Types of Prisms
The unique feature of a prism is if it is cut from the middle or basically at the intersection. The cross-section of the object will be parallel to the base. This means that the cross-section of the prism will be uniform throughout. Given that most prisms are polygonal, depending on the shape of the cross-section, there are mainly two types of prisms, regular and irregular prisms.
A regular prism may be defined as a prism whose base corresponds to a regular polygonal shape, for example, a square, a circle, a triangle, etc. Let’s take the example of a rod. Cylindrical in shape, if this rod is cut in the middle, we will obtain a perfect circle which will be identical to its circular base. Similarly, if you were to cut out a plateau in the middle, you’d get a quadrilateral which would be parallel to its base.
On the contrary, an irregular prism has every other property of a prism but it doesn’t resemble a regular polygon. This is to say that it does not have a rectangle or a triangle or any other regular polygonal shape as its base. Remember those fascinating kaleidoscopes that we all loved to buy from the fairs? Many of them had interesting shapes like hearts, stars, bubbles, etc. Now, these are not regular polygons so prisms with irregular shapes base are called irregular prisms.
Now that we know the definition of a prism, let us move to find out the surface area of the prism.
Area of Prism
As a three dimensional figure, the area of a prism is calculated in terms of the surface area of the prism, which is basically the sum of the area of all its faces. We know in the case of a two-dimensional figure, we find out the area by multiplying the sides of the figure. In case of a three-dimensional figure like a prism, we calculate the area of the prism by including the height of the object as well.
We know that the defining feature of a prism is that it has two identical bases, irrespective of the kind of prism. Now, if the area of prism definition reflects, the sum of the surface area of all its faces, then we can simply double up the area of the bases and add the product of the rest of the dimensions. Let us illustrate this better with the area of the prism formula.
The surface area of prism formula may be defined as 2 x area of base + (perimeter of base x height)
Area of Prism Example
If we take the example of a rectangular prism, then the area of prism formula would read like:
2 x Length x Breadth + 2( Length + Breadth) x Height
Let us now understand the area of prism formula in this case with the figure :
(image will be uploaded soon)
Area of the base is length x breadth, so we simply need to double it up for the top and the bottom surfaces.
Then we sum up the perimeter of the base by adding the length and the breadth and again double it up for the parallel sides. Finally, we multiply this with the height to account for the third dimension and voila! We have our surface area of prism for a rectangular prism.
Area of Prism Solved Example
How can we calculate the surface area of a prism for a triangular prism?
The area of prism formula is 2 x Length x Breadth + 2( Length + Breadth) x Height
On a similar count, we can combine the area of the prism formula to a triangular in order to find the surface area of prism in this case.
The formula to be used here would be Area of the triangular base + 2( perimeter of the sides) x height
Hence, the area of the base can be evaluated as : H x B + 2(S1+S2+S3) x H
(image will be uploaded soon).
FAQs on Area of a Prism
1. What does the surface area of a prism represent?
The surface area of a prism is the total space occupied by all its faces. Imagine unfolding a 3D prism into a flat 2D shape (called a net); the surface area is the total area of that flat net. It includes the area of the two parallel bases and the area of all the rectangular side faces, also known as lateral faces.
2. What is the general formula used to find the total surface area of any prism?
The general formula to calculate the total surface area (TSA) of any prism, regardless of the shape of its base, is:
TSA = (2 × Base Area) + (Perimeter of Base × Height of Prism)
Here, 'Base Area' refers to the area of one of the two identical bases (e.g., a triangle, square, or hexagon), and 'Height' is the perpendicular distance between these two bases.
3. How is the lateral surface area of a prism different from its total surface area?
The main difference lies in which faces are included in the calculation:
- Total Surface Area (TSA) includes the area of all faces: the two bases and all the rectangular side faces.
- Lateral Surface Area (LSA) includes the area of only the side faces (the lateral faces) and excludes the area of the top and bottom bases. The formula for LSA is simply: Perimeter of Base × Height.
4. How do you apply the area formula to a specific type, like a triangular prism?
To find the area of a triangular prism, you adapt the general formula by using the specific measurements of a triangle for the base.
- Base Area: You calculate the area of the triangular base using the formula (1/2 × base of the triangle × height of the triangle).
- Base Perimeter: You find the perimeter by adding the lengths of the three sides of the triangle.
- Final Formula: The total surface area becomes: (2 × Area of triangle) + ((Side1 + Side2 + Side3) × Height of prism).
5. What are some common types of prisms students encounter in the CBSE syllabus?
In the CBSE curriculum, especially in topics related to Mensuration, students primarily work with a few common prisms. These are categorised by the shape of their base:
- Triangular Prism: Has a triangle as its base.
- Rectangular Prism (or Cuboid): Has a rectangle as its base. A cube is a special type of rectangular prism where all faces are squares.
- Square Prism: Has a square as its base.
- Pentagonal Prism: Has a pentagon as its base.
- Hexagonal Prism: Has a hexagon as its base.
6. Why does the prism area formula include both the base perimeter and the base area?
The formula is designed to cover every surface of the prism logically. The term (2 × Base Area) accounts for the flat top and bottom surfaces of the prism. The term (Perimeter of Base × Height) is a clever way to calculate the area of all the side faces at once. Imagine 'unrolling' the side faces; they would form one large rectangle whose width is the perimeter of the base and whose length is the height of the prism.
7. How does calculating the surface area of a prism differ from calculating the surface area of a pyramid?
The key difference is in their structure and the shape of their faces. A prism has two identical bases and rectangular side faces. In contrast, a pyramid has only one base and its side faces are triangles that meet at a single point (the apex). Therefore, their area formulas are different. The pyramid's surface area formula is: Area of Base + Area of all triangular side faces.
8. Can the concept of a prism's surface area be applied to real-world objects?
Yes, absolutely. Calculating surface area is crucial in many real-world scenarios. For example, if you need to paint a rectangular room (a rectangular prism), you would calculate the lateral surface area to find out how much paint is needed for the walls. Similarly, designing packaging like a Toblerone box (a triangular prism) requires calculating the total surface area to determine the amount of cardboard needed.
