What is Prism?

Prism can be described in the following points:

• Congruent parallel bases or ends bound a solid, and the side faces (called the lateral faces) are the parallelograms formed by joining the corresponding vertices of the grounds.
•  A right prism is called if the lateral is a rectangle or an oblique prism. A common side of the two lateral faces is called a lateral edge.
•  They are named according to the shape of their ends.
•  The terms square, rectangular, hexagonal, and parallelepiped describe various prisms with different base shapes.

Definition of Prism

A prism is a geometric object with two parallel and congruent polygonal faces joined by a pair of parallelograms or rectangles known as lateral faces. These faces are often in the shape of triangles or parallelograms. The cross-sectional geometry of the lateral faces is constant along their whole length, and they are parallel to the base faces. According to the design of their base, prisms are given names like triangular, rectangular, hexagonal, etc. They are studied for their volume and surface area formulas in mathematics because they are frequently employed in optics to bend and split light.

Altitude

Axis

In a right prism, the altitude, the axis, and the lateral edge are the same lengths. In the figure, the lateral faces are OAEC, BDGF, and OBCF dan ADEG.
The bases are OABD and ECFG. LM = h is the axis, where L and M are the centers of the bases.

Surface Area of Prism

A prism’s surface area is the sum of all its faces. It is a three-dimensional geometric structure with two congruent parallel bases joined by rectangular or parallelogram sides. It’s surface area may be computed by summing the areas of the bases and lateral faces.

From Fig. 1.2:

Lateral surface area = Area of (OAEC+ ADEG + BDGF + OBCF)
Lateral surface area = (OA)h + (AD)h + (DB)h + (OB)h
Lateral surface area = (OA+ AD + DB + BO)h
Lateral surface area = Perimeter of the base x height

Note: Total surface, surface area, total surface area, and the surface of any figure represent the same meanings.

Volume of Prisms

A sold occupies an amount of space called its volume. Certain solids have an internal volume or cubic capacity. (When not associated with cubic, the term capacity is usually reserved for the volume of liquids or materials which pour, and unique sets of units, e.g., gallon, liter, are used).

The volume of the solid is measured as the total number of unit cubes it contains. If the solid is a Prism, the volume can be computed directly from the formula:

V = l.b.h

Where, l, b and h denoted the length, breadth and height respectively. Also l. b denotes the area A of the base. Then,

Formula for volume of Prism

Volume = Area of the base x height 
l = 6 units
b = 3 units
h = 5 units
The total number of unit cubes = l.b.h
The total number of unit cubes = 6x3x5
The total number of unit cubes = 90 cube units.
Weight of solid = volume of solid x density of solid (density means the weight of unit volume)

Types of Prism

Some most important types are given below:

Rectangular Prism

A three-dimensional solid object with six rectangular faces that are congruent and parallel to one another is known as a rectangular prism. It is also referred to as a box or a rectangular parallelepiped. It has 12 edges and eight vertices (corners). It’s length, width, and height can vary, but all of its faces are rectangles.

• The volume of a rectangular prism
  Volume = abc cu. Unit
• Lateral surface area
  Lateral surface area = Area of four lateral faces
  Lateral surface area = 2 ac + 2bc sq. unit
  Lateral surface area = (2a + 2b)c = Perimeter of base x height
• Total surface area
  Total surface area = Area of six faces
  Total surface area = 2ab + 2ac + 2ca
  Total surface area = 2 (ab + bc+ca) sq. unit
• Length of the diagonal OG
  In the light triangle ODG, by the Pythagorean theorem,
  OG² = OD² +DG² ………eq.1
  OG² = OD² + C² ………..eq. 2
  Also in the right triangle OAD,
  Or
  OD² = OA²+ AD²
  =a² + b²
  Put OD² in equation (1)
  (the line joining the opposite corners of the rectangular prism is called
  it’s diagonal).
  OG² =a²+ b²+c²
  |OG| = √a² + b² +c²

Cube

A cube prism, often referred to as a rectangular prism, is a three-dimensional solid form enclosed by six rectangular faces, each of which has four right angles, and the faces on either side are congruent (equal in size and shape).

The three dimensions of a cube prism—length, width, and height—define it. The base’s length and width are its measurements, while the size is the separation between its parallel counterparts. It differs from a cube in that it has various lengths, widths, and heights as opposed to a cube, which has equal sides on all sides.

It is a familiar geometric shape frequently used in engineering, design, and architecture. Silica can be found in commonplace items like cardboard boxes, literature, and structures.

• Lateral surface area
  Lateral surface area = Area of four lateral faces
  Lateral surface area = 2a. a + 2a. a
  Lateral surface area =4a² sq. unit
• Total surface area
  Total surface area = Area of six faces
  Total surface area =6a² sq. unit
• The length of the diagonal
  |OG| = √a² + a² +a²
  |OG| =a√3

Polygonal Prism

A polygonal prism is a three-dimensional geometric structure consist up of a polygon and pressing it in a single direction perpendicular to its plane. The resultant shape comprises two identical polygonal bases and a set of rectangular or parallelogram-shaped faces connecting the two bases. The number of sides in the polygonal base defines the name of the prism, which might be rectangular, pentagonal, or hexagonal.

• The volume of the polygonal prism
  The volume of the polygonal prism when the base is a regular polygon of n sides and h is the height = Area of the base x height
  V = (na²/4) cot (180º/n)xh…….when side a is given.
  V = (nR²/2) Sin (360º/n)xh……when radius R of the circumscribed circle is given.
  Volume = nr² tan (180º/n)xh…..when radius r of the inscribed circle is given.
• Lateral surface area
  Lateral surface area = Perimeter of the base x height
  Lateral surface area = na x height, a is the side of the base
• Total surface area
  Total surface area = Lateral surface area + Area of bases

Frustum of a Prism

When a plane parallel cuts a solid to its base (or perpendicular to its axis), the section of the concrete is called a cross-section. A frustum is the region of the prism between the plane section and the base of the plane section that is not parallel to the bases.

In the given figure, ABCDEIGH represents a Frustum of a prism whose cutting plane EFGH is inclined at an angle θ to the horizontal. In this case, the Frustum whose base ABEF is a Trapezium and height BC.

• The volume of the Frustum
  Volume of the Frustum = Area of the Trapezium ABEF x BC
  Volume = ((AE+BF/2)* AB) * (BC)
  Volume = (h₁ +h₂ / 2) * (AB+ BC)
  Volume = average height x area of the base
  OR
  Volume = average height x area of the cross-section
• Area of the Section EFGH
  Since the cutting plane is inclined at an angle θ to the horizontal.
  (FI/EF) = Cos θ
  (AB/EF) = Cos θ as FI = AB
  Or
  (AB/EF) x (BC/FG) = Cos θ as BC = FG
  Or
  (Area of the base/ Area of the section EFGH) = Cos θ
  Or
  Area of the section EFGH = (Area of the base/ Cos θ)
• Lateral surface area
  Lateral surface area = Area of rectangle ADEH + BCFG
  Lateral surface area = 2(Area of Trapezium ABEF)
  Lateral surface area = Perimeter of the base x average height
• Total surface area
  Total surface area = Area of the base + Area of the Section
  Total surface area = EFGH + Lateral surface area

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