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# There are 3 identical circles inside a rectangl . Each circle touches other two circles and the sides of the rectangle . The radius of each circle is 2m . what’s the exact area of the rectangle . (in the form of a √3+b where A and B are integers?

This is hard to illustrate without drawing, but essentially here’s the solution. One of the rectangle lengths will be twice the width of the circle. The width of a circle is 2*2m = 4m, so the side length of the rectangle is 8m. There is an equilateral triangle of side-length 2m formed by the centre of these circles. Each internal angle of such a triangle is 60 degrees. We can use trigonometry to show that the remaining side of the triangle is 2*2m + tan(60)*2m = (4+√3)m The area is the product of these two lengths: 8m*(4+√3)m = (8√3+32)m^2 See the following image for a visual depiction: https://i2.wp.com/mindyourdecisions.com/blog/wp-content/uploads/2019/05/rectangle-area-from-3-circles-hard-gcse-maths-problem-solution.png?w=600&ssl=1  James Wright
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