# Proportional Change of Dimensions

## Proportional Change of Dimensions

Say you’re dealing with a rectangle that has an area of 216 centimeters squared. Now, say you have to find the area of a new rectangle. If both of the dimensions of this older rectangle are multiplied by a factor of one third.

You probably know that you find the area of a rectangle by multiplying length times width. If both of these dimensions are multiplied by 1/3, and to find the area of this new triangle, you’re going to multiply 1/3 times length, times 1/3 times width.

Overall, that’s going to equal 1/9 times length, times width. Because 1/3 times 1/3 equals 1/9, you’re finding 1/9 of the shape you had overall. Say for example, before that you had a length of 9 and a width of 9, you were dealing with a square, and you get an area of 81.

Then you multiply both of the dimensions by 1/3. Then you get 3 here, and 3 here, 3 times 3 equals 9. 9 is 1/9 of 81. You can see how much smaller the numbers are going to be. The way we’re going to calculate this right here, the new area equals 1/9 of the original area.

We don’t have to try to figure out what the exact length and width of the original rectangle was, because you may be thinking, “Okay, I have to find 1/3 the length and 1/3 the width but I don’t know what the length and are.”

From this information you can’t determine exactly what the length and the width were. That doesn’t matter, because we just know that you need to find 1/9 of the overall area. All we need to do here is plug in that original area and multiply it by 1/9, and we will be all set.

1/9 times 216 centimeters squared. 1/9 times 216 is 24 centimeters squared. That’s the area of the new rectangle. That’s how you use- or that’s how you find the proportional change of dimensions.