Find the Perimeter of the Square that has 7 identical rectangles | 112 perimeter for each rectangle

Learn how to calculate the Perimeter of the square which consists of seven identical rectangles. Each rectangle has a perimeter of 112. Simple and step-by-step explanation!

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Find the Perimeter of the Square that has 7 identical rectangles | 112 perimeter for each rectangle

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Olympiad Mathematics
pre math
Po Shen Loh

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[Waldlaeufer]

This is a nice problem. I solved it in my head last night while lying in my hammock:

Since there are seven rectangles and all seven together form a square, the aspect ratio of each rectangle must be 1:7.

Each rectangle has two long sides and two short sides: two long sides, each seven times as long as the two short sides: 2a + 14a = 112 => 16a = 112 => a = 7

The whole square therefore has four sides of length 49 (<= 7 * 7), so it must have a perimeter of 196.

You can also say that a single rectangle has a perimeter of 112, so it is missing 2 * 6 * 7 units compared to the square. So 84 units are added to the 112 to complete the perimeter to a square, which also ends up with 196.

[philipkudrna]

The longer side of each “strip” is seven times the shorter side. Thus the perimeter of the whole strip is 16 times the shorter side (let’s call it x), 16x=112 or x=7. The perimeter of the square is 4*7*x or 4*7*7=196 units.

[procash]

Simple but simply superb. Good little problem to keep the mind active

Thanks PreMath Guruji

[math]

love the question, you explained the solution very well, thanks for sharing

[Aditya_Senpai]

Such a amazing solution. This was a easy one!

[davidfromstow]

At last - one I could solve - many thanks to you, as always!

[Copernicusfreud]

Solved it. I had to use a pen and paper for the 112 / 16, but other than that I could do it in my head. For the perimeter of the square, I used (50 - 1) x 4, which is 200 - 4 = 196. That was easier for me than 49 x 4, which is also 196.

[anatoliy]

Thank you for a nice striped task, sir! God bless America and you as well<))

[AnonimityAssured]

This problem lends itself to a bit of generalization by examination:
112∙4 = 224.
1 rectangle = 224∙1/2 = 112;
2 rectangles = 224∙2/3 = 149 + 1/3;
3 rectangles = 224∙3/4 = 168;
4 rectangles = 224∙4/5 = 179 + 1/5;
5 rectangles = 224∙5/6 = 186 + 2/3;
6 rectangles = 224∙6/7 = 192;
7 rectangles = 224∙7/8 = 196 (our answer);
8 rectangles = 224∙8/9 = 199 + 1/9;
9 rectangles = 224∙9/210 = 201 + 3/5.
So, if Q represents the perimeter of each rectangle, and N represents the number of rectangles:
P = 2Q∙N/(N + 1).
The first eleven whole-number answers are 112, 168, 192, 196, 208, 210, 216, 217, 220, 222 and 223 corresponding, respectively, to 1, 3, 6, 7, 13, 15, 27, 31, 55, 111 and 223 rectangles. In fact, these are the only whole-number answers for rectangles with perimeter 112, as 224/4 = 56, which is already half of 112, so the rectangles would have no width.

[alster]

Got it an few secs.

Alternate way

Take the perimeter of square first.

Since one small side of rectangle is x and there are 7 identical rectangles, the side length of the square is 7x

AB = 7x

Since the square has all 4 sides equal

P(square) =4(7x)

P(square)= 28x (eq 1)

Focusing on the small rectangle

112= 2(L+W)

L= x (small side of rectangle)
W = 7x (obtained from the large square's side length)

2(7x+x)=112 (eq 2)

Solving further...

8x=56
x= 7 units

Subtituting this to the square perimeter (eq 1)


P(square) = 28(7)
P(square) =196 units

[DjappoMKS]

It's easy if u know the basic rules of Square. Solved in 3 minutes. But my kids are struggling 😅

Best regards from Indonesia

[mustafizrahman]

The answer will 196. Good question it was but solved it in less than 4 minutes.

[Ivlook]

Total of all rectangles’ perimeters:
112 * 7 = 784

Determine how many sides were part of that calculation:
• 4 exterior sides
• 6 interior sides which must each be counted twice to account for inclusion in two rectangles’ perimeters: 6 x 2 = 12
• Total = 16

784 / 16 = 49
49 * 4 = 196

[bentels]

Before watching:

Let the height of each rectangle be y and the width be x. Now

2y+2x=112

The full figure is a square consisting of 7 rectangles. So y=7x. And the perimeter is 4y=4*7x=28x.

2y+2×=112
2*7x+2x=112
16x=112
x=7

The perimeter is 28x=28*7=196

After watching: 🙂

[devondevon]

Answer 196
Since perimeter of each rectangle = 112
then L + W=56
Since the total width of the 7 rectangles= 1 side of the square, then the length of each rectangle= 1 side
of the square hence, hence the length of each rectangle is 7 times its width
so since L+W= 56 then 7W+W= 56
8W=56
W= 7 the width of each rectangle
L=49 the length of each rectangle
hence the square side =49 hence it perimeter = 49 x 4 =196 Answer

[joelleking]

I have use another way soI found an abswer différent of yours sir but nearly thé same thing

[sandanadurair]

Alas! I missed to note ABCD is a Square and struggled. Very sad.

[Teamstudy]

Ans :Perimeter = 49 x 4 =196 unit. (*Super duper bumper easiest question ever)