# Total Surface Area: Definition & Formula

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• 0:05 Total Surface Area Defined
• 0:54 How Is Area Different…
• 3:17 Practical Use of Surface Area
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Lesson Transcript
Instructor: Flint Johnson

Flint has tutored mathematics through precalculus, science, and English and has taught college history. He has a Ph.D. from the University of Glasgow

This lesson will define and explain what total surface area is and provide the formulas for calculating total surface area, as well as an explanation about why they work.

## Total Surface Area Defined

Surface area means all the area that you can see in two dimensions - this means the length and width. The surface area of a wall is everything you can paint. The surface area of a floor is everything you can walk on. So, the surface area of a rectangle, circle, triangle, or other shape is simply a measurement of everything within the lines of the shape. For a two-dimensional object, that's also its total surface area.

In three dimensions, like a cube, a sphere, or a pyramid, the surfaces can't all be seen at one time. Total surface area in that case means adding up the areas of all the surfaces. For a cube, that means adding up the surface area of all six sides. For a sphere like a baseball, you want to know how much area the leather casing measured.

## How is Area Different From Other Measurements?

When thinking about an object, there are only three ways of measuring its size - perimeter, area, and volume. With perimeter, you are thinking about the distance around the edges of the object. So if you had a square and wanted to know its perimeter, you would measure how long each side was, and then add up your total measurements. Because you would be measuring distance, your result would simply be in meters or feet, a one-dimensional result. For example, with four sides that are each ten meters long, you would have a perimeter of 40 meters.

For area, you would want to know the total surface area, or amount of space within the perimeter. Returning to the square, you would want to measure its two-dimensional size. To do that, you would multiply the length by the width. Your answer would still be in length units, like meters or feet, but now it would be a measurement squared, representing that it was two dimensions. The square would measure 100 meters^2 for area. If you were trying to measure the area of a cube, you would find the area of one side, and then multiply by 6 for each side of the object - 600 meters^2.

To measure volume, you would again need a three-dimensional object like a cube. You would multiply length times width again, but now instead of multiplying your total by 6 for each side of the cube, you would multiply by depth. Your answer would be represented as meters or feet cubed, to signify that it was a three-dimensional measurement. Now your result would be 1000 meters^3.

## Calculating for Area

There are a number of formulas to find area, depending on the shape of the object you are looking at. These include:

Rectangle = Length x Height

Triangle = 1/2(Base x Height)

Cube = 6 x Base x Height

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