This figure has a length of 10 cm, a width of 7 cm, and a height of 3 cm. what is its surface area? (if you can fully explain this to me so i can do it on my own)

  242 cm^2

Step-by-step explanation:

The pattern for the box shows tabs of unspecified width. (They have dotted line edges and trapezoidal shape.) We presume you're to ignore those. The remaining 6 rectangles each have dimensions that are 2 of the 3 given dimensions. There are 2 rectangles with each pair of dimensions. The total surface area is the sum of the areas of the 6 rectangles.

We suppose that the width (left to right) of the bottom rectangle is 10 cm, and that its height (up/down) is 3 cm. Thus its area will be ...

  A = bh = (10 cm)(3 cm) = 30 cm^2

The rectangle immediately above that has another rectangle hanging off to its left. You can compute their areas separately or together. The width of the two of them together (left-right) will be 3 cm + 10 cm = 13 cm. The height of that pair of rectangles will be the 7 cm dimension. So, the area of that rectangle pair is ...

  A = bh = (13 cm)(7 cm) = 91 cm^2

The three rectangles in the top half of the pattern are identical in size to those we just figured, so the total surface area is ...

  total area = 2·(30 cm^2 + 91 cm^2) = 242 cm^2

We have successfully computed the value of the formula ...

  prism surface area = 2(LW +H(L+W))

for L=10 cm, W=3 cm, H=7 cm. The naming of the dimensions of a prism (length, width, and height) is completely immaterial. The formula gives the same result regardless of how you assign the numbers to the letters (as long as the same letter always stands for the same number).

Comment on area formula for a rectangle

I have written the formula for the area of a rectangle as ...

  A = bh

In this formula, b stands for the base dimension and h stands for the height dimension. Essentially, we're multiplying length times width. What you call base, height, length, width, depth, or whatever, may depend on the orientation of the rectangle in space and the relative sizes of the dimensions. Regardless of what you call the dimensions, the area is the product of the measure of one side of the rectangle and the measure of an adjacent side, which will be orthogonal to the first one.

step-by-step explanation:

in which figure is point g a centroid