{"id":437,"date":"2020-01-11T21:35:40","date_gmt":"2020-01-11T21:35:40","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=437"},"modified":"2020-10-26T03:22:59","modified_gmt":"2020-10-26T03:22:59","slug":"circles-in-a-square-part-11","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=437","title":{"rendered":"Circles in a Square – Part 11"},"content":{"rendered":"\n

8 semi-circles are drawn along the side lines inside of a unit square as shown in the following figure, with another circle centered at the center of the square and tangent to all of 8 semi-circles. What is the area of the square not covered by the circle and semi-circles (i.e. the green area)? Click here for the solution.<\/a><\/p>\n\n\n\n

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Lets draw line $\\overline{GI}$, between the midpoint of a side of the square, and the center of the circle; $\\overline{IJ}$, between the center of circle and that of a semi-circle\u00a0on the same side of point $G$, thru the tangent point of the circle and the semi-circle; line $\\overline{JK}$, between the center of the semi-circle and the intersect point $K$ with its adjacent semi-circle; and line $\\overline{DK}$, between the corner of the square adjacent to point $G$ and $K$.<\/p>\n\n\n\n

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Obviously, the radius of the semi-circles is $$s=\\dfrac{1}{4}$$ The area marked in blue color is $$A_{blue}=\\dfrac{1}{4}\\pi s^2+\\dfrac{1}{2}s^2=\\dfrac{2+\\pi}{64}$$ Therefore, the total area of overlapping semi-circles is $$A_s=8\\cdot A_{blue}=8\\cdot\\dfrac{2+\\pi}{64}=\\dfrac{2+\\pi}{8}$$<\/p>\n\n\n\n

Assume the\u00a0radius\u00a0of the circle centered at $I$ is $r$. Because $\\triangle{GIJ}$ is a right triangle, we have: $$s^2+(\\dfrac{1}{2})^2=(r+s)^2$$ i.e $$(\\dfrac{1}{4})^2+(\\dfrac{1}{2})^2=(\\dfrac{1}{4}+s)^2$$<\/p>\n\n\n\n

Solve the above equation and ignoring the negative\u00a0$r$ value, we have: $$r=\\dfrac{\\sqrt{5}-1}{4}$$ Therefore, the area of the circle is $$A_{r}=\\pi r^2=\\pi(\\dfrac{\\sqrt{5}-1}{4})^2=\\dfrac{3-\\sqrt{5}}{8}\\pi$$ And the area of not covered by the circles is $$A_{green}=1-A_{s}-A_{r}=1-\\dfrac{2+\\pi}{8}-\\dfrac{3-\\sqrt{5}}{8}\\pi=\\dfrac{3}{4}-\\dfrac{1}{2}\\pi+\\dfrac{\\sqrt{5}}{8}\\pi \\approx 0.0573$$<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"

8 semi-circles are drawn along the side lines inside of a unit square as shown in the following figure, with another circle centered at the center of the square and tangent to all of 8 semi-circles. What is the area of … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[12,7],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/437"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=437"}],"version-history":[{"count":18,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/437\/revisions"}],"predecessor-version":[{"id":914,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/437\/revisions\/914"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=437"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=437"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=437"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}