Question

A square of side 25cm is divided into n^2 equal small squares.if circle is drawn in each if these small squares touching all the sides,then find the area of given square not covered by these circles?

Answer 1

Find the area of the square and subtract out the area of the circles. 

Area of the square is easy: 

A = s² 
A = 25² 
A = 625 cm² 

The area for the circles takes a little thinking. 

Let's say that there is 1 circle that is inscribing the square. So, n = 1 

Then the radius is half the width of the box, which is 25/2. 

If there are 4 circles inside the square (n = 2), the the radius of each circle is 25/4 

So based on our observations, we can determine the the radius of the circles will be following this equation: 

r = 25/2n 

So now to find the area of 1 circle, in terms of n, then multiply it by n² circles for the total area of the circles. 

A = πr² 
A = π(25/2n)² 
A = π(625/4n²) 
A = 625π/4n² 

Now multiply that by n² circles: 

n² * 625π/4n² 

625π/4 cm² 

Which oddly enough is no longer in terms of n, but a constant. This shows that no matter how many circles you put in the square, the total area remains the same (if it's 1 circle or 100). I honestly wasn't expecting that. 

Now we have the area of the square and the area of n² circles. Subtract the latter from the former to get the area of the square not inscribed by a circle: 

625 - 625π/4 cm²

Hope This Helps :)