I need to solve this geometry problem. There are six circles, all the same size, touching each other and the sides of a large rectangle. Each corner of a small rectangle is the center of one of these circles. The small rectangle's perimeter is 60 cm, and I need to find the perimeter of the large rectangle.

First, since the circles are identical and touch each other and the rectangle sides, the distance between centers of adjacent circles should be equal to the diameter of one circle. Let the radius of each circle be $r$. Then the diameter is $2r$.

Now, the small rectangle has its corners at the centers of four of these circles. There are six circles: probably two rows of three, arranged in a grid. Since it's a rectangle, and circles touch adjacent ones and the sides, it must be a 2x3 grid or 3x2. But the small rectangle's corners are centers, so likely it's a rectangle formed by four circles, but there are six, so maybe the other two are in the middle.

Looking at the diagram (though I can't see it, I need to visualize). The small rectangle has corners at circle centers, so if I have a grid, say two rows and three columns of circles, then the corners of the small rectangle might be the centers of the four corner circles of this 2x3 grid. But there are two more circles, probably in the center of the rows or something.

Assume the large rectangle has circles arranged in 2 rows and 3 columns. So, centers at positions: let's say row 1: center A, B, C; row 2: center D, E, F.

The small rectangle has corners at centers of circles. Probably, it's the rectangle formed by A, B, D, E or something. But that might not be a rectangle. Perhaps it's A, C, D, F or something.

Since it's a rectangle, and corners are circle centers, likely the small rectangle is aligned with the large one, so its sides are parallel.

Suppose the large rectangle has width W and height H.

With two rows of circles, the height of the large rectangle should be equal to the diameter times the number of rows, but since circles touch the top and bottom, and there are two rows, but they might be offset or something.

In a standard packing, for circles in a rectangle, if they are in a grid, and touch sides, then for m rows and n columns, the width of the rectangle is n times diameter, and height is m times diameter, but only if the circles are arranged in a square grid where rows and columns are aligned.

But in that case, for circles to touch adjacent circles, if it's a square grid, then yes, distance between centers is 2r, so if n columns, width is (n) * 2r, but since they touch the sides, the distance from first center to left side is r, and last to right side is r, so total width is (n-1)*2r + 2r? No.

Let's think: for n circles in a row, touching each other and left/right sides. The distance from left side to first center is r, then centers are spaced 2r apart, so from first to last center is (n-1)*2r, then from last center to right side is r, so total width W = r + (n-1)*2r + r = 2r + (n-1)*2r = 2r * n.

Similarly for height with m rows.

In this case, with six circles, if arranged in 2 rows and 3 columns, then W = 3 * 2r = 6r, H = 2 * 2r = 4r.

But now, the small rectangle has corners at circle centers. Which four centers? Probably not all corners, since there are six circles.

The problem says "each of the corners of the small rectangle is the centre of one of the circles." And there are six circles, so likely four corners are used, and two are not, but probably the small rectangle is formed by the centers of the four corner circles of the grid.

In a 2x3 grid, there are four corner circles: top-left, top-right, bottom-left, bottom-right. But that's only four, but there are six circles, so the middle two are not corners. But the small rectangle would be formed by these four corners.

But in a 2x3 grid, the four corners form a rectangle themselves. Let's see: centers at (0,0), (2r,0) for first row? Set coordinates.

Set the bottom-left corner of large rectangle at (0,0). Then, with 3 columns and 2 rows of circles.

Centers: for row 1 (bottom row): x= r, r+2r, r+4r? Distance between centers is 2r.

So, center A: (r, r) // bottom-left

Center B: (r + 2r, r) = (3r, r) // bottom-middle

Center C: (r + 4r, r) = (5r, r) // bottom-right? 3 columns, so x-coordinates: first center at x=r, second at x=r+2r=3r, third at x=3r+2r=5r. Yes.

Similarly, row 2 (top row): since height, distance from bottom to first row center is r, and between rows: if rows are aligned, distance in y is 2r, so center D: (r, r + 2r) = (r, 3r) // top-left

Center E: (3r, 3r) // top-middle

Center F: (5r, 3r) // top-right

Now, the four corner circles are A(r,r), C(5r,r), D(r,3r), F(5r,3r).

The small rectangle has corners at these centers? But these four points: A(r,r), C(5r,r), D(r,3r), F(5r,3r). Now, these form a rectangle: width 5r - r = 4r, height 3r - r = 2r. So it is a rectangle with width 4r, height 2r.

But the problem mentions a small rectangle, and also there are two more circles, B and E, at (3r,r) and (3r,3r), which are not corners.

The small rectangle is defined by these four points, but in the diagram, there is a small rectangle drawn, and its corners are these centers.

The perimeter of this small rectangle is given as 60 cm.

The small rectangle has sides: two widths of 4r, two heights of 2r? From A to C is from (r,r) to (5r,r), so distance is 4r, which is horizontal.

From A to D: (r,r) to (r,3r), distance 2r, vertical.

So yes, the small rectangle has width 4r, height 2r.

Perimeter is 2*(width + height) = 2*(4r + 2r) = 2*6r = 12r.

Given as 60 cm, so 12r = 60, thus r = 5 cm.

Now, the large rectangle: width is distance from left to right. Left side at x=0, right side at x= position of F, which is 5r, but since center is at 5r, and radius r, so right side is at 5r + r = 6r.

Similarly, bottom at y=0, top at y= position of D or F, y=3r, but since radius, top side is at y=3r + r = 4r.

Earlier I said H=4r, W=6r.

From calculation: bottom-left corner (0,0), top-right corner: since F is at (5r,3r), and it touches the top and right sides, so right side is at x=5r + r = 6r, top side is at y=3r + r = 4r.

So width W = 6r, height H = 4r.

Perimeter of large rectangle is 2*(W + H) = 2*(6r + 4r) = 2*10r = 20r.

With r=5, 20*5=100 cm.

But is this correct? The problem says "the small rectangle is the centre of one of the circles", and in this case, the small rectangle's corners are the centers of A,C,D,F, which are four circles, and there are two more, B and E.

But in the diagram, it might be different. Also, the small rectangle is probably inscribed or something, but in this case, it seems fine.

But let me check the perimeter: small rectangle perimeter 12r=60, r=5, large perimeter 20r=100.

But is the arrangement correct? The circles touch adjacent circles: for example, A and B: distance between (r,r) and (3r,r) is 2r, which is diameter, so they touch. Similarly, A and D: (r,r) and (r,3r), distance 2r, so they touch. But in a grid, if it's a square grid, adjacent circles include diagonal, but distance to diagonal is sqrt((2r)^2 + (2r)^2)=sqrt(8r^2)=2√2 r ≈ 2.828r > 2r, so they don't touch diagonally, only horizontally and vertically. But the problem says "touch adjacent circles", which probably means sharing an edge, so horizontal and vertical adjacency, not diagonal. So in 2x3 grid, each circle touches up to four others: left, right, up, down, but corner circles have fewer.

For example, A touches B (right) and D (up), and if there were left or down, but it's corner, so only two adjacent in this grid. But it also touches the sides, but "adjacent circles" probably means other circles.

The problem says "touch adjacent circles and the sides", so adjacent circles are other circles they touch.

In this grid, inner circles have more adjacent: B touches A, C, and D? B is at (3r,r), D at (r,3r)? Distance is sqrt((3r-r)^2 + (r-3r)^2)=sqrt((2r)^2 + (-2r)^2)=sqrt(8r^2)=2√2 r > 2r, so not touching. Similarly, E is at (3r,3r), distance to B is |3r-r| in y? B(3r,r), E(3r,3r), same x, dy=2r, so distance 2r, so yes, they touch if directly above.

In my setup, B(3r,r) and E(3r,3r), same x, dy=2r, so distance 2r, so they touch vertically. Similarly, A and D touch, etc.

But what about diagonally? A and E: (r,r) to (3r,3r), dx=2r, dy=2r, distance 2√2 r > 2r, not touching. Similarly, no diagonal touches.

So each circle touches the circles adjacent horizontally and vertically. Corner circles like A touch two circles (B and D), edge circles like B touch three (A, C, and E? B is bottom-middle, so left A, right C, and above E? But E is top-middle, same x, so yes. In 2x3 grid:

• A (corner): touches B (right) and D (up)

• B (edge): touches A (left), C (right), and E (up)? E is directly above? B(3r,r), E(3r,3r), yes, same x, so touches E.

But is there a circle below? No, so B touches three circles: A, C, E.

Similarly, D (top-left): touches A (down), E (right), and if there were left or up, no, so only two? D(r,3r), touches A below? A(r,r), same x, dy=2r, so yes. And to the right, E(3r,3r), dx=2r, dy=0? Same y, dx=2r, so touches E. And no up or left, so two adjacent circles? But in a 2x3, top-left should touch down and right.

But earlier I thought E is at (3r,3r), which is adjacent to D.

But what about the middle of top row? E(3r,3r) touches D (left), F (right), and B (down)? B(3r,r), same x, dy=2r, so yes, so E touches D, F, B. Three circles.

Similarly, all non-corner circles touch three others, corners touch two.

But the problem doesn't specify that all circles touch the same number, so it's fine.

Now, the small rectangle: corners at A(r,r), C(5r,r), D(r,3r), F(5r,3r). As I said, width 4r, height 2r, perimeter 2*(4r+2r)=12r=60, so r=5.

Large rectangle: from x=0 to x=6r (since F at 5r, plus r to right side), y=0 to y=4r (D at 3r, plus r to top). So width 6r, height 4r, perimeter 2*(6r+4r)=20r=100.

But is this the small rectangle? The problem says "the small rectangle", and in the diagram, it might be different. Also, it says "each of the corners of the small rectangle is the centre of one of the circles", which is satisfied, but there are two more circles, B and E, which are not corners of the small rectangle.

But perhaps the small rectangle is something else. Maybe it's a different rectangle.

Another possibility: perhaps the small rectangle is formed by the centers, but not the outer corners. For example, maybe it's a rectangle inside.

But the problem says "the small rectangle" and "each corner is a center", and with six circles, likely it's the bounding rectangle of four corner centers.

But in this case, the perimeter is 100, but let me see if it makes sense.

I recall that in some such problems, the arrangement might be different. For example, if the circles are arranged in a different pattern, but for a rectangle, 2x3 grid is common.

But in this 2x3 grid, the small rectangle has height 2r and width 4r, which is not a square, but that's fine.

Perimeter 12r=60, r=5, large perimeter 20r=100.

But is there a small rectangle that includes other centers? The problem says "the small rectangle", implying it's specific, and probably it's the one in the diagram, but since I don't have it, I need to infer.

Another thought: the small rectangle might be the one whose corners are the centers, but perhaps it's not the outer rectangle. For example, in the diagram, it might be that the small rectangle is inside, and its corners are centers of the circles, but not necessarily the corner circles.

But with six circles, if it's 2x3, the centers are at (r,r), (3r,r), (5r,r), (r,3r), (3r,3r), (5r,3r).

Now, any four that form a rectangle. For example, A(r,r), C(5r,r), F(5r,3r), D(r,3r) as I had.

Or perhaps A(r,r), B(3r,r), E(3r,3r), D(r,3r) — this would be a rectangle with width 2r, height 2r, a square.

Similarly, B(3r,r), C(5r,r), F(5r,3r), E(3r,3r) — same size.

But in this case, the small rectangle is a square of side 2r, perimeter 8r.

But the problem gives perimeter 60, so 8r=60, r=7.5.

Then large rectangle: width 6r, height 4r, perimeter 2(6r+4r)=20r=150.

But which one is it? The problem likely specifies which small rectangle, but since it's not specified, and the diagram shows one, but I need to see.

The problem says "the small rectangle", and in the context, probably it's the one that is mentioned, and likely it's the bounding rectangle of the whole set or something.

But in the first case, with the large rectangle, the small rectangle I defined has corners at the large rectangle's corners? No, in my coordinate, large rectangle corners at (0,0), (6r,0), (0,4r), (6r,4r). The small rectangle corners at (r,r), (5r,r), (r,3r), (5r,3r), which are not the same.

But the small rectangle is inside.

Now, the key point is that the small rectangle's corners are centers, and there are two more centers at (3r,r) and (3r,3r), which are the midpoints of the sides of the small rectangle? Let's see: the small rectangle has bottom side from (r,r) to (5r,r), so midpoint (3r,r), which is center B. Similarly, top side from (r,3r) to (5r,3r), midpoint (3r,3r), center E. And left side from (r,r) to (r,3r), but no center there, unless, but at x=r, y=r and y=3r, but at y=2r, but no circle center there. Similarly for right side.

But in the diagram, probably the small rectangle is this one, with B and E at the midpoints of the bottom and top sides.

The problem is to find the large rectangle's perimeter.

But in this case, with r=5, large perimeter 100, or if the small rectangle is the square one, then r=7.5, large perimeter 150.

Which one is it? I need to see which small rectangle is intended.

The problem says "a triangulated hollow hexagon" but that was in previous problems, this is a new problem. Looking back, the user changed the problem, but in the last message, it's about circles and rectangles.

In the current problem, it's about circles in a rectangle, and a small rectangle with corners at circle centers.

But to decide, I recall that in such arrangements, often the small rectangle is the one formed by the outer circles, but in this case, for 2x3, the outer circles' centers form a rectangle of size 4r by 2r, as I had.

But let me think about the "triangulated" part, but that was previous, this is different.

Another way: the large rectangle contains the circles, and the small rectangle is inside, and its corners are circle centers, but in the 2x3 grid, there are multiple possible rectangles.

But the problem likely implies a specific one from the diagram.

I could calculate the distance.

Another important point: the small rectangle's perimeter is 60, and I need large perimeter.

But in my first calculation, with small rectangle 4r by 2r, perimeter 12r=60, r=5, large 6r by 4r, perimeter 20r=100.

But is the height correct? In 2 rows, with circles touching top and bottom, the distance from bottom to top should be the diameter times number of rows, but since two rows, and they are separated by dy=2r, and each row is at height r and 3r, so from y=0 to y=4r, which is 4r, and since diameter is 2r, 2 * 2r = 4r, yes.

Similarly for width, 3 circles, 3 * 2r = 6r.

Now, the small rectangle from (r,r) to (5r,3r), so width 4r, height 2r, but 4r is the distance between the first and last center in x, which is (3-1)*2r = 4r for 3 columns, but from first to last center is (n-1)*2r for n circles, but here n=3, (3-1)*2r=4r, yes.

But the small rectangle includes the entire set? No, it's defined by the corners.

But in this rectangle, the points (3r,r) and (3r,3r) are on the bottom and top sides, but not corners.

Now, the problem is that the large rectangle is 6r by 4r, small is 4r by 2r, so ratio of perimeters is 20r / 12r = 5/3, so large perimeter 60 * 5/3 = 100, same as before.

But if the small rectangle is the square one, say from A(r,r), B(3r,r), E(3r,3r), D(r,3r), then it's a square of side 2r, perimeter 8r=60, so r=7.5.

Then large rectangle is 6r by 4r = 45 by 30, perimeter 2(45+30)=150.

Now, which one is it? I need to see which one makes sense with the diagram description.

The problem says "the small rectangle", and "each of the corners is the centre of one of the circles", and there are six circles, so in the first case, the small rectangle uses four corners, and the other two are on the sides.

In the second case, the small rectangle uses A,B,D,E, and the other two are C and F, which are also corners of the large rectangle? C(5r,r), F(5r,3r), which are the bottom-right and top-right corners.

But in this small rectangle, it is a square, and the large rectangle has these.

But the small rectangle is not the one formed by the outer circles; it's an inner rectangle.

The problem might intend this, but I think in most such problems, the small rectangle is the bounding box of the centers or something.

I recall that in the diagram, it might be shown, but since I can't see, I need to think differently.

Another way: the small rectangle's sides are parallel to the large rectangle, which is likely.

Now, the key is that the circles touch the sides, and the small rectangle has corners at circle centers.

Also, the small rectangle is "small" compared to large, but both are possible.

But let's see the answer; perhaps I can see from the calculation.

I am considering the distance between centers.

The small rectangle has diagonal or something, but not given.

Another thought: in the arrangement, the two middle circles are on the small rectangle's sides, as in my first setup, where B and E are on the bottom and top of the small rectangle.

In the second setup, if small rectangle is the left square, then B and E are corners, but C and F are not on it.

But the problem doesn't specify.

Perhaps the small rectangle is the one that is formed by the four circles that are not the middle ones, but same as first.

I think the first one is more standard.

But let me try to calculate the perimeter for large rectangle.

In 2x3 grid, large perimeter is 2(6r + 4r) = 20r.

Small perimeter depends on which rectangle.

If small is 4r by 2r, perimeter 12r.

If small is 2r by 2r (the square), perimeter 8r.

Given 60, so r=5 or r=7.5.

Large perimeter 100 or 150.

Now, which one? I need to see which small rectangle is intended.

The problem says "the small rectangle", and in the context, probably it's the one that is shown in the diagram, and likely it's the one that is not a square, but a rectangle with different side lengths.

In the diagram, it might be clear that the small rectangle is longer.

But to decide, I recall that for such problems, often the small rectangle is the one whose sides are parallel and it contains the circle centers or something.

Another way: the small rectangle's corners are circle centers, and the large rectangle's sides are touched by circles, so the distance from large rectangle sides to circle centers.

For example, the left side of large rectangle is at x=0, and the left-most circle centers are at x=r, so the small rectangle has left side at x=r (since A and D at x=r), so the small rectangle's left side is at x=r, which is distance r from left side.

Similarly, small rectangle right side at x=5r, while large is at 6r, so distance r from right side.

Small rectangle bottom at y=r, large at y=0, distance r.

Small rectangle top at y=3r, large at y=4r, distance r.

So the small rectangle is exactly in the middle, with distance r to each side.

Now, its size is width 4r, height 2r.

The circle centers on its bottom and top sides: bottom side from (r,r) to (5r,r), and at (3r,r) there is a circle center B.

Similarly on top at (3r,3r) circle E.

So the small rectangle has circle centers on its bottom and top sides, at the midpoints.

The problem doesn't mention that, but in the diagram, it might be shown.

For the other small rectangle, the square one, from (r,r) to (3r,3r), then its bottom side from (r,r) to (3r,r), and at (3r,r) there is circle B, which is a corner, not on the side.

Similarly, no circle center on the sides except at corners.

But in the first one, there is a circle center at the midpoint of the bottom and top sides.

Perhaps that is intended.

Moreover, in the problem, it says "the small rectangle", and with six circles, it makes sense.

But to confirm, let's see if there is another arrangement.

Suppose the circles are arranged in a 3x2 grid, but that's the same as 2x3, just rotated, but since the rectangle is large, probably it's oriented with the long side.

In 3x2, if 3 rows and 2 columns, then width 4r, height 6r, small rectangle if corners, width 2r, height 4r, perimeter 2(2r+4r)=12r same as before.

Large perimeter 2(4r+6r)=20r same.

If small is a square, but in 3x2, the small square might be different.

For example, centers at (r,r), (3r,r), (r,3r), (3r,3r), (r,5r), (3r,5r) for 2 columns, 3 rows.

Then corner centers A(r,r), C(3r,r), D(r,5r), F(3r,5r), small rectangle width 2r, height 4r, perimeter 12r same.

Or if I take a square, say A(r,r), C(3r,r), F(3r,5r), D(r,5r)? That's not a square, distance A to C is 2r, A to D is 4r.

To have a square, I need to take four centers that form a square. For example, A(r,r), B(r,3r), C(3r,3r), D(3r,r) but D is (3r,r), which is C in my list? Let's list: row1: A(r,r), B(3r,r) — but if 2 columns, so x=r and x=3r, y for rows.

Set: column 1: y=r, y=3r, y=5r for three rows.

So centers: (r,r), (3r,r) for first row; (r,3r), (3r,3r) for second; (r,5r), (3r,5r) for third.

Now, to form a square, for example, (r,r), (3r,r), (3r,3r), (r,3r) — this is a square of side 2r, since from (r,r) to (3r,r) is 2r, to (r,3r) is 2r, and (3r,r) to (3r,3r) is 2r, etc.

Similarly, I can have another square from (r,3r) to (3r,5r), but that's not a single rectangle.

The small rectangle could be this square: corners at (r,r), (3r,r), (3r,3r), (r,3r), which are circles A, B, and the second row circles, say E and F, but in my notation, (r,r) is A, (3r,r) is B, (r,3r) is C, (3r,3r) is D.

So corners are A,B,D,C.

Perimeter 8r.

Then large rectangle: width from x=0 to x=4r (since B at 3r, plus r), height from y=0 to y=6r (since bottom row at y=r, but top row at y=5r, so top side at y=5r + r =6r).

So width 4r, height 6r, perimeter 2(4r+6r)=20r same as before.

So in both orientations, large perimeter is 20r, and small perimeter is either 12r or 8r.

Now, given small perimeter 60, so if 12r=60, r=5, large=100; if 8r=60, r=7.5, large=150.

Now, to decide, I think the intended one is the first, where the small rectangle is not a square, and it's the one with the outer circles.

In the problem, it says "the small rectangle", and in many such problems, it's the bounding rectangle of the circle centers or something.

But in this case, for 2x3, the bounding rectangle of circle centers is from x=r to x=5r, y=r to y=3r, which is exactly the small rectangle I defined first, with size 4r by 2r.

And the large rectangle is larger by r on each side.

So that makes sense.

Moreover, in that case, the small rectangle has the property that its sides are at the levels of the circle centers, and the large rectangle is offset by r.

Also, the two additional circle centers are on the midpoints of the horizontal sides.

I think this is the intended configuration.

To confirm, if it were the square, then the small rectangle is only part of it, and the large rectangle extends more.

But in the diagram, it might be clear.

Since the problem is symmetric, but in this case, for the small rectangle to be "small", and in the first case, it is smaller than large, while in the second, if I take the square, it is also smaller, but perhaps not.

Another way: the problem might imply that the small rectangle is the one that is formed by the corners, and it's not the large one.

But both are valid, but I recall that in some similar problems, the answer is 100.

I could calculate the number.

The perimeter of small is 60, and for the large, it's 20r, small is 12r in first case, so large is 20r = (20/12)*60 = (5/3)*60 = 100.

In second case, small 8r=60, large 20r = (20/8)60 = 2.560=150.

Now, which one is more likely? I think 100 is a nice number, 150 is also nice, but let's see the context.

The previous problems were about diagonals and cutting, but this is different.

I think I should go with the first one, as it's more standard.

But let me double-check with the arrangement.

In the 2x3 grid, with small rectangle defined by the four corner centers, size 4r by 2r, perimeter 2(4r+2r)=12r=60, so r=5.

Large rectangle: to contain the circles, the leftmost point is the left side of left circles, at x=0, but circle A center at (r,r), so leftmost point of circle is x=0, similarly for others.

The rightmost point is circle C or F at x=5r + r =6r.

Similarly, bottom at y=0, top at y=3r + r =4r for circle D or F.

D is at (r,3r), so top is y=3r + r=4r.

So large rectangle from (0,0) to (6r,4r), width 6r, height 4r, perimeter 2(6r+4r)=20r=100.

The circles touch the sides: for example, circle A touches left and bottom, B touches bottom? B at (3r,r), so it touches bottom side at y=0? Distance from center to bottom is y=r, which is radius, so yes, it touches the bottom side. Similarly, all circles touch the sides: corner circles touch two sides, edge circles touch one side and other circles.

For example, B(3r,r) touches left? No, x=3r, left side at x=0, distance 3r > r, so not. It touches bottom side (y=0), and touches circles A and C horizontally, and E vertically.

Similarly, E(3r,3r) touches top side (y=4r), and circles D and F horizontally, B vertically.

So all good.

Now, the small rectangle has corners at A,B? No, at A(r,r), C(5r,r), D(r,3r), F(5r,3r). So not including B and E.

But B is at (3r,r), which is on the bottom side of the small rectangle, since bottom side is from A(r,r) to C(5r,r), so y=r, x from r to 5r, so B(3r,r) is on that line.

Similarly, E(3r,3r) on the top side.

So the small rectangle's bottom side contains the center of B, top side contains center of E.

I think this is correct.

Perhaps the small rectangle is meant to be the one that has the circles at its corners, and in this case, it does.

So I think r=5, large perimeter 100.

But earlier I thought about the other small rectangle, but that would be a different shape.

To be thorough, suppose the small rectangle is the square one: say from A(r,r), B(3r,r), E(3r,3r), D(r,3r). Then corners are A,B,E,D. Perimeter 8r=60, r=7.5.

Then large rectangle same, 6r by 4r? In 2x3, if I have circles at (r,r),(3r,r),(5r,r),(r,3r),(3r,3r),(5r,3r), then if small rectangle is only the left part, but the large rectangle still goes to x=6r, y=4r, since circle C at (5r,r) and F at (5r,3r) are still there, and they touch the right side.

So large is still width 6r, height 4r, perimeter 20r=20*7.5=150.

But in this case, the small rectangle is only covering the left two columns, but the circles C and F are not corners of the small rectangle; they are outside.

The problem says "each of the corners of the small rectangle is the centre of one of the circles", which is satisfied for A,B,D,E, but there are two more circles, C and F, which are not mentioned, but that's fine, since it doesn't say that all circles are corners.

However, in the diagram, it might show that the small rectangle is inside, and the large one contains all, but probably the small rectangle is intended to be the one that is defined by the outer path or something.

Moreover, in the first configuration, the small rectangle includes all circle centers in a way, but not exactly, since the midpoints are on the sides.

But I think the first one is more appropriate.

Perhaps the problem implies that the small rectangle is the minimal rectangle containing the circle centers or something, but in this case, the minimal rectangle containing all circle centers is from x=r to x=5r, y=r to y=3r, which is exactly the small rectangle I defined first, with size 4r by 2r.

And the large rectangle is that minimal rectangle plus a border of width r on all sides, since the circles have radius r, so to touch the sides, the large rectangle is offset by r.

In that minimal rectangle, the circles are inscribed in some way, but not exactly, but the centers are at the corners and midpoints.

Specifically, the minimal rectangle has corners at (r,r), (5r,r), (r,3r), (5r,3r), and the other centers are on the sides: (3r,r) on bottom, (3r,3r) on top, and on the left and right sides? Left side x=r, y=r and y=3r, but at y=2r, is there a center? In this grid, no, only at the corners.

So on the left side, only at (r,r) and (r,3r), no other centers.

Similarly for right side.

But the circles at (r,r) and (r,3r) are at corners, so they touch the left side.

Now, the large rectangle is outside, distance r.

So in this view, the small rectangle is the minimal bounding rectangle for the circle centers, and the large rectangle is the minimal bounding rectangle for the circles themselves, which is the center bounding rectangle plus r on each side, so size increases by 2r in each direction.

Minimal center rectangle: width 4r, height 2r.

Then large rectangle: width 4r + 2r = 6r? No: the minimal center rectangle has width from x=r to x=5r, so width 4r. The circles extend to the left of x=r, since leftmost circle center at x=r, but circle extends to x=0, so the large rectangle's left side is at x=0, while minimal center rectangle left side at x=r, so the large rectangle is shifted left by r, and similarly.

The minimal center rectangle has left side at x=r, large rectangle left side at x=0, so distance r to the left.

Similarly, right side: minimal center at x=5r, large at x=6r, distance r to the right.

Similarly for top and bottom.

So the large rectangle is wider by 2r (r on left, r on right), and taller by 2r (r on top, r on bottom).

Minimal center rectangle width W_c = 4r, height H_c = 2r.

Large rectangle width W_l = W_c + 2r = 4r + 2r = 6r.

Height H_l = H_c + 2r = 2r + 2r = 4r.

Same as before.

Perimeter small: 2(W_c + H_c) = 2(4r + 2r) = 12r = 60.

Perimeter large: 2(6r + 4r) = 20r.

So 20r = (20/12)*60 = (5/3)*60 = 100.

If I had chosen a different minimal rectangle, but in this grid, the minimal rectangle for all centers is indeed 4r by 2r.

The other rectangle I considered, the square one, is not minimal for all centers; it only covers four centers.

So probably the intended small rectangle is the minimal bounding rectangle for all circle centers.

Therefore, the answer should be 100 cm.

So I'll go with that.