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In the following article, we’ll explore the fundamental principles defining **3D rectangles**, delve into their fascinating properties, and illustrate their indispensable roles in **simple** and **complex structures** around us.

**Definition**

A **“3D rectangle”** is more formally known as a **rectangular prism** or a **cuboid** in **geometry**. It is a **three-dimensional solid shape** bounded by **six faces**, all **rectangles**. The defining properties of a rectangular prism are:

- It has
**six faces**, each of which is a**rectangle**. - The
**opposite faces**of the**prism**are**identical**. - It has
**12 edges**and**8 vertices**.

Figure-1.

The term “**3D rectangle**” isn’t often used in formal geometric contexts, but it’s typically understood to mean a **rectangular prism** or **cuboid**. This shape is common worldwide, seen in items like **books**, **boxes**, and **buildings**.

**Ralevent Formulas **

A **3D rectangle**, or a **rectangular prism (a cuboid)**, has several key formulas. Here are those formulas and their explanations:

**Volume (V)**

The **volume** of a **rectangular prism** can be calculated by multiplying the **length (l)**, **width (w)**, and **height (h)** of the prism. The formula is:

V = lwh

**Surface Area (A)**

The **surface area** of a **rectangular prism** is the sum of the areas of all six faces. Since opposite faces on a rectangular prism have the same area, this can be calculated by finding the area of each unique face (**length x width**, **length x height**, **width x height**), and then multiplying by **2**:

A = 2lw + 2lh + 2wh

**Space Diagonal (d)**

The **space diagonal** of a **rectangular prism** is the line segment that passes through the interior of the prism, connecting two opposite corners (or vertices). This can be calculated using the three-dimensional version of the **Pythagorean theorem**:

d = sqrt(l² + w² + h²)

**The perimeter of Base (P)**

The **perimeter** of the base of a **rectangular prism** (if considering one of the rectangular faces as a base) can be found by adding up all the sides of the rectangle (which is 2 times the sum of the **length** and the **width**):

P = 2l + 2w

**Lateral Surface Area (LA)**

The **lateral surface area** of a **rectangular prism** is the area of the four faces not considered the base or top. It can be found by multiplying the **perimeter of the base** by the **height**:

LA = Ph = 2lh + 2wh

**Exercise **

**Example 1**

Suppose a **3D rectangle** has a length of** 5 cm**, a width of **4 cm**, and a height of **3 cm**. What is its **volume**?

Figure-2.

### Solution

The volume can be computed as follows:

Volume = length x width x height

Volume = 5 cm x 4 cm x 3 cm

Volume = 60 cubic cm.

**Example 2**

Suppose a **3D rectangle** has a length of** 7 m**, a width of** 3 m**, and a height of **6 m**. What is its** surface area**?

Figure-3.

### Solution

Surface Area = 2lw + 2lh + 2wh

Surface Area = 2(7 m x 3 m) + 2(7 m x 6 m) + 2(3 m x 6 m)

Surface Area = 42 m² + 84 m² + 36 m²

Surface Area = 162 square m

**Example 3**

Suppose a **3D rectangle** has a length of **8 m**, a width of** 5 m**, and a height of **4 m.** What is the length of its **space diagonal**?

### Solution

Space Diagonal = sqrt(l² + w² + h²)

Space Diagonal = sqrt((8 m)² + (5 m)² + (4 m)²)

Space Diagonal = sqrt(64 m² + 25 m² + 16 m²)

Space Diagonal = sqrt(105 m²)

Space Diagonal = 10.25 m

**Example 4**

Suppose a **3D rectangle** has a length of **8 cm** and a width of** 5 cm**. What is the **perimeter** of its **base**?

### Solution

Perimeter = 2l + 2w

Perimeter = 2(8 cm) + 2(5 cm)

Perimeter = 16 cm + 10 cm

Perimeter = 26 cm

**Example 5**

Suppose a **rectangular prism** has a length of **7 m**, a width of **3 m**, and a height of **4 m**. What is its** lateral surface area**?

### Solution

Lateral Surface Area = 2lh + 2wh

Lateral Surface Area = 2(7 m x 4 m) + 2(3 m x 4 m)

Lateral Surface Area = 56 m² + 24 m²

Lateral Surface Area = 80 square m

**Applications**

**Mathematics and Physics**

**Rectangular prisms** are foundational in understanding **three-dimensional space** and concepts like **volume** and **surface area**. They are often used in **problem-solving** and **mathematical proofs**. In **physics**, they can represent objects in **kinematics problems** or even areas of space in **field problems**.

**Engineering and Architecture**

**Rectangular prisms** are a common shape in **buildings** and **engineered products**, thanks to their stability and ease of construction. **Architects** use them to model rooms and buildings, while **engineers** might design anything from a small tool to a large machine. Understanding the properties of a rectangular prism is crucial for determining **load capacities**, **space utilization**, and **structural integrity**.

**Computer Graphics and Gaming**

In **3D modeling** and **computer graphics**, complex shapes are often broken into small **rectangular prisms** (or **cuboids**) to simplify rendering. This is particularly true in **voxel-based games** like Minecraft, where the world comprises many small **cubes**, a type of rectangular prism.

**Logistics and Shipping**

**Rectangular prisms** are the most common shape for **boxes** and **shipping containers**. Understanding these prisms’ properties helps optimize **packing** and **stacking arrangements**, maximizing the use of space and potentially saving significant costs.

**Geology and Geophysics**

In disciplines like **geology** and **geophysics**, rectangular prisms can represent rock or soil volumes in models of the Earth’s crust. This is particularly useful in fields like **resource exploration** or **seismic analysis**, where understanding the properties of different volumes of earth is important.

*All images were created with GeoGebra.*