Not a homework question: Doesn't the (sqrt of A) x 4 equal the perimeter regardless of length and width?

Question

If l is 3w, and A is 75, find P. Unless one is asked to find l and w, doesn't P = (sqrt A) x 4?

Funny, I just figured if P is 2l+2w for a square or a rectangle, and A is l times w, then the sqrt of A would equal a non-specified side and ..., Thanks for the clarification!

I tried examples while you were answering my question. Great illustrations everyone!

Answer 1

Not at all.

Think about a square with sides of length 4. Then the area would be 16 and the perimeter would match your formula, 4 x 4 = 16.

Now instead think about a rectangle with width 2 and length 8. The area would still be 16. Your formula would say that the perimeter was still 16, but in fact, it would be 2(2) + 2(8) = 22.

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For this specific question: if l = 3w then:

l*w = 75
(3w)(w) = 75
3w² = 75
w² = 25
w = 5

So l = 3(5) or 15.

The perimeter is then 2(5) + 2(15) = 40.

Your formula would say it was 4√75 = 12√5

Answer 2

No. For example, take a rectangle with width=1 and length=25. Then A = 1*25 = 25, so by your measure, P = sqrt(25)*4 = 5*4 = 20. However, the actual perimeter is 1+1+25+25 = 52.

In general for rectangles, if you keep the area constant, the more square you make it, the lower the perimeter will be. The more stretched you make it, the higher the perimeter will be.

Answer 3

Here's a related example that might be helpful:

Consider a 4-by-4 square and a 2-by-8 rectangle.
Both have area 16. Do they both have perimeter (sqrt 16) x 4?

In general, a good way to test a universal statement like this one is to try to think of a counterexample.

Answer 4

For a square, P = (sqrt A)x4, or A = P²/4. For rectangles that are not squares, A < P²/4, or P > (sqrt A)x4. E.g a rectangle 1x100 will have area 100 and perimeter 202. A square 10x10 will also have area 100, but perimeter 40. If you have multiple rectangles with the same area, the one closest to a square will have the smallest perimeter. Put reversely, if you have multiple rectangles with the same perimeter, the one closest to a square will have the largest area.

Answer 5

L = 3w
A = L*w = 3w*w
Area is 75, so 3w*w = 75, or w^2 = 25
so w = 5, and L = 15.
P = 2*5 + 2*15 = 40

the square root of 75 is 5 roots of 3, and multiplying that by 4 is 20 roots of 3, so the answer to your second question is NO.

Answer 6

No. That only works if it's a square.

If L = 3w, then A = 3ww or 3wsquared (wouldn't let me do a superscript here.) Therefore, 3ww = 75. Divide both sides by 3, then ww (or w squared) = 25. Take the square root of both sides and w = 3 so L = 9. You can do the rest.

If P= (sqrtA) x 4 then your P would = 34.64... because sqrt 75 = 8.66....