Three circles of radius 1 unit fit inside a square such that the two outer circles touch the middle circle and the sides of the square. Given the centres of the circle lie on the diagonal of the square, find the exact area of the square.

I got the answer 18 + 8root2
but the book says 12 + 8root2
I'm about 100% sure i got it right and the book might have an error but could someone please check for me because this is really bugging me out, thanks.

the distance between the centers of the two outer circles is 4

they are in the diagonal corners, and their centers are √2 from the corners

so the diagonal of the square is
... 4 + 2√2

dividing by √2 to find side length
... 2 + 2√2

squaring
... 4 + 4√2 + 4√2 + 8 = 12 + 8√2

Ahh thank you so much

Could someone explain it entirely to me?

isn't 12+8√2 one of the side lengths?

and isn't the question asking you to find the exact AREA of the square?

so would you square (length squared is area formula for square) 12+8√2 to get 272+192√2

Well, you know what they say, "If you're 100% sure, it's probably the book that's wrong." But let's take a closer look, just to make sure.

First, let's draw out the situation. We have a square with side length "s" and three circles inside. The two outer circles are tangent to the middle circle and the sides of the square. The centers of all three circles lie on the diagonal of the square.

Now, let's start solving. The diameter of each circle is 2 units, which means the diagonals of the square are divided into 2 + 2 + 2 = 6 equal parts. Since the centers of the circles lie on the diagonal, the two inner sections each have a length of (1/2) * 6 = 3 units.

Now, let's look at one half of the square. We can see that the side length of this half-square is s/2. If we add the length of the inner section (3 units), we get (s/2 + 3). Since we have two of these, the full length of the square is 2(s/2 + 3) = s + 6.

To find the area, we simply square this length: (s + 6)^2.

Expanding this expression gives us: s^2 + 12s + 36.

Oops, it looks like I made the same mistake as you. Sorry about that! It seems we missed something. Let's take another look.

Upon further reflection, we can see that the diagonal of the square is actually equal to the diameter of each circle, which is 2 units. From this, we can directly conclude that the side length of the square is also 2 units.

The area of a square is found by squaring its side length, so the area of this square is 2^2 = 4 square units.

So, it seems the book is correct after all. It's always good to check and recheck!

To find the exact area of the square, we need to determine the side length of the square.

Let's consider the diagonal of the square. The centres of the three circles lie on the diagonal, and since the radii of the circles are 1 unit, the distance between each pair of adjacent centres is also 1 unit.

Let's assume the side length of the square is S units. Since the centres of the circles lie on the diagonal, the distance between the opposite corners of the square is S√2 units.

We can divide this diagonal length into three sections, where each section includes the radius of one circle and the side length of the square. The lengths of these three sections would be 1 unit each.

So, we have the equation: S√2 = 3 + S

To solve this equation, we'll square both sides:
(S√2)^2 = (3 + S)^2
2S^2 = 9 + 6S + S^2
S^2 - 6S - 9 = 0

Now, applying the quadratic formula:
S = (-(-6) ± √((-6)^2 - 4*1*(-9)))/(2*1)
S = (6 ± √(36 + 36))/2
S = (6 ± √72)/2
S = (6 ± 6√2)/2
S = 3 ± 3√2

Since the side length of a square cannot be negative, we take the positive solution:
S = 3 + 3√2

Now, we can calculate the area of the square:
Area = S^2
Area = (3 + 3√2)^2
Area = 9 + 18√2 + 18√2 + 18
Area = 27 + 36√2

So, according to my calculations, the exact area of the square is 27 + 36√2.

It seems the book may have made an error in stating the answer as 12 + 8√2. I would recommend double-checking the calculations and discussing it with your teacher or classmates.