# 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 correspondingĀ
**verticesĀ**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)

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,

OGĀ² = ODĀ² +DGĀ² ā¦ā¦ā¦eq.1

OGĀ² = ODĀ² + CĀ² ā¦ā¦ā¦..eq. 2

Also in the right triangle OAD,

Or

ODĀ² = OAĀ²+ ADĀ²

=aĀ² + bĀ²

Put ODĀ² in equation (1)

(the line joining the opposite corners of the rectangular prism is called

it’s diagonal).

OGĀ² =aĀ²+ bĀ²+cĀ²**|OG| = āaĀ² + bĀ² +cĀ²**

### 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 prismālength, width, and heightādefine 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 =4aĀ² sq. unit****Total surface area**

Total surface area = Area of six faces**Total surface area =6aĀ² sq. unit****The length of the diagonal**

|OG| = āaĀ² + aĀ² +aĀ²**|OG| =aā3**

### 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 = (naĀ²/4) cot (180Āŗ/n)xhā¦ā¦.when side a is given.

V = (nRĀ²/2) Sin (360Āŗ/n)xhā¦ā¦when radius R of the circumscribed circle is given.**Volume = nrĀ² tan (180Āŗ/n)xh**ā¦..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 Īø 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 = (hā +hā / 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 Īø to the horizontal.

(FI/EF) = Cos Īø**(AB/EF) = Cos Īø as FI = AB**

Or**(AB/EF) x (BC/FG) = Cos Īø as BC = FG**

Or**(Area of the base/ Area of the section EFGH) = Cos Īø**

Or**Area of the section EFGH = (Area of the base/ Cos Īø)****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**

## FAQ’s

### What are the different ways to find the volume of a prism?

The volume is the product of the area, base and its height. There are many methods for calculating the volume.

1- **Using the formula (v=bxh)**

2- **Water displacement method**

3- **Cavalieri’s principle**

4- **Integration**

5- **Counting unit cubes**

### What is the formula of all types of prism?

The formula for the volume (V) is given below:**V = Bh**

where** B **is the **base** and **h is its height.**

The formula for the surface area (SA) is given below:

S**A = 2B + Ph**

where **B is the base area**, **P is the base perimeter**, and** h is the height**.

### What methods are used to find the volume of irregular objects?

There are several methods for determining the volume of irregular objects. Here are some standard methods:

1- **Displacement method**

2- **Geometric approximation**

3- **3D scanning and modeling**

4- **Calculus integration**

5- **Laser scanning**

### Who is the father of a prism?

Many mathematicians and scientists have studied prisms and their properties throughout history, so there is no single “**father**” of it.

However, the ancient Greek mathematician **Euclid **was one of the most well-known prism researchers in his book “**Elements**,” **published around 300 BCE.**

### What is the volume formula for a Triangular prism?

Volume:

**V = (1/2)Bh**

Surface area: **SA = B + Ph**

where **B is the base area** of the triangular base and** P is the perimeter** of the base

### What is the volume formula for a Pentagonal prism?

Volume: **V = (5/2)asl**

Surface area:

SA = 5B + 5Pl

where B is the **base area **of the pentagonal base, **P is the perimeter **of the base, and **l is the slant height**.

### What is the volume formula for a Hexagonal prism?

Volume: **V = 3ā3/2a^2h**

Surface area: **SA = 2B + 6al**

where **B is the base area **of the hexagonal base and **l is the slant height**.

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