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CAHSEE Math Exam: Help and Review21 chapters | 242 lessons

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Lesson Transcript

Instructor:
*Joseph Vigil*

In this lesson, we'll look at what makes cubes unique among three-dimensional shapes, and we'll put that uniqueness to use in finding an easy way to find the surface area of cubes.

Before we determine the surface area of a cube, let's determine exactly what surface area is.

For a two-dimensional shape, the **area** is the space that it covers. Think about calculating the number of tiles needed for a floor or the amount of space inside a rectangular fence. We have formulas to calculate the areas of different shapes, like squares, triangles, and circles.

But what about three-dimensional shapes? Since they're three-dimensional, they have an outside, or surface. **Surface area** is just what it sounds like - the area of the shape's surface, which is its entire outside. Think about determining the amount of paint needed to cover a house or the total surface space of a box. Just like there are different ways to find the areas of different two-dimensional shapes, there are also ways to find the surface areas of different three-dimensional shapes like cubes, pyramids, and spheres.

Let's look at how to find the surface area of a cube.

**Cubes** are three-dimensional shapes that have the same dimensions all over. The length, width, and height are identical, and every edge meets every other edge at the same angle. They are highly regular!

Finding the surface area of a cube is nice and convenient because by definition its surface consists of **congruent**, or equal-sized, squares. So, once we find the area of one of the squares, we know the area of all of the squares. But how many squares are there?

From this illustration of a cube, we can see that it has a front and a back, two sides, a top, and bottom. So the cube's surface consists of **six congruent squares**. In fact, if we unfold the cube, we can clearly see the six squares that make up its surface.

All we really need is the length of a side of one of these squares, which would also be one edge of the cube. Let's say that the cube's height is two inches. Because the cube's surface is made of squares, all of its edges will be the same length. So if the cube's height is two inches, its length and width will also be two inches.

To find the area of one of the squares, we go back to basic geometry and recall that Area (A) = s^2, where s is the length of one of the sides. In this case, *s* = 2 inches, so A = 2 in^2 = 2 in * 2 in = 4 in^2.

The area of one of the surface squares is four square inches.

Since the entire surface consists of six congruent squares, we just have to multiply the area of that single square by six to find the total surface area. So:

4 in^2 * 6 = 24 in^2

The cube's surface area is 24 square inches.

Now that we've determined that the surface area of a cube is six times the area of one of its surface squares, we can say that the equation for a cube's surface area is:

A = 6 * *s* * *s* or 6*s*^2

Let's apply this to a bigger cube that's ten feet wide.

As we've discovered, if the cube is ten feet wide, it's also ten feet high, so we can find the area of one of these squares:

A = 10 ft * 10 ft = 100 ft^2

The area of one square is 100 square feet.

To find the entire surface area, we need to take into account all six squares that make up the cube's surface:

100 ft^2 * 6 = 600 ft^2

This cube's surface area is 600 square feet. The method works the same way no matter how small or large the cube is.

A cube's surface is made up of **six congruent squares**, so we can follow a simple formula to find any cube's surface area. Since the area of one square with length *s* is *s* times *s*, to find the entire surface area of any cube, we can just multiply that by 6. Our formula for the surface area of a cube, then, is:

Surface Area = 6 * *s* * *s* = 6*s*^2

The only number we need to know to find a cube's surface area is the length of one of its edges! Cubes are fairly simple after all.

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13 in chapter 18 of the course:

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CAHSEE Math Exam: Help and Review21 chapters | 242 lessons

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