Introduction to Prisms
Sometimes we see rainbows in the sky after it rains. A rainbow appears as a consequence of the prismatic effect of water droplets in the air. This indicates that prisms have the capability of creating an optical phenomenon like the rainbow. In mathematics, however, the prism is a unique three dimensional object. The term prism primarily refers to an optical prism. An optical prism is a transparent three-dimensional optical object. It has flat, transparent, or polished surfaces that can refract or reflect beams of light that are incident on it.
Traditionally, it was only the optical prism that was popular. Known as the triangular prism, it has a triangular base and all its sides are rectangular. The prisms available today are generally made of glass, fluorite, acrylic, and so on.
A prism can break white light into its constituent spectral colors. This phenomenon of the breaking of light using a prism is called the dispersion of light.
Sir Issac Newton proved that light is a combination of multicolor beams using a prism. In 1665, Newton experimented with light and prisms. He darkened the room and made a hole in his window. Then he placed a glass prism in the middle of the beam of sunlight. He observed that the light broke into seven multicolor beams and made a band like a rainbow. After this, he placed a prism upside down in front of the color spectrum and observed that all the colored rays recombined and formed a white beam of light.
What is a Prism?
A prism is a three-dimensional object or shape that has two identical surfaces facing each other. These faces are called the bases of a prism. The base of the prism can be a polygon, like triangles, rectangles, squares or pentagons or any other regular polygon. The rest of the faces of the prism are parallelograms or rectangular.
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What is the Volume of a Prism?
The volume of a three-dimensional figure is the measure of the amount of space that it occupies. Similarly, the volume of a prism is the amount of space that the prism can hold within its boundary. It is measured in cubic units like cubic inches, cubic meters, cubic feet, and so on. The following example shows how to find the volume of prism.
Formula for the volume of a prism:
The volume of a prism is the product of the area of the base and the height of the prism.
Prism volume (V) = B × h, where, B is the area of the base and h is the height of the prism.
Formula for the volume of a rectangular prism:
Volume of a rectangular prism (V) = B × h
= l × w × h, where l, w, and h are the length, width, and height of the rectangular prism.
Formula for the volume of a triangular prism:
Volume of a triangular prism (V) = B × h
= \(\frac{1}{2}\times b\times h\times l\)
Here, the length of the prism can be taken as the height of the prism.
Explore Word Problems on Volume of Prisms
through this Video
Example 1:
Find the volume of the prism shown in the figure.
Solution:
V = B × h [Volume of a prism formula]
= l × b × h [Area of rectangle formula]
= 6 × 8 × 15 [Substitute values of l and b]
= 720 [Simplify]
The volume of the prism is 720 cubic yards.
Example 2: Find the volume of the triangular prism shown in the figure.
Solution:
V = B × h [Volume of a prism formula]
= \(\frac{1}{2}\times b\times h\times l\)[Area of a triangle formula; replace h with l]
= \(\frac{1}{2}\times 5.5\times 2\times 4\) [Substitute values of l, b, and h]
= 22 [Simplify]
The volume of the prism is 22 cubic inches.
Example 3:
A box of popcorn holds 96 cubic inches of popcorn. The length and width of the base of the bag are 4 inches and 3 inches, respectively. Find the height of this box of popcorn.
Solution:
The height of the bag can be easily calculated by applying the equation for the formula of the volume of a prism.
V = B × h [Formula for the volume of a prism]
96 = l × b × h [Formula for the area of a rectangle; replace V with 96]
96 = 4 × 3 × h [Substitute values of l and b]
96 = 12 × h [Simplify]
8 = h [Divide each side by 12]
The height of the popcorn bag is 8 inches.
Example 4: A building is in the shape of a triangular prism. The height of the building is 190 meters. The base of the building is in the form of an equilateral triangle whose side is 45 meters. Find the lateral surface area of the building.
Solution:
The base of the building is in the form of an equilateral triangle. And there are three rectangular lateral surfaces of this building. The lateral surface area of the building can be calculated by multiplying the area of each rectangle by 3, as:
A = 3 × area of each face
= 3 × b × h [Area of a rectangle formula]
= 3 × 45 × 190 [Replace b with 45 and h with 190]
= 25650
The lateral surface area of the building is 25650 square meters.
No, the base of the prism should be a polygon. If we change the base of a prism into a circle, the prism will be converted into a three-dimensional cylinder.
A triangular prism has five faces. Two of its faces are in the shape of a triangle, and the rest of the three faces are in the shape of a rectangle.