Volume of Rectangular Pyramid: Formula, Examples

What Is the Volume of a Rectangular Pyramid?

A rectangular pyramid is a three dimensional object with a rectangle as its base and triangular lateral faces. A rectangular pyramid is crowned at the top at a point known as the apex. Except for the base, all the faces connect at a vertex at the top called the apex.

Thus, a rectangular pyramid has these main parts: a rectangular base, four triangular faces, five vertices, and eight edges. 

Depending on the position of the axis of the rectangular pyramid, it is classified into two types. 

• Right rectangular pyramid: The apex is aligned with the center of the base. 
• Oblique rectangular pyramid: The apex is not aligned with the center of the base. The perpendicular line drawn from the apex to the base of the pyramid determines the height of the oblique rectangular pyramid.

The amount of unit cubes that can fit within a rectangular pyramid is called its volume. The volume is measured in cubic units. For example, it can be expressed as $in^{3},\; ft^{3},\; unit^{3}$, etc., depending upon the given units. The general formula to calculate the volume of a pyramid is equal to one-third the product of the area of the base and the height of the pyramid.

Volume of Rectangular Pyramid Formula

The volume of a rectangular pyramid depends on the area of the base and its height. It is measured in cubic units. The formula for the volume of the rectangular pyramid is as follows:

Volume $= \frac{1}{3}\times$ Base Area$\times$ Height

Here,

l $=$ length of the rectangular base

w $=$ width of the rectangular 

h $=$ perpendicular height of rectangular pyramid

Base area of the rectangular pyramid $=$ Area of rectangle $=$ Length $\times$ Width $= l \times w$

Thus,

Volume of rectangular pyramid $= \frac{1}{3} \times $length $\times$width $\times$ height $= \frac{1}{3} \times$ l $\times$ w $\times$ h

How to Find the Volume of a Rectangular Pyramid 

The volume of a rectangular pyramid is calculated using the formula: 

Volume $= \frac{1}{3} \times$ Base Area $\times$ Height $= \frac{1}{3} \times$ l $\times$ w $\times$ h

Let’s look at the steps to find the volume of a rectangular pyramid.

Step 1: Find the area of the base of the pyramid using the formula of area of rectangle.

Area of the base $=$ Length $\times$ Breadth

Step 2: Find the volume using the formula: Volume $= \frac{1}{3} \times $ Base Area $\times$ Height 

Step 3: The volume is expressed in cubic units, like $in^{3},\; ft^{3},\; unit^{3}$, etc. 

Fun Facts!

• The rectangular pyramid is one of the most common pyramids we encounter the first time we learn about three-dimensional figures. 

• A rectangular pyramid is a three-dimensional figure that has triangles as surfaces and a rectangle as its base.  

• The volume of a rectangular pyramid is simply equal to the amount of space that can be occupied within the rectangular pyramid.

• The volume of a rectangular pyramid is a third of the corresponding rectangular prism.

• Three rectangular-based pyramids fill one rectangular cuboid (with the same size base and height).

Conclusion

In this article, we learned the definition, meaning, and formula of the volume of a rectangular pyramid. Let’s solve some examples to understand it better.

Solved Examples

1. Determine the volume of a rectangular pyramid shaped tank whose base area and height are $60\; ft^{2}$ and 10 ft respectively.

Solution: 

Area of the rectangular base $= 60\; ft^{2}$

The height of the pyramid shaped tank $= 10\; ft$

We know that,

The volume of the rectangular pyramid (V) $= \frac{1}{3} \times$ Base Area $\times$ Height

                                                                     $= \frac{1}{3} \times60 \times10$

                                                                     $= 20 \times 10$

                                                                     $= 200\; ft^{3}$

Hence, the volume of the given rectangular pyramid shaped tank  is $200\; ft^{3}$.

2. Find the volume of a rectangular pyramid if the base length is 10 inches and the base width is 6 inches, and the height of the pyramid is 14 inches.

Solution:

Base length (l) $= 10$ inches

Base width (w) $= 6$ inches

The height of the pyramid (h) $= 14$ inches

We know that,

Base area of rectangular pyramid $=$ length $\times$ width 

                                                      $= 10 \times 6$ 

                                                      $= 60\; in^{2}$

Now, The volume of the rectangular pyramid (V) $= \frac{1}{3} \times$ Base Area $\times$ Height

                                                                             $= \frac{1}{3} \times 60 \times 14$

                                                                             $= 13 \times 840$

                                                                             $= 280\; in^{3}$

Hence, the volume of the given rectangular pyramid is $280\; in^{3}$.

3. Determine the height of a rectangular pyramid whose base area and volume are 120 $ft^{2}$ and 360 $ft^{3}$, respectively.

Solution: 

Area of the rectangular base $= 120\; ft^{2}$

The volume of the rectangular pyramid $= 360\; ft^{3}$

We know that,

The volume of the rectangular pyramid (V) $= \frac{1}{3} \times$ Base Area $\times$ Height

                                                               $360 = \frac{1}{3} \times 120 \times$ Height

                                                               $360 = 40 \times$ h

                                                                  h  $= \frac{360}{40}$ 

                                                                  $= 9$ ft.

Hence, the height of the given rectangular pyramid is 9 ft.

4. The base of rectangular pyramid has a dimension of $8\; ft \times 10\;ft$ and its height is 12 ft. What is the volume of the rectangular pyramid?

Solution: 

h $= 12$ inches.

The base area $= 8 \times 10 = 80\; ft^{2}$.

The volume of the rectangular pyramid (V) $= \frac{1}{3} \times$ Base Area $\times$ Height

                                                                    $= \frac{1}{3} \times (80) \times 12$

                                                                    $= \frac{960}{3}$

                                                                    $= 320\; ft^{3}$
Volume of the given rectangular pyramid $=  320\; ft^{3}$

Practice Problems on Volume of Rectangular Pyramid

Frequently Asked Questions on Volume of Rectangular Pyramid