaning if you narrow through it you'd see an equivalent 2D shape as on either end.

Formulas (Surface Area & Volume)

The formulas are defined for the area and volume of the prism. As the prism may be a three-dimensional shape, so it's both the properties, i.e., surface area and volume.


Volume of a Prism

The volume of the prism is defined because of the product of the bottom area and therefore the prism height.

Therefore,

  • The Volume of Prism is equal to Base Area × Height


For example, if you want to find the volume of a square prism, then it is important to know the area of a square, then its volume can be calculated as follows:

The volume of the square Prism equals the Area for square × height.

V = s2 × h cubic units

Where “s” is the side of a square.


Surface Area of a Prism

The surface area of the prism is equal to the total area covered by the faces of the prism.

For any kind of prism, the surface area can be found using the given formula;

  • Surface Area of the Prism equals 2(Base Area) + (Base perimeter × height)


Solved Problems

Example 1: Find the volume of a given triangular prism whose area is given to be 60 cm2 and given height is 7 cm.

Solution: Given,

Base area = 60 cm2

Height = 7 cm

We know that,

The volume for the prism equals (Base area × Height) cubic units

Therefore, V = 60 × 7 = 420

Hence, the volume of a triangular prism = 420 cm3.


Example 2: Find the height of the square prism whose volume is equal to 360 cm3 and the base area is 60 cm2.

Solution: Given,

The volume for the square prism = 360 cm3

Base Area = 60 cm2

So, the height for the square prism is calculated as given below:

The volume for the square prism equals the Base area × height

360 = 60 × prism height

Therefore, the height, h = 360/60

Prism Height, h = 6 cm.

FAQs (Frequently Asked Questions)

Question 1. What Does Prism Mean?

Answer. Let’s define a prism! A transparent solid body normally having a triangular base, which is used for dispersing light into the spectrum or for reflecting rays of the light is known as a prism.

Question 2. How Many Triangles Does a Prism Have?

Answer. Two triangles

The Different Parts of a Triangular Prism - There are three rectangles and two triangles. The 2-D shapes which form a 3-D shape are called the faces. The top as well as the bottom, which are triangles, are bases.

Question 3. What is the Angle of a Prism?

Answer. The angle between a prism’s two lateral faces is called the angle of the prism or the prism angle. When the light ray is allowed to pass through the prism, it makes the emergent ray bend at an angle to the direction of the ray that is incident. This angle is also known as the angle of the deviation for the prism.

Question 4. What is the Function of a Prism?

Answer. Prism, in optics or a piece of glass or any other transparent material, can cut with a precise angle and the plane faces, it is also very useful for analyzing and reflecting the light. An ordinary triangular prism can also separate white light into its constituent colours, known as a spectrum.

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