{"id":16919,"date":"2013-02-22T09:00:40","date_gmt":"2013-02-22T16:00:40","guid":{"rendered":"https:\/\/gmatclub.com\/blog\/?p=16919"},"modified":"2013-02-17T07:06:04","modified_gmt":"2013-02-17T14:06:04","slug":"slicing-up-gmat-circles-arclength-sectors-and-pi","status":"publish","type":"post","link":"https:\/\/gmatclub.com\/blog\/slicing-up-gmat-circles-arclength-sectors-and-pi\/","title":{"rendered":"Slicing up GMAT Circles: Arclength, Sectors, and Pi"},"content":{"rendered":"

\"\"<\/a>Few topics make students let out an audible sigh like\u00a0circles on the GMAT<\/a>. \"What was the formula for area of a circle, again? I know pi is in there somewhere!\" Then they throw sectors, proportions, and arclengths at you. Now you're really rolling your eyes.<\/p>\n

Slicing up circles on the GMAT? Arclength, sectors, and pi! Oh my!<\/h2>\n

First, let's induce some of that anxiety. How do you find the do calculations for something like this?<\/p>\n

\"sop_img1\"<\/a><\/p>\n

Given that O is the center of the circle, and given the info in the diagram, how do we find\u00a0(1)<\/strong>\u00a0the length of arc AB? , and\u00a0(2)<\/strong>\u00a0the area of sector AOB (i.e. the shaded region)?\u00a0 BTW, the word\u00a0sector<\/strong>\u00a0is the name for a \"slice of pie\" piece of a circle, and\u00a0arclength<\/strong>\u00a0is the curved length along the \"crust\" of the pie.\u00a0\u00a0 You can find a little more about arclength at\u00a0this post<\/a>.<\/p>\n

 <\/p>\n

Proportions<\/h2>\n

One of the greatest math \"tricks\" is proportional reasoning, and this is precisely what unlocks all the questions posed above.\u00a0 Basically, to find either the length of an arc or the area of sector, we set up a part-to-whole ratio, and set this equal to a ratio for the angle.\u00a0\u00a0 In the diagram, the angle at the center of the circle is 80\u00b0.\u00a0\u00a0 A whole circle, all the way around, is always 360\u00b0, so this 80\u00b0 angle takes up 80\/360 = 8\/36 = 2\/9 of a circle.\u00a0\u00a0 Therefore, both the arclength and the area of the sector are 2\/9 of their respective \"wholes\".<\/p>\n

 <\/p>\n

Arclength<\/strong><\/h2>\n

If arclength is the \"part\", then the \"whole\" is the length all the way around the circle --- we call that length circumference,\u00a0 and its formula is<\/p>\n

\"sop_img2\"<\/a><\/p>\n

The part-over-whole of the angles has to equal the part-over-whole of these lengths:<\/p>\n

\"sop_img3\"<\/a><\/p>\n

For this diagram, this proportion becomes:<\/p>\n

\"sop_img4\"<\/a><\/p>\n

 <\/p>\n

Area of a sector<\/h2>\n

If area of the sector is the \"part\", then the area of the whole circle is the whole.\u00a0 Of course, for the area of a circle, we use\u00a0Archimedes<\/a>' remarkable formula:<\/p>\n

\"sop_img5\"<\/a><\/p>\n

As with arclength, the part-over-whole of the angles has to equal the part-over-whole of these areas:<\/p>\n

\"sop_img6\"<\/a><\/p>\n

For this diagram, this proportion becomes:<\/p>\n

\"sop_img7\"<\/a><\/p>\n

BTW,\u00a0a mental math tip<\/em>\u00a0--- how did I just know, in my head, that 144\/9 = 16, or in other words, that 9*16 = 144?\u00a0 Well, think about it.<\/p>\n

\"sop_img8\"<\/a><\/p>\n

That logic makes it very easy to find factors of perfect squares.<\/p>\n

 <\/p>\n

Summary<\/h2>\n

If you remember these simple proportional reasoning tricks, you will be able to figure out anything the GMAT might ask about a part of a circle.\u00a0\u00a0 Here's a free practice question on which to apply these skills.<\/p>\n

1)\u00a0https:\/\/gmat.magoosh.com\/questions\/87<\/a><\/p>\n

What about GMAT circles slices you the wrong way? Let us know below. \ud83d\ude42<\/p>\n

This post was written by Mike McGarry, GMAT expert at Magoosh<\/a>, and originally posted here<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"

Few topics make students let out an audible sigh like\u00a0circles on the GMAT. “What was the formula for area of a circle, again? I know pi is in there somewhere!”…<\/p>\n","protected":false},"author":133,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9,783,243,736],"tags":[],"class_list":["post-16919","post","type-post","status-publish","format-standard","hentry","category-gmat","category-magoosh-blog","category-blog","category-quant-gmat","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/16919","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/users\/133"}],"replies":[{"embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/comments?post=16919"}],"version-history":[{"count":4,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/16919\/revisions"}],"predecessor-version":[{"id":17031,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/16919\/revisions\/17031"}],"wp:attachment":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/media?parent=16919"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/categories?post=16919"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/tags?post=16919"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}