Rectangular Prism Calculator

What is a Rectangular Prism?

A rectangular prism, also known as a cuboid or rectangular cuboid, is a three-dimensional geometric shape with six rectangular faces. It is characterized by its length (l), width (w), and height (h). Rectangular prisms are common in everyday life, from boxes and buildings to books and refrigerators.

How to Calculate Rectangular Prism Properties

To fully understand a rectangular prism, we need to calculate several key properties: its volume, surface area, and diagonal length. Each of these properties provides unique information about the prism's dimensions and characteristics.

Formulas

Here are the essential formulas for a rectangular prism:

1. Volume (V):

\[ V = l \times w \times h \]

2. Surface Area (SA):

\[ SA = 2(lw + lh + wh) \]

3. Diagonal Length (d):

\[ d = \sqrt{l^2 + w^2 + h^2} \]

Where:

• \(l\) is the length of the rectangular prism
• \(w\) is the width of the rectangular prism
• \(h\) is the height of the rectangular prism

Calculation Steps

1. Determine the length (l), width (w), and height (h) of the rectangular prism
2. Calculate the volume using \(V = l \times w \times h\)
3. Calculate the surface area using \(SA = 2(lw + lh + wh)\)
4. Calculate the diagonal length using \(d = \sqrt{l^2 + w^2 + h^2}\)

Example and Visual Representation

Let's calculate the properties of a rectangular prism with length 5 units, width 3 units, and height 4 units:

1. Given: \(l = 5\) units, \(w = 3\) units, \(h = 4\) units
2. Volume: \(V = l \times w \times h = 5 \times 3 \times 4 = 60\) cubic units
3. Surface Area: \(SA = 2(lw + lh + wh) = 2(5 \times 3 + 5 \times 4 + 3 \times 4) = 2(15 + 20 + 12) = 2(47) = 94\) square units
4. Diagonal Length: \(d = \sqrt{l^2 + w^2 + h^2} = \sqrt{5^2 + 3^2 + 4^2} = \sqrt{25 + 9 + 16} = \sqrt{50} \approx 7.07\) units

Here's a visual representation of this rectangular prism:

In this diagram, you can see a 3D representation of our rectangular prism with length 5 units, width 3 units, and height 4 units. The blue surface represents the prism's faces. The red dashed line shows the length, the green dashed line shows the height, and the blue dashed line shows the width. The length (l), width (w), height (h), volume (V), surface area (SA), and diagonal length (d) are labeled.