This triangle has an area of 10.
The area of a triangle is ½ times base times height. So, this triangle's area is ½ x 5 x 4, which is 10. Let's make a new triangle using a scale factor of 3. This new triangle has a base of 15 and a height of 12. Its area is ½ x 15 x 12, or 90.
This triangle has an area of 90.
Okay, remember that our scale factor was 3. And, 3^2? That's 9. So, the change in area should be 9 times the original. Does 10 x 9 = 90? Yes! So, it works..
When you're working with scale factors, square the scale factor to determine the area of the new figure. If you think about it, it makes sense why area would be the scale factor squared. Area involves two dimensions multiplied together. With scale factor, all you're really doing is multiplying the scale factor times itself.
Volume Examples
Next, there's volume. Like going from perimeter to area, going from area to volume means adding a layer. In this case, it's a third dimension. Instead of squaring the scale factor, guess what? We're going to cube it! So, the change in volume is equal to the scale factor cubed. Cubing a number is raising it to the third power. So, if you remember that volume involves three dimensions, you can remember to cube the scale factor. Let's try this out.
Here's a rectangular prism:
The volume of this rectangular prism equals 24 cubic inches.
Let's make this interesting. Let's say it's a box of cookies. It's 4 inches by 2 inches by 3 inches. The volume of a rectangular prism will be length times width times height. So, its volume is 4 x 2 x 3, or 24 cubic inches. That's not going to hold a lot of cookies, even if they're small. So, let's scale it! Let's use a scale factor of 3.
Here's our new box:
This is the box of cookies scaled by three.
It's 12 by 6 by 9. The volume of this box will then be 12 x 6 x 9, which is 648 cubic inches. Now that box will hold a lot of cookies. We just need some milk.
Oh, but what about the change in volume? I said it's the scale factor cubed. But, what's 3 cubed? It's 27. And, what happens if we multiply the original volume, 24, times 27? Yep. It's 648.
But, what about that milk? Let's do one more volume example. Here's a kid's size glass of milk:
The volume of this glass of milk is 12.6 cubic inches.
The volume of a cylinder is pi times r^2 times h, where r is the radius of the circle on the top and h is the height. This glass has a radius of 1 inch and a height of 4 inches. So, its volume is pi x 1^2 x 4, or about 12.6 cubic inches.
We have a huge box of cookies, so we need a bigger glass of milk. What scale factor should we use? Since scale factors are cubed with volume, remember that even a small change will have significant ramifications.
Let's try a scale factor of 2. That will make our radius 2 inches and our height 8 inches. That doesn't seem unreasonable, right? Now, we don't need to do the volume formula. We can just cube the scale factor. 2^3 = 8. If our original volume was 12.6, then our new volume is 12.6 x 8, or 101 cubic inches. For those of you who don't think of your milk in cubic inches, that's about 1.75 quarts of milk. Holy cow!
Lesson Summary
In summary, a scale factor is simply a number that multiplies the dimensions of a shape. It's a way of describing the relationship between similar shapes, whether it's a house and a dollhouse-size replica or a pair of triangles.
The perimeter of a scaled object will be equal to the scale factor. If the scale factor is three, then the perimeter of the new object will be three times the original perimeter. The area of a scaled object will be equal to the scale factor squared. Again, if the scale factor is three, the area of the new object will be nine times, or three squared, the area of the original object. Finally, the volume of a scaled object will be equal to the scale factor cubed. So, if the scale factor is three, the volume of the new object will be three cubed, or 27 times, the volume of the original object.