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| Math Central | Quandaries & Queries |
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Question from Riley, a student: I have a math problem, and a big test coming up on Friday or Thursday (my teacher hasn't decided yet.) I need help understanding how I would do this problem. Here it is: they wanted me to find the perimeter and area of a rectangle when they only give me the scale factor. I've dealt with scale factors before, but this is quite confusing. I have done the scale factor way by having a rectangle that maybe has a width of 10 and a height of 20, and I needed to find what the scale factor would be for a rectangle with the width of 5 and a height of 10. I know that they scale factor would be 2, right? But this just gives me the scale factor, and I have to FIND the perimeter and area. Hopefully you can get back to me tonight. Thanks in advance. :-) |
Hi Riley,
You are correct that in your example the scale factor is 2. The scale factor refers to length (a one dimensional concept) and in your example the 5 by 10 rectangle has a perimeter of 5 + 10 + 5 + 10 = 30 units which is a length measurement. You didn't give units but I am going to assume the units are centimeters so the perimeter is 30 centimeters. Thus the larger rectangle has a perimeter of (the scale factor) × 30 = 2 × 30 = 60 centimeters. You can check this since the larger rectangle has perimeter 10 + 20 + 10 + 20 = 60 centimeters.
Area is a two dimensional concept and the units are square centimeters. The area of the small rectangle is 5 × 10 = 50 cm2 so the area of the large rectangle is
[(scale factor) × 5 cm] × [(scale factor) × 10 cm]
=(scale factor)2 × 5 × 10 × cm2
= 22 × 50 cm2
= 200 cm2
Again in your example you can check this since the large rectangle is 10 cm by 20 cm so its area is 10 × 20 = 200 cm2.
The result is that for length (one dimensional) measurements you multiply by the scale factor and for area (two dimensional) measurements you multiply by the square of the scale factor.
Good luck on the test,
Harley
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