# of rhombuses to fit in a rectangle

[Peter_Hls]

Hi there!
I deem myself so dumb, because I can't solve this:

How many rhombuses/isosceles triangles(side/base=4,height=3) can fit in a rectangle(40 x 30 [cm] ) when they must be separated by a space wide 0.5 cm?

I calculated 88 pieces, but according to book, it should be approx. 64 rhombuses.

I don't get what kind of shape I have to divide by whole area of the rectangle

I have tried adding 0.5 of length to the side length side + 0.5 = 4.5 and also extending the height by 0.5 making it 3.5.... which results in 76 pieces... really have no idea how to do it

 

[Dr.Peterson]

Please explain what you mean by

rhombuses/isosceles triangles(side/base=4,height=3)

Are they rhombuses, or are they triangles?

Are the sides and bases all supposed to be 4? Then the height for a triangle will be 2\sqrt{3} = 3.46, not 3; the rhombus will work, but it's an unusual shape.

Then, is the width of the space between measured in one of the ways you show, or as one would usually take it, perpendicular to the sides? The problem has to tell you that!

Please show the exact wording of the original problem and tell us where it came from. What topic is being taught, at what level?

 

[Peter_Hls]

this problem with rhombuses is from test A and there is also similar problem in test B, but with isosceles triangle(base is 4, height 3)

Well, the book is written in Slovak language, I apologize for not being taught proper English, but I will try to shed some light onto this stuff.
This problem is for 6th graders at primary school, chapter being about area and perimeter of parallelogram.

Every rhombus has a side length of 4 cm and height is 3. Every iso. (Separate problem, but similar) triangle has base equal to 4 and height is 3.

well, the original wording is in Slovak, but... this my literal translation:
How many cakes(rhombuses) fit onto a baking pan(that's the rectangle) , if the width of the space between them(cakes) must be 0.5 cm? Just this 0.5 width of space is mentioned.

 

[Peter_Hls]

maybe like in the picture, but the result from book is 64 cakes.... meaning that the 1200 / x = 64 cakes... the area of the shape x is 18.75

 

[Dr.Peterson]

I'll try to look at this later, but it can be helpful to show us an image of the problem, even though it is not in English, so we can see any subtleties in how it is presented (and try to translate it ourselves). Especially, I'd like to see if there is a picture in the problem, which there should be.

 

[Peter_Hls]

Unfortunately, this (and some other I have) book has a few pictures. Alright, you don't have to waste time on this problem, It is not properly stated in the book maybe.
The photos I send you all just for curiosity

 

[Dr.Peterson]

Thanks. Google translates the actual problem pretty much the way you say:

Mom bakes small pastries, which she places on a rectangular plate with dimensions of 40 cm and 30 cm. The pastry has the shape of diamonds with a side 4 cm long and a height of 3 cm. How many cakes will fit on a baking sheet if there is to be a gap of 0.5 cm between them?​

At this level, I would expect that approximate, technically invalid methods might be used (and the "approximate" answer supports that). Properly, you can't use area at all, but need to determine the actual arrangement of pieces to see how they fit in the dimensions; this requires at least a couple applications of the Pythagorean theorem and geometrical thinking that seems beyond that level.

We can include just one of the 0.5 cm spaces next to each cake (say, the spaces below and to the right), since neighbors would share those spaces:


If we change the problem to be about rectangular cakes (with the same area), we can think of each cake as 3.5 by 4.5 cm. (For a careful calculation I would make some additional adjustments even in this simplified case.) This gives an approximate area of 3.5*4.5 = 15.75 per cake, and purely in terms of area, we could fit 1200/15.75 = 76.

But if we try to actually fit them into rows, as you need in real life, we could fit about 40/4.5 = 8 per row, and 30/3.5 = 8 rows (rounding down!), for a total of 64. So maybe that's what they expect: enough awareness of real issues to fit squares accurately, but not enough to be really careful with rhombuses. (Fitting in the other direction, however, I get 66: 30/4.5 = 6, 40/3.5 = 11.)

The actual shape means they might not actually fit; here is an approximate suggestion of what it looks like fitting 64:



So if I worked this out carefully, I might well find it can fit only 56.

Of course, I really dislike problems that ask for an estimate without being clear what parts you need to be careful about.

 

[Peter_Hls]

Wow, thank you very much!
In what program have you done the,, canvas with shapes"?

 

[Dr.Peterson]

Wow, thank you very much!
In what program have you done the,, canvas with shapes"?

I just used Microsoft Word, because I do a lot of drawing there, though it is far from ideal for this purpose!