There are 4 squares. Each square has circles which touch each other, and are shaded. The first square has one large circle touching all 4 edges of the square, the 2nd square has 4 circles, the 3rd square has 9 circles and the last square has 16 circles.
All circles within a square are equal in size and symmetrically placed.
Which of these squares has the maximum shaded (purple) area? How can you best explain your answer?
Post different ways of thinking about this problem in the comments.
It’s the same. All others are created by using scalled down copies of the first.
When you scale a shape, ratio of area stays constant.
If you divide the square in 4 copies that re scalled down by 2, then you keep the same area.
That holds true because if you have n copies along the side, you will scale down by n (so, area will scale down by n^2), but will have n^2 copies. Note that if you focus on area ratio, it will stay constant!
The ratio of circle to square area stays the same because that’s how scalling work.
So, when you make smaller copies of the square and inscribed circles and glue them together, the ratio stay the same.
All circles have same shaded area(purple area);
Analysis:
let the side of square= d;
If there are nXn circles inside this square;let the radius of each small circle=r;
then d=n*(2*r); or r= d/(2*n); Number of circles inside square has area=n^2X[π X{d/(2*n)}^2]=π* (d^2/4);
In previous post : last line It is independent of number of circles inside the square(n) should be read as :It is independent of number of circles inside the square(n^2)
All circles have same shaded area(purple area);
Analysis:
let the side of square= d;
If there are nXn circles inside this square;let the radius of each small circle=r;
then d=n*(2*r); or r= d/(2*n); Number of circles inside square has area=n^2X[π X{d/(2*n)}^2]=π* (d^2/4);
It is independent of numer of circles inside the square(n)
Same in all cases. I have done it purely mathematically. Considering the square to be of 24 units (so that radius of circle is a whole number in all cases), the total shaded area comes to be π(12)^2 in all 4 cases.
Great! Can you think of another way to explain this to a person who doesn’t know the formula for the area of a circle?
Lets take a circle from the rest(Fig 2) and imagine the little square around it, in which the circle is inscribed. As we can see that the little square has 1/4 th the area of original square we can also say that the area of chosen circle is also 1/4 th of the larger circle from fig 1. Thus summation of area of all the four circles give the area of larger circle from fig one. Hence area of shaded region of fig 2 is same as fig 1.
We can also say the same for all figures.