Finding the area

Geometry Level 2

Four identical circles each with radius 5 are fitted exactly inside a square as shown above. If the area of the circle that can be fitted exactly between the four circles can be expressed as $25\pi(a-b\sqrt{b})$ , where $a$ and $b$ are coprime positive integers, find $a^2$ .

The answer is 9.

1 solution

A Former Brilliant Member
Apr 12, 2017

Connect the centers of the circle to form a square. Let $d$ be the diameter of the small circle. By pythagorean theorem, we have

$(10+d)^2=10^2+10^2$ $\implies$ $d=10\sqrt{2}-10$ $\implies$ $r=\frac{d}{2}=5\sqrt{2}-5$

The area of the small circle is

$A=\pi r^2 = \pi (5\sqrt{2}-5)^2=25\pi(3-2\sqrt{2})$

Finally,

$a^2=3^2=$ $\boxed{\color{#20A900}9}$

@Marvin Kalngan I did the same way..👍 Is there any other method?

Toshit Jain - 4 years, 1 month ago

Finding the area

The answer is 9.

1 solution

0 pending reports