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# Oblique Prism|Definition & Meaning

## Definition

The word “**oblique**” means “**slanting**”. An oblique prism is a type of prism based on its **alignment**. It is a slanted or **tilted** prism as its **bases** are not perfectly aligned. Its lateral faces are **parallelograms** that do **not** form **90°** angles with its base. The **height** of an oblique prism is measured **externally** and is not the height of the **lateral** face.

**Figure 1** shows an **oblique prism** with height **h** which is the **perpendicular** length.

To understand an **oblique prism** comprehensively, first, we need to know about the “**prism**.”

## Prism

A three-dimensional object with **two** same **ends** is known as a prism. It is a **polyhedron** with flat **faces**, straight **edges**, and sharp **vertices**. It has equal **cross-sections** with identical bases. The edges of a prism are **parallel** to each other, extending from one **base** to the other.

### Base

The two **bases** are the two **unique** sides of a **prism** which defines the type of prism according to its **shape**. These bases are **parallel** to each other regardless the prism is right or **oblique**.

For example, a **pentagonal** prism has two pentagons as **bases** on each end of the prism.

### Lateral Faces

A prism’s **lateral** faces are the **faces** that are on its **sides** instead of its top and bottom. The faces or sides of the **oblique** prism are parallelograms which is a four-sided polygon with **parallel** opposite** sides**.

**Figure 2** shows a **hexagonal** oblique prism with a demonstration of its **bases** and lateral** faces**.

## Right Prism

A **right** angle is a **90°** angle. A right **prism** is a type of prism in which the two **bases** and the lateral **faces** are joined by **perpendicular** lines. The base and the face form a **90°** angle with each other. If a prism is not right, then it is **oblique**.

**Figure 3** compares a **pentagonal right** prism and an **oblique** prism.

The **bases** of an **oblique** and a **right** pentagonal prism are the **same; what’s** **different** is the **alignment** of the two bases with each other. A right prism has **rectangular** faces, whereas an oblique prism has lateral faces as **parallelograms**.

Notice that the **height** of a **right** prism is the **same** as the height of an **oblique** prism. To understand this, consider two **decks** of the same number of **cards**.

One deck **aligned** perfectly, and the other slightly **tilted**. Both piles of cards will have the **same** height. Hence, the height of an **oblique** prism is the **perpendicular** length, not the **slanting** one.

## Surface Area of a Prism

A prism’s **surface** area is the **area** of the **outer part** of a prism. Its **formula** is:

Surface Area = 2(B) + P(l)

Where **P** is the base** perimeter**, **B** is the base **area**, and **l** is the prism’s **length** or height. The base area, base perimeter, and length of a **rectangular** prism are illustrated in **figure 4**.

The surface area of an **oblique** prism is **not easy** to calculate. It is **not common** practice to **calculate** the surface area of an oblique prism as there is no direct **formula**.

## Volume of an Oblique Prism

The **volume** is the **space** occupied by a solid object. The **formula** for a prism’s volume is as follows:

Volume of Prism = h.B

Where **h** is the **height** and **B** is the **Area** of the base of the prism. The SI **unit** of volume is **cubic meters**.

An **oblique** prism’s **volume** is equal to the volume of a **right** prism if both have the same **bases**. This is because both have the same **perpendicular** height.

## Volume of Different Types of Oblique Prisms

Different types of **prisms** have different **bases** and hence have different **base areas**. By knowing the base area, one can calculate the **volume**. The volume of different kinds of **oblique** prisms is discussed below:

### Square Prism

A **square** prism has a square as its **base** which has **four **equal sides. A **cube** is also a form of a square prism. The **Base Area** of a square prism is given by:

Base Area = s ✕ s = $s^2$

Where **s** is the **length** of the side of the **square** base.

The **volume** of an **oblique** and non-oblique square **prism** is:

Volume = $s^2$ ✕ h

### Rectangular Prism

A rectangular prism has a **rectangular** base. A rectangle is a **quadrilateral** having equal **opposite** sides. If the **length **of the rectangle is **L** and the **width** is **W**, then its **base area** is given as:

Base Area = L ✕ W

So, a **rectangular** prism’s **volume** will be:

Volume = L ✕ W ✕ h

### Triangular Prism

A triangular prism has a **triangle** as its base. A triangle is a **three-sided** polygon whose three **angles** sum to **180°**. If the **base of the triangle **is **B** and the **height** is **H**, then its **base area** is:

Base Area = B.H / 2

Then, the **triangular** prism’s **volume** will be:

Volume = (B.H ✕ h) / 2

**Figure 5** shows **oblique** square, rectangular and triangular **prisms** with their **dimensions**.

## Examples

### Example 1

Find the **volume** of an **oblique,** rectangular prism of height **16 cm**. The length of the rectangular base is **10 cm,** and its width is **6 cm**.

### Solution

A **rectangular** prism’s **volume** is given as:

Volume = L ✕ W ✕ h

An **oblique,** rectangular prism has the **same** volume as a **right** rectangular prism. Here:

L = 10 cm, W = 6 cm, h = 16 cm

Putting the **values** in the above **equation** gives:

Volume = 10 ✕ 6 ✕ 16

**V = 960 cm$^3$**

So, the **volume** of the oblique, **rectangular** prism is **960** cubic centimeters.

### Example 2

The **volume** of an **oblique,** triangular prism is **400** cubic meters, and its height is** 16 m**. If the length of the **triangular** base is **10 m**, what is the **height** of its base?

### Solution

The **volume** for an oblique, triangular **prism** is given as:

V = (B.H ✕ h) / 2

Multiplying **2** on both sides gives:

2V = B.H ✕ h

Dividing **B.h** on both sides gives the equation for the **height** of the **triangular** base **H** as:

H = 2V / B.h

Here:

V = 400 cubic meters, B = 10 m, h = 16 m

Putting the **values** in the above **equation** gives:

H = 2(400) / 10(16)

H = 800 / 160

**H = 5 m**

So, the **height** of the **triangular** base of an **oblique** prism is **5 m**.

*All the images are created using GeoGebra.*