# Three Circles +-- {.image} [[ThreeCircles.png:pic]] > The small circles each have area $1$. What's the area of the large circle? =-- ## Solution by [[Similar Triangles]] and [[Angle Between a Radius and Tangent]] +-- {.image} [[ThreeCirclesLabelled.png:pic]] =-- As the two small circles have the same area, they are the same size. Let $r$ be the radius of the small circles and $R$ of the larger one. Then $\pi r^2 = 1$. With the points labelled as above, angles $O \hat{D} A$ and $O \hat{C} B$ are [[right-angles]] as they are the [[angle between a radius and tangent]]. So triangles $O D A$ and $O C B$ are [[similar]]. In particular, the ratios of the lengths of $A D$ to $O A$ and $B C$ to $O B$ are the same. In terms of the radii of the circles, this means that $R : R + 3 r = 1 : 2$. Equivalently, $R + 3 r = 2 R$ and so $R = 3 r$. The area of the larger circle is then $\pi R^2 = \pi (3 r)^2 = 9 \pi r^2 = 9$.