The Chinese Taijitu symbol, also known as the Yin and Yang, symbolizes the duality and balance of everything. It appears in many contexts, including ancient Confucian writings, martial arts, and even the national flag of South Korea. The Taijitu is always shown with the yin (dark side) and the yang (light side) in balance. But, what happens when they are unbalanced?
The typical Taijitu has the dark (yin) and light (yang) side separated by a specific curve that is related to the apparent path of the sun around the earth (the ecliptic), but for the purposes of this problem, we'll simply use two connected semicircles as shown below. In the image on the left, the Taijitu is balanced, with an equal area of yin and yang. The image on the right has significantly more yang (because the upper semicircle has a larger radius than the lower semicircle), and is therefore unbalanced.
Note: Each side is usually depicted with a "seed" (a small dot) of the opposite color. For simplicity, we will ignore this.
Given the overall radius (in centimeters) of the Taijitu, and the radius (in centimeters) of the upper semicircle (as shown in the images above), determine the total area of yin and yang in the corresponding Taijitu. Note that the yang (the light side) always appears on the left side of the Taijitu, and the two semicircles defining the division are always oriented as shown above.
Use value of 3.14159 for π (pi).
2 10 5 12 7
Taijitu #1: yin 157.08, yang 157.08 Taijitu #2: yin 188.50, yang 263.89