{"id":331,"date":"2019-11-30T21:10:18","date_gmt":"2019-11-30T21:10:18","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=331"},"modified":"2020-10-14T21:11:38","modified_gmt":"2020-10-14T21:11:38","slug":"circles-in-a-square-part-4","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=331","title":{"rendered":"Circles in a Square &#8211; Part 4"},"content":{"rendered":"\n<p>In the\u00a0previous\u00a0post of this series, we asked if the area of the region can be solved without using\u00a0previous\u00a0calculation.\u00a0Look at the figure below, after drawing\u00a0several lines by connecting several points:<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" src=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2020\/10\/Screen-Shot-2019-11-17-at-4.01.15-PM.png\" alt=\"\" class=\"wp-image-332\" width=\"296\" height=\"296\" srcset=\"https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2020\/10\/Screen-Shot-2019-11-17-at-4.01.15-PM.png 592w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2020\/10\/Screen-Shot-2019-11-17-at-4.01.15-PM-300x300.png 300w, https:\/\/mathfun4kids.com\/mlog\/wp-content\/uploads\/2020\/10\/Screen-Shot-2019-11-17-at-4.01.15-PM-150x150.png 150w\" sizes=\"(max-width: 296px) 100vw, 296px\" \/><\/figure><\/div>\n\n\n\n<p>The fan area of $[AEF]$ is $\\dfrac{\\pi}{12}$, because line $\\overline{AE}$ and $\\overline{AF}$ trisect the right angle at $A$.<\/p>\n<p>Additionally,<\/p> \n$$\\overline{EG}=\\overline{FG}=\\overline{EH}-\\overline{GH}=\\dfrac{\\sqrt{3}}{2}\u2212\\dfrac{1}{2}=\\dfrac{\\sqrt{3}-1}{2}$$\n$$[\\triangle{AGE}]=[\\triangle{AGF}]=\\dfrac{1}{2}\\cdot\\overline{EG}\\cdot\\overline{AH}=\\dfrac{1}{2}\\cdot\\dfrac{\\sqrt{3}-1}{2}\\cdot\\dfrac{1}{2}=\\dfrac{\\sqrt{3}-1}{8}$$\n$$[GEF]=[AEF]-[\\triangle{AGE}]-[\\triangle{AGF}]=\\dfrac{\\pi}{12}-2\\cdot\\dfrac{\\sqrt{3}-1}{8}=\\dfrac{1}{4}-\\dfrac{\\sqrt{3}}{4}+\\dfrac{\\pi}{12}$$\n<p>The area of the middle region bounded by the four quarter circles is:<\/p>\n$$ 4\\cdot [GEF]=4\\cdot(\\dfrac{1}{4}-\\dfrac{\\sqrt{3}}{4}+\\dfrac{\\pi}{12})=1-\\sqrt{3}+\\dfrac{\\pi}{3}$$\n\n\n\n<p>The above shows the same answer as given in the previous post. <em>To be continued&#8230;<\/em><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In the\u00a0previous\u00a0post of this series, we asked if the area of the region can be solved without using\u00a0previous\u00a0calculation.\u00a0Look at the figure below, after drawing\u00a0several lines by connecting several points: The fan area of $[AEF]$ is $\\dfrac{\\pi}{12}$, because line $\\overline{AE}$ and &hellip; <a href=\"https:\/\/mathfun4kids.com\/mlog\/?p=331\">Continue reading <span class=\"meta-nav\">&rarr;<\/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":[12,9],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/331"}],"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=331"}],"version-history":[{"count":9,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/331\/revisions"}],"predecessor-version":[{"id":341,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/331\/revisions\/341"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=331"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=331"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=331"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}