💯 Maths

# 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?  Verified
James WrightExperienced tutor and undergraduate at University of St Andrews1 students helped

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 