What is a Prism?
A three-dimensional solid object called a prism has two identical ends. It consists of equal cross-sections, flat faces, and identical bases. Without bases, the prism’s faces are parallelograms or rectangles. And any n-sided polygon, including a triangle, square, rectangle, or other, could serve as the prism’s base. For instance, a pentagonal prism has five rectangular faces and two pentagonal bases.
Cross Section of a Prism
The shape formed when an object and a plane intersect along its axis is known as a cross-section. It is also described as using a plane to slice a three-dimensional object into a different shape. A prism’s cross-section will have the same shape as the base if it is intersected by a plane that is parallel to the base.
Types of Prism
Prisms are named based on their cross-sections.
There are two categories:
- Regular Prism
- Irregular Prism
Regular Prisms
The term ‘regular prism’ refers to a prism whose bases are shaped like regular polygons.
Irregular Prism
A prism is referred to as an irregular prism if its bases have an irregular polygonal shape.
The prism can be divided into two additional categories in addition to regular and irregular and that are as follows:
- Oblique Prism
- Right Prism
A prism with joining edges and faces that aren’t parallel to the base faces is referred to as an oblique prism.
For instance, a parallelepiped is an oblique prism with a parallelogram as its base, or, alternatively, a polyhedron with six parallelograms as faces.
A right prism is a geometric solid with vertical sides that are perpendicular to the base and a base that is a polygon. The top surface and base have the same size and shape. The fact that the angles between the base and sides are right angles gives rise to the term ‘right’ prism.
Surface Area of a Prism
The total area that the faces of a prism enclose is its surface area.
The formula applied to find the surface area of a prism is;
Surface Area of Prism = 2 (Base Area) + (Perimeter of the Base \( \times \) Height)
Volume of a Prism
The volume of a prism is calculated by multiplying its height by the area of its base.
Prism volume \( (V)~=~B~\times~h \), where B is the area of the base and h is the height of the prism.
Therefore,
- Volume of prism = Base Area × Height
Solved Prism Examples
Example 1: Find the volume of a triangular prism with a 30 \(cm^2 \) base area and a 5 cm height.
Solution: We have,
Base area = 30 \( cm^2 \)
Height = 5 cm
The volume of a prism = (Base area × Height) cubic units
Therefore, V = 30 \( \times \) 5 = 150 \( cm^3 \)
Hence, the volume of a triangular prism is 150 \( cm^3 \).
Example 2: Determine the surface area of a prism with a 30 \(cm^2 \) area, 5 cm height, and a 15 cm base perimeter.
Solution: We have,
Base area = 30 \( cm^2 \)
Height = 5 cm
Perimeter of the base = 15 cm
The surface area of a prism = 2 (Base Area) + (Perimeter of the Base \( \times \) Height)
Therefore, Surface Area = 2(30) + (15 \( \times \) 5) = 4500
= 60 + 75 = 135
Hence, the surface area of the prism is 135 \( cm^2 \).
Example 3: Find the height of the square prism whose volume is 400 \( cm^3 \) and the base area is 50 \( cm^2 \).
Solution:
The height of the square prism is calculated as follows:
The volume of square prism = Base area × height
400 = 50 × prism height
Therefore, the height h = \( \frac{400}{50} \)
Prism Height, h = 8 cm
Hence, the height of the prism is 8 cm.
No, the prism’s lateral surfaces have to be rectangular.
Yes, prisms are present in every parallelepiped. The bases of the prism can be any two opposite pairs of opposite faces. In the case of a parallelepiped, the remaining four faces can be thought of as the lateral faces of the prism.
We have triangular, square, rectangular, pentagonal, and hexagonal prisms based on the shapes of the base of such prisms.
Pyramids and prisms are both three-dimensional solids with flat faces and bases. However, a pyramid has one base, whereas a prism has two bases that are identical.
When a plane intersects a prism, the shape formed is known as the cross section. The cross section of the prism will have the same shape as the base if it is divided by a plane that runs horizontally and parallel to the base.