What is a Square Prism Formula for Volume and Surface Area
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 Shape Definition and Formulas
1. What is a square prism?
A square prism is a three-dimensional solid that has two parallel square bases and four rectangular lateral faces. It is a type of prism where the cross-section is a square.
- The two square faces are congruent and parallel.
- The side faces are rectangles.
- All cross-sections parallel to the base are squares.
2. How many faces, edges, and vertices does a square prism have?
A square prism has 6 faces, 12 edges, and 8 vertices. These can be counted as follows:
- Faces: 2 square bases + 4 rectangular faces = 6 faces.
- Edges: 4 edges on the top square + 4 on the bottom square + 4 vertical edges = 12 edges.
- Vertices: 4 on the top square + 4 on the bottom square = 8 vertices.
3. What is the formula for the volume of a square prism?
The volume of a square prism is given by the formula V = s²h, where s is the side length of the square base and h is the height of the prism.
- Step 1: Find the area of the square base: s × s = s².
- Step 2: Multiply by the height: s² × h.
4. How do you find the surface area of a square prism?
The surface area of a square prism is calculated using the formula SA = 2s² + 4sh. Here, s is the side of the square base and h is the height.
- 2s² represents the area of the two square bases.
- 4sh represents the area of the four rectangular side faces.
5. What is the difference between a square prism and a cube?
The key difference is that a cube has all edges equal, while a square prism only requires the base to be square.
- In a cube, length = width = height.
- In a square prism, the height can be different from the base side length.
- Every cube is a square prism, but not every square prism is a cube.
6. What is the lateral surface area of a square prism?
The lateral surface area of a square prism is 4sh, where s is the base side length and h is the height.
- It includes only the four rectangular side faces.
- It does not include the top and bottom square bases.
7. How do you find the diagonal of a square prism?
The space diagonal of a square prism is found using the formula d = √(2s² + h²). This uses the Pythagorean theorem twice.
- First, find the base diagonal: √(s² + s²) = √(2s²).
- Then apply Pythagoras with height h.
8. Is a square prism a type of rectangular prism?
Yes, a square prism is a special type of rectangular prism where the base is a square instead of a general rectangle.
- All square prisms have rectangular side faces.
- The base rectangle has equal sides, making it a square.
9. What is an example of a square prism in real life?
A real-life example of a square prism is a tall box with a square base, such as certain storage containers or pillars.
- Square columns in buildings.
- Gift boxes with square bottoms.
- Some water tanks with square bases.
10. How do you solve a word problem involving a square prism?
To solve a square prism word problem, first identify whether you need volume, surface area, or another measurement, then apply the correct formula.
- Step 1: Identify given values (side length s and height h).
- Step 2: Choose the formula (V = s²h or SA = 2s² + 4sh).
- Step 3: Substitute values and calculate carefully.
