Is there a formula to work out the height and width of a rectangle (in 2:3 ratio) from a given area?

2 x 3 = 6 … but how do i get 2 & 3 from 6?

It's been a long time since i did any algebra.

I think you need 2 knowns in a three value equation . . ?

Since the ratio is known, then length = A/2 and width = A/3. Or am I missing something?

So:

a*(2/3)*a = b

where a is the length of the longest side of the rectangle and b is the area.

This then becomes:

(2/3)*a^2 = b

where ^2 means squared.

Rearrange this to get "a" by itself:

a = sqrt((3/2)*b)

where sqrt means square root.

This gives you your long side based on the area, and by your original definition, the short side is just 2/3 of this.

If the ratio is always 2:3 surely then to work out the length of the sides, divide the area by 6 to give you a number 'a'.

One side will be 2 x the square root of 'a' the other will be 3 x the square root of 'a'.

In your example above a = 1 so square root of a also = 1, but it should work for any size rectangle that is of the ratio 2:3…I think…seems logical to me.

e.g. area = 24, then 24/6 = 4, so one side is 2 x 2 = 4 and the other is 3 x 2 = 6.

a * b = C the Area
b = 3/2 * a

Thus, a * 3/2 *a = C
and a*a = 2/3*C

Thus a = SQRT(2/3*C)

Now I feel silly. GJP seems to explain it best!

GJP and I are saying the same thing except he has "a" as the short side and I have "a" as the the long side.

I think we are both correct (but I would say that).

So…

Width = SQRT (2/3 x Area)

Then Width x 3/2 = length

Cheers.

LabWormy – Member
GJP and I are saying the same thing except he has "a" as the short side and I have "a" as the the long side.

I think we are both correct (but I would say that).

We are both correct. It is quite easy to conceptualize this by simply drawing the rectangle and adding a line in the appropriate place to create a square. Clearly then the length of the short side is equal to the SQRT of two thirds of the area. I am thinking "within the box"

LabWormy, however to conceptualize his/her approach you need to extend the rectangle to make a square and then clearly the length of the long side is equal to the SQRT of one and a half times of the area. LabWormy is thinking "out of the box" 😆

I use the word "clearly" to mean it should be obvious to someone of average intelligence! 😆

See you don't really need any algebra – easiest explained with pictures.

Well I would not dare to argue with one who is "too intelligent for mensa".

My previous post was just to highlight that our solutions are algebraically equivalent so that the OP had faith in our inputs.

Quoting Harry Hills TV Burp
Who is right?
GJP or LabWormy
Only one way to find out………. FIGHT!

Soops – as LabWormy stated both his/hers and my algebraic solution are correct – they will both give the same answer.

I can see it is a little confusing as one solution is taking the SQRT of 2/3 of something and another the SQRT of 3/2 of something.

My previous post attempted to explain the difference in the approaches by graphical illustration. If the rectangle has sides of length a and b respectively, where b is greater than a.

Then my approach is equivalent to shrinking the rectangle to a square with sides of length a. This would require a reduction in area by two thirds (2/3) in the OPs example. Whereas LabWormy approach is equivalent to extending the rectangle to create a square with sides of length b. This would require an increase in area of one and a half (3/2).

As LabWormy stated several posts ago. I solved for the shorter side a; whereas he/she solved for the longer side b.

You say "tomato I say tomato"

😆

He/She is he at the moment and hopes to stay that way!

And apologies for any assumption on the gender of GJP.