How to Calculate the Surface Area of a Rectangle

A rectangular box is a shape with 6 faces that people often call a box. Think of a brick, or a shoebox so you can figure out exactly what a rectangular box is. Surface area is the total area of the outer part of the object. “How much paper do I need to wrap this shoebox” sounds much less complicated, but it’s exactly the same math.

Table of Contents

Find Surface Area

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/e/e7/Find-the-Surface-Area-of-a-Rectangular-Prism-Step-1-Version-3.jpg/ v4-460px-Find-the-Surface-Area-of-a-Rectangular-Prism-Step-1-Version-3.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/ e/e7/Find-the-Surface-Area-of-a-Rectangular-Prism-Step-1-Version-3.jpg/v4-728px-Find-the-Surface-Area-of-a-Rectangular-Prism- Step-1-Version-3.jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:”<div class=”mw-parser-output “></div>”}

Name the length, width, and height of the rectangular box. Each rectangular box has a length, a width, and a height. Draw a box, and write the symbols l , w , and h next to the three different sides of the shape.

If you don’t know which edge to mark, choose any corner. Mark the three lines that intersect at that corner.
Example: A box has a bottom of 3 cm x 4 cm, and a height of 5 cm. The long side of the base is 4 cm, so l = 4, w = 3, and h = 5.

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/f/f9/Find-the-Surface-Area-of-a-Rectangular-Prism-Step-2-Version-3.jpg/ v4-460px-Find-the-Surface-Area-of-a-Rectangular-Prism-Step-2-Version-3.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/ f/f9/Find-the-Surface-Area-of-a-Rectangular-Prism-Step-2-Version-3.jpg/v4-728px-Find-the-Surface-Area-of-a-Rectangular-Prism- Step-2-Version-3.jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:”<div class=”mw-parser-output “></div>”}

Look at the six sides of the box. To cover the entire surface area, you need to paint all six sides.

There is one top and one bottom. These two sides are the same size. ^{[1] X Research Source}
There is one front and one back. These two sides are the same size.
There is a left side and a right side. These two sides are the same size.
If it’s hard to visualize, cut a paper box into sections along the edges and spread it out. ^{[2] X Research Source}

READ More: How to Grow Avocados

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/b/b1/Find-the-Surface-Area-of-a-Rectangular-Prism-Step-3-Version-3.jpg/ v4-460px-Find-the-Surface-Area-of-a-Rectangular-Prism-Step-3-Version-3.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/ b/b1/Find-the-Surface-Area-of-a-Rectangular-Prism-Step-3-Version-3.jpg/v4-728px-Find-the-Surface-Area-of-a-Rectangular-Prism- Step-3-Version-3.jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:”<div class=”mw-parser-output “></div>”}

Find the area of the bottom surface. To start, let’s find the area of just one face: the base. This is a rectangle, like the other faces. To find the area of a rectangle, simply multiply the two sides together. Area (bottom) = length multiplied = lw .

Going back to the example above, the area of the base is 4 cm x 3 cm = 12 square cm .

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/b/bc/Find-the-Surface-Area-of-a-Rectangular-Prism-Step-4-Version-3.jpg/ v4-460px-Find-the-Surface-Area-of-a-Rectangular-Prism-Step-4-Version-3.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/ b/bc/Find-the-Surface-Area-of-a-Rectangular-Prism-Step-4-Version-3.jpg/v4-728px-Find-the-Surface-Area-of-a-Rectangular-Prism- Step-4-Version-3.jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:”<div class=”mw-parser-output “></div>”}

Find the area of the top surface. We already know the bottom and top are the same size so they have the same area.

Thus, the area of the top surface is also equal to 12 square centimeters .

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/1/11/Find-the-Surface-Area-of-a-Rectangular-Prism-Step-5-Version-3.jpg/ v4-460px-Find-the-Surface-Area-of-a-Rectangular-Prism-Step-5-Version-3.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/ 11/1/Find-the-Surface-Area-of-a-Rectangular-Prism-Step-5-Version-3.jpg/v4-728px-Find-the-Surface-Area-of-a-Rectangular-Prism- Step-5-Version-3.jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:”<div class=”mw-parser-output “></div>”}

Find the area of the front and back sides. The front has one side marked width and one side marked height. Area of the front = width times height = wh . The area of the back is also equal to wh .

In the above example, w = 3 cm and h = 5 cm, so the frontal area is 3 cm x 5 cm = 15 square cm . The area of the back is also equal to 15 square centimeters .

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/a/ac/Find-the-Surface-Area-of-a-Rectangular-Prism-Step-6-Version-3.jpg/ v4-460px-Find-the-Surface-Area-of-a-Rectangular-Prism-Step-6-Version-3.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/ a/ac/Find-the-Surface-Area-of-a-Rectangular-Prism-Step-6-Version-3.jpg/v4-728px-Find-the-Surface-Area-of-a-Rectangular-Prism- Step-6-Version-3.jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:”<div class=”mw-parser-output “></div>”}

Find the area of the left and right faces. We are left with only two faces of the same size. One side is the length of the box, and the other side is the height of the box. The area of the left side is lh and the area of the right side is also lh .

In the above example, l = 4 cm and h = 5 cm, so the area of the reverse side = 4 cm x 5 cm = 20 square cm . The area of the right face is also equal to 20 square centimeters .

Shorten the Formula

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/e/e8/Find-the-Surface-Area-of-a-Rectangular-Prism-Step-8-Version-3.jpg/ v4-460px-Find-the-Surface-Area-of-a-Rectangular-Prism-Step-8-Version-3.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/ e/e8/Find-the-Surface-Area-of-a-Rectangular-Prism-Step-8-Version-3.jpg/v4-728px-Find-the-Surface-Area-of-a-Rectangular-Prism- Step-8-Version-3.jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:”<div class=”mw-parser-output “></div>”}

Shorten the formula. To make it faster, we’ll shorten the formula by using some basic algebra. Start with the above equation: Area of rectangular box = lw + lw + wh + wh + lh + lh. If we add all the same terms, we get:

Area of rectangular box = 2lw + 2wh + 2lh

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/2/25/Find-the-Surface-Area-of-a-Rectangular-Prism-Step-9-Version-3.jpg/ v4-460px-Find-the-Surface-Area-of-a-Rectangular-Prism-Step-9-Version-3.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/ 2/25/Find-the-Surface-Area-of-a-Rectangular-Prism-Step-9-Version-3.jpg/v4-728px-Find-the-Surface-Area-of-a-Rectangular-Prism- Step-9-Version-3.jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:”<div class=”mw-parser-output “></div>”}

Common factorial group. The formula will be even shorter if you factor them by two:

Area of rectangular box = 2lw + 2wh + 2lh = 2(lw + wh + lh) .

{“smallUrl”:”https://www.wikihow.com/images_en/thumb/2/2d/Find-the-Surface-Area-of-a-Rectangular-Prism-Step-10-Version-3.jpg/ v4-460px-Find-the-Surface-Area-of-a-Rectangular-Prism-Step-10-Version-3.jpg”,”bigUrl”:”https://www.wikihow.com/images/thumb/ 2/2d/Find-the-Surface-Area-of-a-Rectangular-Prism-Step-10-Version-3.jpg/v4-728px-Find-the-Surface-Area-of-a-Rectangular-Prism- Step-10-Version-3.jpg”,”smallWidth”:460,”smallHeight”:345,”bigWidth”:728,”bigHeight”:546,”licensing”:”<div class=”mw-parser-output “></div>”}

Check again on the example. Going back to the original example with a length of 4, a width of 3, and a height of 5. Substitute the above numbers into the formula:

Area = 2(lw + wh + lh) = 2 x (lw + wh + lh) = 2 x (4×3 + 3×5 + 4×5) = 2 x (12 + 15 + 20) = 2 x (47) = 94 cm square. The result is the same as the original, but this calculation is much faster.

Advice

Always use “square units”, like square centimeters or square meters. ^{[3] X Research Sources} . If a box has a surface area of 50 square meters, that means that 50 square meters is needed to cover the entire surface of the box.
Some teachers use “width” or “thickness” instead of the above names. This doesn’t matter as long as you name each edge explicitly.
If you don’t know what the height of the box is, you can call any side the height. The length is usually the longest side, but even that doesn’t really matter. It is important that the name you choose is consistent throughout the problem. ^{[4] X Research Sources}

Steps

Find Surface Area

Shorten the Formula

Advice

Related Posts