Rectangular Pyramid

The rectangular pyramid is one of the most common pyramids we encounter the first time we learn about three-dimensional figures. Understanding the properties and formulas involving three-dimensional shapes open a wide range of concepts and applications in Geometry. This includes learning key concepts about rectangular pyramids.

In this article, we’ll cover the basic definition and components of rectangular pyramids. We’ll also break down the process of finding the volume and surface area of a rectangular pyramid. Keep your notes ready and by the end of this discussion, you’ll feel even more confident when working with problems involving rectangular pyramids!

What Is a Rectangular Pyramid?

A rectangular pyramid is simply a pyramid with a rectangular base. You’ve probably encountered what pyramids are when studying about Egypt or pharaohs in your history or geography classes.  These are in fact a special type of pyramid – the rectangular pyramid.

In geometry, we define pyramids as three-dimensional figures that have triangles as lateral surfaces and their bases can vary. The rectangular pyramid is a special type of pyramid that has a rectangle for its base. 

This means that a rectangular pyramid is a three-dimensional figure that has triangles as surfaces and a rectangle as its base.  Now that we’ve established what makes rectangular pyramids unique, it’s time for us to learn the terms that are essential when dealing with rectangular pyramids.

Understanding the Components of a Rectangular Pyramid

A rectangular pyramid will always have four lateral surfaces that are triangular and a base that is rectangular. This means that the rectangular pyramid is made up of four triangular faces for a rectangle. The four lateral faces all meet at the vertex or the apex. It’s the highest point or the peak of a rectangular pyramid.

Its height represents the distance between the pyramid’s vertex and its rectangular base. Here’s a fun fact: when the height and the base form a right angle, the rectangular pyramid is an oblique pyramid. Now, the slant height represents the height of the triangular lateral surface and is important when finding its surface area.

 Knowing the terms of the rectangular pyramid will make it easier for one to solve problems involving rectangular pyramids – especially when finding its volume and surface.

How To Find the Volume of a Rectangular Pyramid?

The volume of a rectangular pyramid is simply equal to the amount of space that can be occupied within the rectangular pyramid. In general, we calculate the volume of a pyramid by taking the third of the base area times its height. This, of course, applies to finding the volume of rectangular pyramids and we know that the area of a rectangle is equal to the product of its length and width.

\begin{aligned}V &= \dfrac{1}{3} Bh\\&= \dfrac{1}{3}lwh\\\\\ V&: \text{Volume}\\ B&: \text{Base Area}\\ l&: \text{length}\\w&: \text{width}\\ h&: \text{height}\end{aligned}

The volume of a rectangular pyramid will have a unit represented as cubic units such as cm3, ft3, and yards3. Let’s go ahead and break down the steps when calculating the rectangular pyramid’s volume:

  1. Find the base’s area by multiplying the rectangle’s length and width.
  2. Multiply the base area by the rectangular pyramid’s height. 
  3. Take one third of the result to find the volume and write the final answer with the appropriate unit.

Of course, the best way to know the process by heart is to keep on trying out different examples and word problems. 

Problem 1

Suppose that we have a rectangular pyramid that has a height of 12 m and a base with length and width of 8 m and 5 m, respectively. Find its volume by applying the steps we’ve just discussed.

We begin by multiplying the length and the width of the base area then multiply the product by the pyramid’s height. Once we have this result, we now multiply one third of it to find the rectangular pyramid’s volume.

\begin{aligned}V &=\dfrac{1}{3}Bh\\&= \dfrac{1}{3}(l \times w)\times h\\&= \dfrac{1}{3}(8 \times 5) \times 12\\&= \dfrac{1}{3}(40 \times 12)\\&= 160\end{aligned}

Keep in mind to include the unit of the volume, so the rectangular pyramid will have a volume of 160 cubic meters or 160 m3.

The volume of three-dimensional figures allows us to measure the amount of space that can be filled within the figure or simply, its capacity.

Problem 2

Suppose that we have a rectangular pyramid for a water tank. The tank has a base area of 240 ft2 and it can be filled up to a height of  40 ft. We can find the amount of water that can be filled within the tank by finding its volume.

\begin{aligned}V &= \dfrac{1}{3} Bh\end{aligned}

For this case, B = 240 ft2 and h = 40 ft, so the rectangular pyramid has a volume of \dfrac{1}{3}(240)(40) = 3200 ft3. This means that the water tank has a capacity to store a total of 3200 cubic feet of water.

We’ve now shown you how to calculate the rectangular pyramid’s volume and how to use the volume to solve world problems. It’s also important to learn about the rectangular pyramid’s total surface and we’ll learn more about this in the next section. 

What Is the Surface Area of Rectangular Pyramid?

The surface area of a rectangular pyramid represents the area covered by the external surface of the rectangular pyramid. It represents the total area of the surfaces that form the rectangular pyramid: four triangular surfaces and one rectangular base. 

When dealing with the rectangular pyramid’s surface area, we simply add all the areas of its lateral surfaces and its rectangular base. If these values are already given, then simply add them. 

\begin{aligned}\text{Surface Area} &= 4 \times \text{Triangular Area} + \text{Base Area} \end{aligned}

However, there are instances when we’ll have to find the triangular surfaces’ slant heights first. There will be two varying slant heights – one for each pair of triangles facing each other and sharing the same base.

Use the Pythagorean theorem when finding each pair of triangles’ slant height. The slant height becomes the hypotenuse while the pyramid’s height and half of either the length or width become the legs of the right triangle. The two figures break down the process for finding the slant heights of the two pairs of triangles.

Apply the formulas for finding the areas of the surfaces. This table summarizes the areas of the lateral surfaces and the rectangular base. Adding them all up will lead to the total surface area of the rectangular pyramid.

Triangles with \boldsymbol{l} as the Base Triangles with \boldsymbol{w} as the BaseRectangular Base
\begin{aligned}A = \dfrac{l}{2}\sqrt{\left(\dfrac{w}{2} \right )^2 + h^2}\end{aligned} \begin{aligned}A = \dfrac{w}{2}\sqrt{\left(\dfrac{l}{2} \right )^2 + h^2}\end{aligned} \begin{aligned}A = lw\end{aligned}
\begin{aligned} \textbf{Total Surface Area }&= lw + 2\left(\dfrac{l}{2}\sqrt{\left(\dfrac{w}{2} \right )^2 + h^2} \right )+ 2\left(\dfrac{w}{2}\sqrt{\left(\dfrac{l}{2} \right )^2 + h^2} \right )\\ &= lw + l\sqrt{\left(\dfrac{w}{2} \right )^2 + h^2}+ w\sqrt{\left(\dfrac{l}{2} \right )^2 + h^2}\end{aligned}

The formula may seem intimidating at first, but by learning how it was derived and applying the steps needed, you’ll actually find that calculating for the rectangular pyramid’s surface area is not as complicated.

Problem 3

Let’s try calculating the total surface area of a rectangular pyramid with the following dimensions: length of 8 ft, width of 6 ft, and a height of 12 ft.

This means that we have triangular lateral surfaces where the back and front triangles have a base of 8 ft. The triangles to the left and right sides have a base of 6 ft. 

To find the slant height of these pairs of triangular surfaces, we’ll set up the corresponding right triangles and apply the Pythagorean theorem (c^2 = a^2 + b^2) for each.  Use the slant height and the bases to find each pair’s area.

  • The front and back of the triangles will have a slant height of \sqrt{3^2 +12^2} = 3\sqrt{17} \approx 12.369 feet and an area of (1/2)(3\sqrt{17})(8) = 12\sqrt{17} \approx 49.477 squared feet.
  • The left and right sides of the triangles will have a slant height of \sqrt{4^2 +12^2} = 4\sqrt{10} \approx 12.649 feet and an area of (1/2)(4\sqrt{10})(6) = 12\sqrt{10} \approx 37.947 squared feet.

The rectangular base will have an area of 8(6) = 48 squared feet. Adding all the lateral surfaces (four triangular surfaces) and the rectangular base, the total surface area of the rectangular pyramid is equal to 2(49.477) + 2(37.947) + 48 = 222.848 squared feet.

Finding the total surface areas, in general, helps when finding estimates when working with lateral surfaces and structures. The problem below shows how we can use the rectangular pyramid’s total surface to estimate the cost of painting its external surface.

Problem 4

Suppose that we want to paint the surface of the rectangular pyramid mentioned in Problem 3 and it costs 5.60 dollars per square foot to paint the surface. To find the total cost, we simply multiply the cost per square foot by the total surface area of the pyramid.

\begin{aligned}\text{Total Cost} &= \text{Cost per ft}^2 \times \text{Surface Area}\\&= (5.60)(222.848)\\&=1247. 949 \end{aligned}

This means that it will cost 1247.95 dollars to paint the surfaces of the rectangular pyramid from the previous problem.