The lateral area of a rectangular prism is the sum of the area of the four lateral rectangles without taking into account the surfaces of the top and bottom rectangle (the bases). The lateral area of a rectangular prism can be calculated with the following formula:

$2\times ah+2\times bh$

$a=$ length $b=$ width $h=$ height

Lateral area of a rectangular prism

In this article, we will learn how to find the lateral area of a rectangular prism, quickly and easily. First, it is convenient that we remember certain basic and important details about the rectangular prism and understand how its lateral sides look.

Here is a rectangular prism

What is the lateral area of the rectangular prism?

The lateral faces are all the faces of the rectangular prism except for the two bases (top and bottom). How can you remember it? Imagine that you have to wrap a ribbon around the rectangular prism (a gift box, for example). All you would need to do is turn around the box with a ribbon, and thus you would cover its entire lateral area. The bottom and top bases are excluded.

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What is the lateral area of a rectangular prism?

The lateral area of a rectangular prism is the sum of the area of the four lateral rectangles (excluding the bases). To calculate the lateral surface area, we will use the following formula: $2 \times ah+2 \times bh$ That is, length times height will give us the surface area of the rectangle in front of us (we will multiply it by $2$ to also get the rectangle hidden on the opposite side) Plus Width times height will give us the surface area of the rectangle on one side. This result must also be multiplied by $2$ to obtain the surface area of the opposite side.

Let's look at 2 examples

We have here a rectangular prism

Width of the rectangular prism: $6$ cm Length of the rectangular prism: $4$ cm Height of the rectangular prism: $1$ cm larger than the length of the rectangular prism.

What is the lateral area?

Solution: We can mark the data on the rectangular prism to understand it better or, simply, to act according to the formula, do it as you find comfortable. Let's mark the data on the rectangular prism. Notice that the height was given as larger than the length of the rectangular prism by $1$, meaning, the height is $4+1=5$

Now we have to calculate the area of the lateral rectangles. Let's calculate $width \times height \times 2$ $+$ $length \times height \times 2$

Notice -> Even if you got confused between the width and the length the result will be the same.

We will obtain: $6 \times 5 \times 2+4 \times 5 \times 2=$ $60+40=100$ The lateral area of the rectangular prism is $100$ cm^{2}.

Another question

Given the following rectangular prism. The areas of its surfaces are indicated. What is the lateral area of the rectangular prism knowing that the surface of the top rectangle (the base) is $10$ cm^{2}?

Solution: As we have learned, the lateral area is composed of the surfaces of the side rectangles of the rectangular prism without taking into account the top and bottom rectangles. Therefore, the information provided in the question about the surface of the base being $10$ cm^{2} is only meant to confuse. In reality, the corresponding data are already given in the illustration, and all we have to do is correctly add up the surfaces. The area of $20$ cm^{2} we will calculate twice and the area of $10$ cm^{2} we will also calculate twice. We will obtain: $2 \times 20+2 \times 10=$ $40+20=60$ The lateral area is $60$ cm^{2}.

Note: In this exercise, we could not have used the formula since we did not have the data about the width, length, and height, therefore, we acted using our logical understanding of what the lateral area of a rectangular prism is.