Prism - 3D Shape

Prisms are three-dimensional shapes of geometry with similar polygonal bases of different shapes and lateral surfaces, which are generally parallelograms or rectangles. Every prism has its own volume and surface area based on the number of edges in the bases of the prism. Here, we will explore different properties of these prisms along with some of the most important formulas, such as the volume of a prism or the area of a prism that can help in understanding this important geometrical shape and its uses in real life. 

1.0Definition of Prism

A prism is a polyhedron consisting of two identical, congruent, and parallel polygonal bases. The lateral faces linking the bases are parallelograms. The bases are arranged so that the prism's height is the perpendicular distance between them. The form of the polygonal bases classifies the prism into a triangular prism, rectangular prism, pentagonal prism, or hexagonal prism. 

2.0Properties of a Prism

All prisms possess several key properties common to every type, which include: 

1. Faces: A prism will always have two congruent, identical bases and a series of parallelogram-shaped lateral faces. The amount of lateral faces is equal to the number of sides on the polygonal base. 
2. Edges: The faces meet on the edges of a prism. Three types of edges occur on a prism: edges of lateral faces and bases. The total number of edges of a prism is 3n, given that n refers to the number of sides contained within the base polygon. 
3. Vertices: A prism has vertices where the edges intersect. The vertices are twice the number of vertices of the base polygon. 
4. Height: The height (or altitude) of the prism is defined as the perpendicular distance between its two bases. Basically, it is a measure of how tall a prism is. This height plays an important role in calculating the prism's volume and surface area.

3.0Types of Prisms

As mentioned earlier, prisms come in various forms and shapes depending on the number of edges of the polygonal base. Here are some key types of these prisms: 

1. Triangular Prism: As the name suggests, the base of this type is triangular with 5 faces, of which 2 are triangular bases, and 3 are rectangular lateral faces. It has 9 edges and 6 vertices. The volume of any given triangular prism can be found by simply multiplying the area of the base and height of the prism. 

$V=\text{ Base Area }\times \text{ Height }$

Here, the Base is the triangle, so the base area may be found using the formula: 

$\text{ Base Area }=\frac{1}{2}\times \text{ Base }\times \text{ Height }$

The surface area of a triangular prism can be found by summing the areas of the two triangular bases and three rectangular lateral faces. Mathematically, it can be expressed as: 

$\text{ Area }=2\times \text{ Base Area of Triangle }+\text{ Perimeter of Triangle }\times \text{ Height of Prism }$

2. Rectangular Prism (Cuboid): A rectangular prism has 6 faces (base and lateral sides), all in rectangular shape, giving it a cuboidal shape. It has 12 edges with 8 vertices. The volume of a rectangular prism is the same as the volume of a cuboid, meaning it is the product of the length(l), width(w) & height(h) of the prism. Mathematically, it can be written as: 

$\text{ Volume of Rectangular Prism }=l\times w\times h$ , The surface area of a rectangular prism may be calculated using the following formula: 

$\text{ Lateral Surface area of Rectangular Prism }=2lw+2lh+2wh$

3. Square Prism: Just like a rectangular prism, a square prism has 6 faces, of which 2 are square bases and 4 are rectangular lateral faces. It also has 12 edges and 8 vertices. Volume of a square prism can be calculated using the formula for the base area of the prism, i.e square (a²), which is: 

$\text{ Volume of Square Prism }=\text{ Base Area }\times \text{ Height }$, The surface area of a square prism is calculated with the help of the following formula: 

Surface Area = 2 × Base Area + Perimeter of Base × Height

4. Trapezoidal Prism: A prism with bases in the shape of a trapezoid is considered a trapezoidal prism. It has 6 faces, with 2 trapezoidal bases and 4 rectangular lateral faces.  The volume of a Trapezoidal prism is calculated using the formula of the base area of the prism, i.e a trapezium (½ × (a + b) × h₁):

$\text{ Volume }=\text{ Base Area }\times \text{ Height }$, The lateral surface area of a trapezoidal prism is calculated using the formula: 

Surface Area = 2 × Base Area + Sum of Areas of 4 Rectangular Faces

5. Pentagonal Prism: This type of prism has a pentagon as its base with 7 faces (2 pentagonal bases and 5 rectangular lateral faces). It also contains 15 edges and 10 vertices.  The volume of a pentagonal prism is the product of its base area and the perpendicular distance between these two bases, or the height. This can be expressed as: 

$V=\text{ Base Area }\times \text{ Height }$

Here, the base is pentagonal and can be calculated using the formula: 

$\text{ Base Area of Pentagonal Base }=\frac{1}{4}\times \sqrt{5(5+2\sqrt{5})}\times s^2$, The surface area of the pentagonal prism can be calculated by summing up the 2 pentagonal base areas and the area of the 5 rectangular lateral faces: 

$A=2\times \text{ Base Area }+\text{ Perimeter of Triangle }\times \text{ Height of Prism }$

6. Hexagonal Prism: In this prism, hexagon is present as the base with 8 faces (2 hexagonal bases and 6 rectangular lateral faces). It also has 18 edges & 12 vertices. The volume of a hexagonal prism is the product of the area of the base and the height: 

$V=\text{ Base Area }\times \text{ Height }$

The base is hexagonal in shape; hence, the base area is written as: 

$\text{ Base Area }=\frac{3\sqrt{3}}{2}\times s^2$, The surface area of the hexagonal prism is the sum of 2 base areas and 6 rectangular faces. 

$A=2\times \text{ Base area }+\text{ Perimeter of Triangle }\times \text{ Height of Prism }$

4.0Solved Examples 

Problem 1: A triangular prism has a triangular base where the base and the height of the triangle are 5cm & 12 cm. The height (or length) of the prism is 10 cm. Calculate the volume of the triangular prism.

Solution: given the base and height of the triangle are 5cm and 12 cm, and the length or height of the prism is 10cm. 

Here, the base area for the triangular base is, 

$\text{ Base Area }=\frac{1}{2}\times 5\times 12=30{\mathrm{cm}}^2$

Now, the volume of the prism, 

$V=30\times 10=300{\mathrm{cm}}^3$

Problem 2: A hexagonal prism has a side length of 4 cm for its hexagonal base and a height (length of the prism) of 15 cm. Calculate the volume of the hexagonal prism.

Solution: Given the side of the hexagonal base is 4 cm and the length of the prism is 15cm,

$\begin{array}{rl}& \text{ Base Area }=\frac{3\sqrt{3}}{2}\times s^2\\ & \text{ Base Area }=\frac{3\sqrt{3}}{2}\times 4^2=24\sqrt{3}{\mathrm{cm}}^2\end{array}$

Now, the volume for the given prism, 

$\begin{array}{rl}& V=\text{ Base Area }\times \text{ Height }\\ & V=24\sqrt{3}\times 15=360\sqrt{3}{\mathrm{cm}}^3\end{array}$

Problem 3: A triangular prism has a base of 10 cm and a height of 6 cm for the triangular base. The height of the prism is 12 cm. If the two sides of the triangle are 8 cm and 10 cm, calculate the surface area of the triangular prism.

Solution: Given that the base of the triangular base is 10cm, the height is 6cm. The other two sides are 8cm and 10cm. The length of the prism is 12cm. 

The Base area of the triangular base:

$\begin{array}{rl}& \text{ Base Area }=\frac{1}{2}\times \text{ Base }\times \text{ Height }\\ & \text{ Base Area }=\frac{1}{2}\times 10\times 6=30{\mathrm{cm}}^2\end{array}$

Now, the surface area of the prism: 

$A=2\times \text{ Base Area }+\text{ Perimeter of Triangle }\times \text{ Height of Prism }$

Perimeter of triangle = 8 + 10 + 10 = 28cm

$A=2\times 30+28\times 12=396{\mathrm{cm}}^2$