Square Based Prism
As per the square prism definition, it is a three-dimensional geometric solid object that has a base of a square. In a square prism, the sides and angles opposite of each other are congruent. Thus, when a minimum of two of the sides and angles are equal in measurement, it can also be termed as a square prism. You can also consider cuboid whose base is square as a square prism. In the illustration below, you can see that the bases are squared and thus it is a square prism.
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The Volume of a Square Prism
The volume of a square-based prism is a computation of the inhabited units of the solid. The volume of a square prism is the number of units that are used to fill a cube. It is represented in the form of cubic units.
We have the formula to calculate the volume of a square prism, i.e. V= a²h cubic units
Where,
V = volume
A = side of the base
H = height of the prism
How to Calculate Surface Area of a Square Prism
The surface area of a square prism is a quantification of the total area of a surface of a 3-D solid object. For a square-based prism, the surface area is defined as the sum total of twice the base area and the lateral surface area (LSA) of the prism. Its measure is represented in square units.
Formula to find the surface area of a square prism is as follows:
The surface area of the square prism i.e. SA = 2a² + 4ah square units
In which,
a = side of the square prism
h = height of the square prism
Is Square Prism a Cube?
A cube is a unique case of a square prism where the lengths in all the three dimensions are identical. Hence, all cubes make square prisms but not all square prisms are cubes.
Types of Prisms
There are different prisms that are categorized based on the cross-section. These are as given:
Square Prism
Rectangular Prism
Triangular Prism
Oblique Prism
Cubical Prism
Pentagonal Prism
Note: prism with rectangular base is also sometimes called a square prism.
Cross Section in a Prism
A cross section is a shape obtained by cutting straight through an object.
For example, in the image below, the cross-section of the object is a triangle and also contains Identical cross-section all along its length, therefore it is a triangular prism.
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Solved Examples on Prisms
Example:
Find the surface area of a prism where the base area is 20 m², the base perimeter is 22 m, and the length is 10 m.
Solution:
Surface Area of A Prism = 2 × (Base Area + Base Perimeter) × Length
Thus, we get
= 2 × 20 m² + 22 m × 10 m
= 40 m² + 220 m²
= 260 m²
Example:
Calculate the volume and the surface area of a square prism whose side is 7 cm and height is 11 cm.
Solution:
Given:
Side, a = 7 cm
Height, h = 11 cm
The Volume a Square Prism, V = a² h cubic units.
V = (7)² (11)
V = 49 (11)
V = 539 cm³
Hence, the volume of a square prism is 539 cm³
The surface area of the square prism, SA = 2a² + 4ah square units
Substituting the given values, we get
SA = 2(7)² + 4(7)(11) cm²
SA = 2(49) + 4(77) cm²
SA = 98 + 308 cm²
SA = 406 cm²
Thus, the surface area of a given square prism is 406 cm².
FAQs on Square Prism
1. What exactly is a square prism?
A square prism is a three-dimensional shape, also known as a polyhedron, that has two identical and parallel square bases. These two square bases are connected by four rectangular side faces, called lateral faces. It is a type of right prism and also a specific kind of cuboid.
2. How many faces, edges, and vertices does a square prism have?
A square prism has a fixed number of faces, edges, and vertices. Specifically, it has:
- 6 Faces: Two square faces (the top and bottom bases) and four rectangular faces (the sides).
- 12 Edges: The lines where the faces meet.
- 8 Vertices: The corners where the edges meet.
3. What are the formulas for the volume and surface area of a square prism?
The formulas for a square prism depend on the side length of its base (let's call it 'a') and its height ('h').
The Volume (V) is calculated as: V = a²h.
The Total Surface Area (TSA) is the sum of the areas of its two square bases and four rectangular sides, calculated as: TSA = 2a² + 4ah.
4. What is the main difference between a square prism and a cube?
The key difference lies in their faces and dimensions. While both are prisms with square bases, a cube is a special type of square prism where the height is equal to the side length of its base (h = a). This results in all six faces of a cube being identical squares. In a general square prism, only the two bases must be squares; the four side faces are rectangles and can have a different height.
5. What are some real-life examples of a square prism?
Square prisms are common in everyday objects. Some familiar examples include a new, unsharpened pencil with a square cross-section, a block of cheese or butter cut into a bar shape, building bricks, and many types of packaging boxes, such as those for cereal or juice, that have square tops and bottoms.
6. Is a square prism considered a type of cuboid?
Yes, a square prism is a type of cuboid. A cuboid is defined as a 3D shape with six rectangular faces. Since a square is a special kind of rectangle (with all four sides equal), a prism with two square bases and four rectangular sides fits the definition of a cuboid perfectly. It is also known as a right cuboid.
7. How does the 'net' of a square prism look when unfolded?
The net of a square prism is the 2D pattern that can be folded to create the 3D shape. It typically consists of two identical squares (for the top and bottom bases) and four identical rectangles (for the side faces). A common arrangement is to have the four rectangles connected side-by-side in a row, with the two square bases attached to opposite sides of one of the rectangles.
