Max Area of 2 Circles in Square of Side 1

  • Thread starter rudolfstr
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In summary, the conversation discusses a problem involving drawing two circles inside a square without going outside or crossing, and determining the maximum area their sum can make. Possible approaches are suggested, with one being easier than another. The suggestion to look up Malfatti's problem on Google is also mentioned.
  • #1
rudolfstr
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I've been thinking of a solution, but can't find a one. You have a square of side length 1. You have to draw 2 circles inside the square so they wouldn't go outside the square and at the same time wouldn't cross. What is the maximum area their sum can make?
 
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  • #2
hi rudolfstr! :smile:

start by making a guess as to what the line joining their centres will be

(i can only think of two likely candidates)

then find where they touch the sides, and calculate their areas

(and then see if you can improve on that) :wink:
 
  • #3
I think Tiny-Tim's approach will work for this one, it is not as difficult as malfatti's problem, which you might find fun to look up.

There is plenty on Google.
 

1. What is the maximum possible area of two circles that can fit inside a square with a side length of 1?

The maximum possible area of two circles that can fit inside a square with a side length of 1 is approximately 0.7854 square units. This occurs when the two circles are tangent to each other and to the sides of the square.

2. How do you calculate the area of a circle?

The area of a circle can be calculated using the formula A = πr^2, where A is the area and r is the radius of the circle. Alternatively, you can also use the circumference of the circle (C) and the formula A = C^2 / (4π) to calculate the area.

3. Can the two circles be of different sizes?

Yes, the two circles can be of different sizes and still have a combined maximum area of approximately 0.7854 square units. However, the circles must still be tangent to each other and to the sides of the square.

4. What is the significance of finding the maximum area of two circles in a square of side 1?

Finding the maximum area of two circles in a square of side 1 can be useful in various real-world applications, such as optimizing the space utilization in a packing problem or determining the maximum capacity of a circular container that can fit inside a square container.

5. Is the maximum area of two circles in a square of side 1 a unique solution?

Yes, the maximum area of two circles in a square of side 1 is a unique solution. This is because the solution occurs when the two circles are tangent to each other and to the sides of the square, and there is only one way for circles to be tangent to each other at a given point.

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