Ashley C. answered • 04/18/20
Experienced and Enthusiastic Academic and Test Prep Tutor
Hi Eva,
Surface area is just what it sounds like: the area of all of the surfaces of an object. For the first part of your question, you didn't give what that last question was, so I'll use a hypothetical box with side lengths x, y, and z. A box has 3 sets of identical rectangular sides, or faces. Two faces would have the dimensions x * y, two faces would have the dimensions y * z, and two faces would have the dimensions of x * z. The area of each of these rectangles is the side lengths of each multiplied by one another, or length times width.
The surface area of that original box would be 2(x * y) + 2(y *z) + 2(x * z), or 2xy + 2yz + 2xz.
Now if I double each dimension, I use the same equation, but my dimensions are now 2x * 2y, 2y * 2z, and 2x *2z.
This surface area would be 2(2x*2y) + 2(2y*2z) + 2(2x*2z). If I multiply all those 2's, this can be simplified as
8xy + 8yz +8xz. This surface area is 4 times our original surface area.
One thing to note is that in geometry, when you have similar figures (same shape but different sizes), the ratio of the squares of the side lengths is equal to the squares of the areas. So if I have one side that's x and the corresponding side on a similar shape is 2x, they are in a ratio of 1:2. The ratio of the squares would be 1^2:2^2, or 1:4. That's another way to show that the area of the figure with doubled sides would have 4 times the surface area.
The second part of your question deals with the same question, except here we have concrete numbers to work with. If we have side lengths of 3, 4, and 5 units, we have 2 faces with the dimensions 3x4, 2 faces with the dimensions 4x5, and 2 faces with the dimensions 3x5.
Our surface area = 2(3*4) + 2(4*5) +2(3*5)
=2(12) + 2(20) + 2(15)
=24 + 40 + 30
= 94 square units
I hope that helps!
