Introduction of prism
In mathematics the prism is a very special three dimensional object. The prism primarily refers to the optical prism. An optical prism indicates a transparent three-dimensional optical element or object. The prism is generally made up of glass, fluorite or acrylic, etc.
In geometry or even science, a prism primarily refers to the optical prism. Traditionally the optical prism is only referred to as the triangular prism which has a triangular base and all the sides are rectangular. It has flat and transparent or polished surfaces that can refract or reflect the beam of light.
The prism also has the ability to split white light into its constituent spectral colors. This type of splitting of light using a prism is called the dispersion of light. A rainbow of seven colors that is visible after it rains is also an example of the dispersion of light. Water droplets in the atmosphere behave like prisms in this case.
In 1665, Sir Isaac Newton performed an experiment with light and a prism. He darkened the room and made a hole in his window. Then he placed a glass prism in between the beam of sunlight. He observed that the light broke into seven multicolor light beams and made a band like a rainbow. In the next step, he placed a prism upside down in front of the color spectrum. At this point in his experiment, he observed that all the colored rays recombined and formed a beam of white light. Hence with this experiment time Newton proved that with the use of a prism that light is a combination of multiple rays of coloured light.
What is a prism?
The prism is a three-dimensional object or shape that has two identical surfaces facing each other. Each of these faces is called the base of the prism. The base of the prism can be a polygon like a triangle, a rectangle, a square, or even a pentagon. And the rest of the faces of the prism may be a parallelogram or rectangular in shape.
What is the surface area of a prism?
The surface area (S) of a prism is the sum of the areas of the bases and the lateral faces of the prism. It is measured in square units like square inches, square meters and square feet. In the case of a rectangular prism there are two bases and four lateral surfaces. The summation of the areas of the bases and the lateral surfaces gives us the surface area of the prism.
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Formula for the surface area of a prism
The formula for the surface area of a prism is the summation of the areas of bases and the lateral faces.
S = areas of bases + areas of lateral faces
Formula for the surface area of a rectangular prism
A rectangular prism has six faces. All the opposite faces can work like the bases and the rest four as the lateral faces. So, the summation of areas of bases and the four faces can be calculated for a rectangular prism as,
S = 2lw + 2lh+ 2wh
Where, l, w and h are the length, width and the height of the rectangular prism, respectively.
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Formula for the surface area of a triangular prism
A triangular prism has five faces. There are two triangular opposite faces called the bases and the three lateral sides are rectangular. So, the summation of the areas of the two bases and the three lateral faces can be calculated for a triangular prism as,
Check out Volume of triangular prism calculator
S = L × (a + b + c) + 2 × base area
or, S = L × (a + b + c) + b × h
Example 1:
Find the surface area of the prism as shown in this diagram.
Solution:
First imagine the prism is opened as shown in the image. Then we calculate the areas of the bases and the lateral surfaces separately.
The area of base\(=\frac{1}{2}\times 3\times 4\) [Formula for area of a triangle]
= 6
Area of lateral faces = 3 × 6 + 5 × 6 + 4 × 6 [Use formula for area of rectangle]
= 18 + 30 + 24
= 72
Surface area of prism = Area of two faces + lateral surface area
= 6 + 6 + 72 [There are two identical bases]
= 84
The surface area of the prism is 84 square meters.
Example 2: A cube shaped satellite has a side length of 15 inches. What is the minimum amount of aluminium foil required to fully cover the satellite?
Solution:
The satellite is in the shape of a cube. Therefore, the amount of aluminium foil required to cover the satellite is equal to the surface area of the cube. To calculate the area of a satellite using the prism surface area formula we calculate the bases and lateral areas and add them. Each face of a cube is a square. So,
Areas of two bases\(=2\times a^2\) [Area of square formula]
= 2 × \((15)^2\) [Replace a with 15]
= 450
Areas of four lateral faces\(=4\times a^2\)[Area of square formula]
\(=4\times (15)^2\) [Replace a with 15]
= 900
Total surface area of satellite = 450 + 900
= 1350
Therefore, a minimum 1350 square inches of aluminium foil is required to cover the satellite.
Example 3: Find the surface area of a rectangular prism as shown in the figure.
Solution: The prism is in the shape of rectangular parallelopiped. So we can calculate the surface area of the prism using the surface area of a parallelepiped formula as,
Surface area of prism = 2lw + 2lh + 2wh
= 2(5)(3) + 2(5)(4) + 2(3)(4) [Surface area of rectangular prism formula]
= 94
The area of the rectangular prism is 94 square centimeters.
No, the lateral surfaces of the prism must be rectangular. However, the bases of the prism can be triangular.
Yes, all the parallelepiped are prisms. Any two opposite pairs of opposite faces can be the bases of the prism. The rest four faces can represent the lateral faces of the prism in the case of a parallelepiped.