# Surface Area

## How To Find The Surface Area Of Shapes (3D Shapes)?

**What is the definition of Surface Area?** In geometry, a three dimensional shape is a shape with length, width and height. The Surface
Area of a three dimensional shape, is the total area of the outside
of the shape. The unit of measurement for surface area of a shape may
be any of the following: square inches, square feet, square centimeters, square
meters, square kilometers, plus many other measurements.

Let us look at a real example. Look at figure 1. It is a picture of a perfect cube
i.e all sides are equal in length, width, and height. Now we want to determine the
surface area of the cube. A cube is a three dimensional shape. It much easier to
determine the surface area if we change it into a two dimensional shape. See figure
2; If you have a box, you can "flatten" it into a two dimensional shape like figure
2. Now look at the formula below and match it up to the figure. For example purposes,
assume all sides are 10 centimeters in length. That means the surface area is 6 sides
× length of one side × width of one side = 6 × Side × Side = 6 × 10 centimeters ×
10 centimeters = 6 × (10 centimeters)^{2} which equals 600 square centimeters. If you had an
actual box in the shape of figure 2 and each side was ten centimeters, you could
draw exactly six-hundred 1 centimeter-by-1 centimeter squares on the flattened box.

Still having a little difficulty understanding the calculation? Look at figure 2 again. Do you see it is made up of 6 attached squares? A top, bottom, and four sides.
We know the area of one square is one side times one side. That means the area of one square is one side squared: Side × Side = Side^{2}. Well a cube is made up of six squares.
So the total surface area is six times the area of one side: 6 × Side × Side = 6 × Side^{2} = 6S^{2}.

**Example: Surface Area Of A Cube**

Surface Area = (6 Sides To A Cube) × Area Of One Side (Side × Side) =

Surface Area = (6 Sides To A Cube) × Area Of One Side (S^{2}) =

Surface Area = (TOP + BOTTOM + SIDE + SIDE + SIDE + SIDE) × S^{2} =

Surface Area = (6) × S^{2} =

Surface Area = 6 × S^{2}

Figure 1 - Image Of A Three Dimensional Cube

Figure 2 - Image Of A Three Dimensional Cube "Flattened" Into Two Dimensions To See Surface Area