![\"\"](\"wp-content\/uploads\/2020\/10\/Screen-Shot-2019-11-19-at-11.09.09-PM.png\")
\n\n\n\nBy adding one more quarter circle, we have the following figure. The question is: what is the area of the region bounded by arc $\\stackrel \\frown {AE}$, $\\stackrel \\frown {EF}$ and $\\stackrel \\frown {FA}$? <\/p>\n\n\n\n\n
In fact, based on the calculations in the previous posts in this series, we have: $$ [AEF]=1\u2212[BCD]\u2212[ADE]\u2212[ABF] $$\n$$ =1-\\dfrac{\\pi}{4}-2\\cdot(1-\\dfrac{\\sqrt{3}}{4}-\\dfrac{\\pi}{6})=\\dfrac{\\pi}{12}+\\dfrac{\\sqrt{3}}{2}-1 $$<\/p>\n\n\n\n
If one more quarter circle is added, shown as the following figure, what is the area of the region in the middle bounded by arc $\\stackrel \\frown{EF}$, $\\stackrel \\frown{FG}$, $\\stackrel \\frown{GH}$, and $\\stackrel \\frown{HE}$? <\/p>\n![\"\"](\"wp-content\/uploads\/2020\/10\/Screen-Shot-2019-11-20-at-1.04.11-AM.png\")
\n\n
Again, based on the previous calculations, we have: $$[EFGH]=1-4\\cdot([ADE] + [AEF]) $$ $$=1-4\\cdot((1-\\dfrac{\\sqrt{3}}{4}-\\dfrac{\\pi}{6}) + (\\dfrac{\\pi}{12}+\\dfrac{\\sqrt{3}}{2}-1)) = 1-\\sqrt{3}+\\dfrac{\\pi}{3} $$<\/p>\n\n\n\n
Can you solve the problem without using the previous calculations? To be continued…<\/em><\/p>\n","protected":false},"excerpt":{"rendered":" By adding one more quarter circle, we have the following figure. The question is: what is the area of the region bounded by arc $\\stackrel \\frown {AE}$, $\\stackrel \\frown {EF}$ and $\\stackrel \\frown {FA}$? In fact, based on the calculations … Continue reading