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VOOZH | about |
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VOOZH | about |
A prism is a three-dimensional shape with two identical, parallel polygonal bases and rectangular lateral faces connecting the corresponding sides of the bases. Prisms are named after the shape of their base; for example, a hexagonal prism has hexagonal bases, and a rectangular prism has rectangular bases. Prisms do not have curved faces.
In real life, we encounter prisms in various objects like buildings, optical devices, and containers. They are commonly used in construction, architecture, and design due to their simple and practical structure.
Prisms can be classified based on three criteria:
Based on the type of polygon base, the prism can be classified as:
Regular Prism: It is characterized by a base that takes the form of a regular polygon, which means all its sides and angles are equal. This results in a prism with uniform and symmetric properties. The faces and edges are organized in a structured and predictable manner, making calculations and geometric analysis more straightforward.
Irregular Prism: It features a base in the shape of an irregular polygon, where the sides and angles are not equal. This leads to a prism with non-uniform and asymmetric characteristics. The faces and edges exhibit a less predictable arrangement, making geometric calculations and analysis more complex due to the lack of symmetry.
Prisms are named based on the shape of their cross-sections, which means the shape you get when you cut them. The different types of Prism based on the shape of the base are:
Triangular Prism
Square Prism
Rectangular Prism
Pentagonal Prism
Hexagonal Prism
Octagonal Prism
Trapezoidal Prism
Prism can also have more types based on the alignment of the base. Examples of prism based on alignment are:
Right Prism
A right prism is a solid shape with flat ends that align perfectly and create rectangular bases. Its side faces are also rectangular, that gives a consistent, upright structure. This geometric structure plays a crucial role in various mathematical concepts and practical applications.
Oblique Prism
An oblique prism seems slanted because its flat ends aren't perfectly aligned. The sides form parallelograms, creating an inclined shape. This occurs due to the prism's construction. It's this structure that causes the visual effect of tilting when observed from certain angles.
A cross-section forms when a 3D object is sliced by a plane along its axis. In simpler terms, you can think of it as cutting a 3D object with a flat plane to create a different shape.
If a plane parallel to its base intersects a prism, the resulting cross-section will match the shape of the base. For instance, when a plane cuts through a square pyramid in the same direction as its base, the cross-section will also be a square. This means the shape after the cut is the same as the starting shape.
A prism has mainly two formulas, one is the surface area of the prism and another one is the volume of a prism. Let's learn them in detail.
There are two kinds of areas regarding prisms:
Lateral Surface Area of Prism
Lateral Area of a prism is the sum of the areas of all its side faces. On the other hand, the Total Surface Area of a prism is the sum of its lateral area and the area of its bottom and top faces.
To find the lateral surface area of a prism can be calculated using the formula:
Lateral Surface Area = Base Parameter × Height
Total Surface Area of Prism
For the total surface area of a prism, there are two methods to calculate: by adding two times the base area to the lateral surface area, or by adding two times the base area to the product of base perimeter and height.
Total Surface Area = 2 × (Base Area )+ Lateral Surface Area
FThe formulaformulasfor Surface Area of Various Prisms
👁 Cross Section of a PrismThere are seven types of prisms we have discussed earlier, and each type has different base shapes. Therefore, the formulas for finding the surface area of the prism vary depending on the specific type of prism.
Shape | Base | Lateral Surface Area Formula | Total Surface Area Formula |
|---|---|---|---|
Triangular Prism | Triangular | Perimeter of Base × Height (Ph) | 2 × Base Area + Perimeter of Base × Height (2Ab + Ph) |
Square Prism | Square | 4 × Base Side Length × Height (4aH) | 2 × (Base Area) + 4 × Base Side Length × Height (2a² + 4aH) |
Rectangular Prism | Rectangular | 2 × (Base Perimeter × Height) (2(l + w)H) | 2 × (Base Area) + 2 × (Base Perimeter × Height) (2lw + 2(l + w)H) |
Pentagonal Prism | Pentagonal | Base Perimeter × Height (Ph) | 2 × Base Area + Base Perimeter × Height (2Ab + Ph) |
Hexagonal Prism | Hexagonal | Base Perimeter × Height (Ph) | 2 × Base Area + Base Perimeter × Height (2Ab + Ph) |
Octagonal Prism | Octagonal | Base Perimeter × Height (Ph) | 2 × Base Area + Base Perimeter × Height (2Ab + Ph) |
Trapezoidal Prism | Trapezoidal | Base Perimeter × Height (Ph) | 2 × Base Area + Base Perimeter × Height (2Ab + Ph) |
Volume is how much space a prism takes up. To find the volume of a prism, simply multiply the base area by its height. The volume of a prism is represented as V = B × H. The base area is measured in square units (units²), and the height is in linear units (units), so the unit of volume is given as units³.
Volume (V) = Area of Base (A) × Height (H)
Formula for Volume of Various Prisms
Various formulas for calculating the volume of different prisms are:
Shape | Base | Volume of Prism |
|---|---|---|
Triangular Prism | Triangle | Volume = ½ × Base × Height × Length (½bhL) |
Square Prism | Square | Volume = Base Area × Height (a²h) |
Rectangular Prism | Rectangle | Volume = Base Area × Height (lwh) |
Pentagonal Prism | Pentagon | Volume = 5/2 × Base × Height (5/2abh) |
Hexagonal Prism | Hexagon | Volume = 3 × Base × Height (3abh) |
Octagonal Prism | Octagon | Volume = 2 × (1 + √2) × Base × Height (2(1+√2)a²h) |
Trapezoidal Prism | Trapezoidal | Volume = ½ × (Sum of Bases) × Height × Height (½(a + b)h²) |
A prism and a pyramid are distinct three-dimensional geometric shapes. The key difference lies in their base configuration. A prism has two identical parallel bases, which are typically polygons, and its sides are rectangular.
In contrast, a pyramid has a single base, often a polygon, and triangular sides that converge at a common point called the apex.
Characterstics | Pyramid | Prism |
|---|---|---|
Base Shape | Single polygon (usually triangular) | Two congruent polygons (usually rectangular) |
Bases | One triangular base and three triangular faces | Two parallel and congruent bases with rectangular or polygonal side faces |
Edges | Varies depending on the base shape | Consistent number of edges (equal on both bases) |
Vertices | Four or more vertices depending on the base | Six or more vertices, depending on the base |
Volume | V = (1/3) × Base Area × Height | V = Base Area × Height |
Example | The Great Pyramid of Giza | Rectangular Prism, Triangular Prism |
Visual Shape | Pointed top, triangular sides | Rectangular or polygonal sides, parallel bases |
Example 1: Find the volume of a rectangular prism with a length of 8 units, a width of 4 units, and a height of 6 units.
Solution:
Given,
- L = 8 units
- B = 4 units
- H = 6 units
We know that,
Volume of Rectangular Prism = L × B × H
putting the values in formula, we get:
V = 8 × 4 × 6
Volume = 192 units3
Example 2: Calculate the total surface area of a square prism with a side length of 5 units and a height of 10 units.
Solution:
Given,
- Side Length = 5 units
- Height = 10 units
Lateral surface area (LSA) formula for a square prism is:
LSA= 4 × side length × heightPutting the values in formula, we get:
LSA = 4 × 5 × 10
LSA = 200 units2Now,
Total Surface Area = 2 × base area + LSA
TSA = 2 × (side length)2 + 200 units2
= 2 × 52 + 200 units2
= 2 × 25 + 200 units2
= 50 + 200 units2∴ TSA = 250 units2
