![\"\"](\"https:\/\/magoosh.com\/gmat\/files\/2019\/10\/GMAT-Circles.png\")
<\/p>\n<\/p>\n
Are you prepping for geometry questions on the GMAT? Feeling confused or in need of a refresher on circles? Read on to get a rundown of GMAT circles terminology, concepts, and practice questions.<\/p>\n
<\/a> <\/a> A circle is the set of all points equidistant from a fixed point.\u00a0 That means a circle is this:<\/p>\n and not this:<\/p>\n Photo by\u00a0cliparts.co<\/a><\/p>\n In other words, the circle is only the curved round edge, not the middle filled-in part.\u00a0 A point on the edge is “on the circle”, but a point in the middle part is “in the circle” or “inside the circle.”\u00a0 In the diagram below,<\/p>\n point A is on<\/span> the circle, but point B is in<\/span> the circle.<\/p>\n By far, the most important point in the circle is the center<\/strong> of the circle, the point equidistance from all points on the circle.<\/p>\n <\/a> Any line segment that has both endpoints on the circle is a chord<\/strong>.<\/p>\n By the way, the word “chord” in this geometric sense is actually related to “chord” in the musical sense: the link goes back to Mr. Pythagoras<\/a> (c. 570 \u2013 c. 495 BCE), who was fascinated with the mathematics of musical harmony.<\/p>\n If the chord passes through the center, this chord is called a diameter<\/strong>.\u00a0 The diameter is a chord.\u00a0 A diameter is the longest possible chord.\u00a0 A diameter is the only chord that includes the center of the circle.<\/p>\n The diameter is an important length associated with a circle, because it tells you the maximum length across the circle in any direction.<\/p>\n An even more important length is the radius<\/strong>.\u00a0 A radius is any line segment with one endpoint at the center and the other on the circle.<\/p>\n As is probably clear visually, the radius is exactly half the diameter, because a diameter can be divided into two radii.\u00a0 The radius is crucially important, because if you know the radius, it’s easy to calculate not only the diameter, but also the other two important quantities associated with a circle: the circumference<\/span> and the diameter<\/span>.<\/p>\n <\/a> The circumference<\/strong> is the length of the circle itself.\u00a0 This is a curve, so you would have to imagine cutting the circle and laying it flat against a ruler.\u00a0 As it turns out, there is a magical constant that relates the diameter (d) & radius (r) to the circumference.\u00a0 Of course, that magical constant is If you remember the second, more common form, you don’t need to know the first.\u00a0 The number These two formulas follow from the definition of <\/a> Suppose you stand at the center of a circle and turn around that you face each and every point on the circle.\u00a0 You would turn all the way around, which is an angle of 360\u00ba.\u00a0 In this sense, a whole circle has an angle of 360\u00ba.\u00a0 If you divided a circle equally, you could calculate the angle of each “slice”.\u00a0 Here are a few division results that could help you to know on test day (I’m just giving the ones that come out as nice round numbers, not the ones that result in ugly decimals):<\/p>\n 360\/2 = 180<\/p>\n 360\/3 = 120<\/p>\n 360\/4 = 90<\/p>\n 360\/5 = 72<\/p>\n 360\/6 = 60<\/p>\n 360\/8 = 45<\/p>\n 360\/9 = 40<\/p>\n 360\/10 = 36<\/p>\n 360\/12 = 30<\/p>\n Suppose we look at a “slice” of a circle, like a slice of pizza.<\/p>\n The curved line from A to B, a part of the circle itself, is called an arc<\/strong>.\u00a0 This corresponds to the crust of the pizza.<\/p>\n We can talk about the size of an arc in one of two ways: (a) its angle, sometimes called “arc angle” or “arc measure<\/strong>“, and (b), its length, called arclength<\/strong>.\u00a0 The angle of the arc, its arc measure, is just the same as the angle at the center of the circle.\u00a0 Here \u2220AOB = 60\u00b0, so the measure of arc AB is 60\u00b0.<\/p>\n We find the arclength<\/a> by setting up a proportion of part-to-whole.\u00a0 The angle is part of the whole angle of a circle, 360\u00b0.\u00a0 The arclength is part of the length all the way around, i.e. the circumference.\u00a0 Therefore:<\/p>\n Here, let’s say the radius is r = 12.\u00a0 Then, the circumference is Cross-multiply:<\/p>\n In other words, since the angle 60\u00b0 is one sixth of the full angle of a circle, the arclength is one sixth of the circumference.<\/p>\n <\/a> 1) The area <\/p>\n The 8-inch pizza has a radius of r = 4, so the area is <\/p>\n <\/p>\n Statement #1: if you know the circumference, then you can use Statement #2: if you know the area, you can find the radius, and then double that to get the diameter. This statement by itself is also sufficient.<\/p>\n Both statements alone are sufficient. Answer = D<\/strong>. The post GMAT Circles: An Introduction<\/a> appeared first on Magoosh GMAT Blog<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":" Are you prepping for geometry questions on the GMAT? Feeling confused or in need of a refresher on circles? Read on to get a rundown of GMAT circles terminology, concepts,…<\/p>\n","protected":false},"author":133,"featured_media":0,"comment_status":"open","ping_status":"1","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9,783,243,940],"tags":[],"class_list":["post-48312","post","type-post","status-publish","format-standard","hentry","category-gmat","category-magoosh-blog","category-blog","category-gmat-prep-gmat","entry"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/48312","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=48312"}],"version-history":[{"count":0,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/posts\/48312\/revisions"}],"wp:attachment":[{"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/media?parent=48312"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/categories?post=48312"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gmatclub.com\/blog\/wp-json\/wp\/v2\/tags?post=48312"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n <\/p>\nTable of Contents<\/h2>\n
\n
\n <\/p>\nGMAT Circles: Basic Terminology<\/h2>\n
![\"GMAT](\"https:\/\/magoosh.com\/gmat\/files\/2012\/08\/c1_img1.png\")
<\/a><\/p>\n![\"Incorrect](\"https:\/\/magoosh.com\/gmat\/files\/2012\/08\/c1_img2.png\")
<\/a><\/p>\n![\"Points](\"https:\/\/magoosh.com\/gmat\/files\/2012\/08\/c1_img3.png\")
<\/a><\/p>\n![\"Equidistant](\"https:\/\/magoosh.com\/gmat\/files\/2012\/08\/c1_img4.png\")
<\/a><\/p>\n![\"Back](\"https:\/\/magoosh.com\/gmat\/files\/2017\/07\/back-to-top-button-1.png\")
<\/a><\/p>\n
\n <\/p>\nChords<\/h2>\n
![\"GMAT](\"https:\/\/magoosh.com\/gmat\/files\/2012\/08\/c1_img5.png\")
<\/a><\/p>\n![\"GMAT](\"https:\/\/magoosh.com\/gmat\/files\/2012\/08\/c1_img6.png\")
<\/a><\/p>\n![\"GMAT](\"https:\/\/magoosh.com\/gmat\/files\/2012\/08\/c1_img7.png\")
<\/a><\/p>\n<\/a><\/p>\n
\n <\/p>\nCircle Formulas<\/h2>\n
![\"pi\"](\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/magoosh-wpmathpub\/phpmathpublisher\/img\/math_993.5_8edb2cf68079344a2edd739531259f6c.png\" "\\"pi\\"")
.\u00a0 From the very definition of itself, here are two equations for the circumference, c.<\/p>\n![\"c](\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/magoosh-wpmathpub\/phpmathpublisher\/img\/math_993_603abfd5165132d946561a6b0d0d2b54.png\" "\\"c")
<\/p>\n![\"c](\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/magoosh-wpmathpub\/phpmathpublisher\/img\/math_993_2fff2748d85430c2f890de86723948f0.png\" "\\"c")
<\/p>\n is slightly larger than 3 — this means that three pieces of string, each as long as the diameter, together would not be quite long enough to make it all the way around the circle.\u00a0 The number can be approximated by 3.14 or by the fraction 22\/7.\u00a0 Technically, it is an irrational number<\/a> that goes on forever in a non-repeating pattern.<\/p>\n, so basically every culture on earth figured out these.\u00a0 By contrast, the area of circle<\/strong> was discovered by one brilliant mathematician, and everyone on earth has this one man to thank for his formula for the area of a circle.\u00a0 That man was Archimedes<\/a> (c. 287 \u2013 c. 212 BCE).\u00a0 Here is Archimedes’ amazing formula:<\/p>\n![\"A](\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/magoosh-wpmathpub\/phpmathpublisher\/img\/math_993.5_3265fb4fa2bccaef61cf48529b1dd80c.png\" "\\"A")
<\/p>\n\n
<\/a><\/p>\n
\n <\/p>\nCircles and Angles<\/h2>\n
Arcs and Arclength<\/h3>\n
![\"Circles](\"https:\/\/magoosh.com\/gmat\/files\/2012\/08\/c2_img1.png\" "\\"c2_img1\\"")
<\/a><\/p>\n![\"\"](\"https:\/\/magoosh.com\/gmat\/files\/2012\/08\/c2_img3.png\" "\\"c2_img3\\"")
<\/a><\/p>\n![\"c](\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/magoosh-wpmathpub\/phpmathpublisher\/img\/math_993_6536d76179c02e15ff287d757c578044.png\" "\\"c")
.\u00a0 Since the angle is 60\u00b0, the ratio on the left side, angle\/360, becomes 1\/6.\u00a0 Call the arclength x.<\/p>\n![\"{1\/6}](\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/magoosh-wpmathpub\/phpmathpublisher\/img\/math_984_3df9b763a91fca2efd99392934f77684.png\" "\\"{1\/6}")
<\/p>\n![\"{24{pi}}](\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/magoosh-wpmathpub\/phpmathpublisher\/img\/math_993_748cb411b7286ac5753b9753d85292aa.png\" "\\"{24{pi}}")
<\/p>\n![\"{4{pi}}](\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/magoosh-wpmathpub\/phpmathpublisher\/img\/math_993_f706206c6eed967c289d7491d227fe82.png\" "\\"{4{pi}}")
<\/p>\n<\/a><\/p>\n
\n <\/p>\nGMAT Circles Practice Questions<\/h2>\n
![\"\"](\"https:\/\/magoosh.com\/gmat\/files\/2012\/08\/c2_img2.png\" "\\"c2_img2\\"")
<\/a><\/p>\n\n
![\"A](\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/magoosh-wpmathpub\/phpmathpublisher\/img\/math_993_f4439b5fe650d67eb436ba2b4b3fdcb5.png\" "\\"A")
.\u00a0 The perimeter of the shaded region is\n\n
![\"12](\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/magoosh-wpmathpub\/phpmathpublisher\/img\/math_993_02b18ab4742344312be2b1839887c681.png\" "\\"12")
<\/li>\n![\"12](\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/magoosh-wpmathpub\/phpmathpublisher\/img\/math_993_8a914845e8080ffb01122de7c2e40cde.png\" "\\"12")
<\/li>\n![\"24](\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/magoosh-wpmathpub\/phpmathpublisher\/img\/math_993_7a974375069ea63d89196e424259c651.png\" "\\"24")
<\/li>\n![\"24](\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/magoosh-wpmathpub\/phpmathpublisher\/img\/math_993_8a49acd8dc431e47786e2079b5b37524.png\" "\\"24")
<\/li>\n![\"24](\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/magoosh-wpmathpub\/phpmathpublisher\/img\/math_993_6c70ccbefcb3dd224f163484c20caf35.png\" "\\"24")
<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n\nShow answer and explanation<\/summary>\n
![\"144{pi}](\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/magoosh-wpmathpub\/phpmathpublisher\/img\/math_993.5_bacf6a4dfc95bb4860324fdade7718fa.png\" "\\"144{pi}")
, so r = 12.\u00a0 This means KO = 12 and OL = 12, so those two sides together are 24.\u00a0 The remaining side is arc KL.\u00a0 The whole circumference is ![\"c](\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/magoosh-wpmathpub\/phpmathpublisher\/img\/math_993_0d54b11c47cc671183fa41447d35e147.png\" "\\"c")
.\u00a0 The angle of 120\u00b0 is 1\/3 of a circle, so the arclength is 1\/3 of the circumference.\u00a0 This means, ![\"arclength](\"https:\/\/magoosh.com\/gmat\/wp-content\/plugins\/magoosh-wpmathpub\/phpmathpublisher\/img\/math_993_fdcbe1da5eb51423173049af7bf2a673.png\" "\\"arclength")
, and therefore the entire perimeter is .\u00a0 Answer = C<\/strong>.<\/details>\n\n
\n
\nShow answer and explanation<\/summary>\n
![\"A](\"https:\/\/2aih25gkk2pi65s8wfa8kzvi-wpengine.netdna-ssl.com\/gmat\/wp-content\/plugins\/magoosh-wpmathpub\/phpmathpublisher\/img\/math_993_06a3d301dfd180e30c9572444e59ed60.png\" "\\"A")
. That area is how much pizza you get. The 12-inch pizza has a radius of r = 6 and an area of ![\"A](\"https:\/\/2aih25gkk2pi65s8wfa8kzvi-wpengine.netdna-ssl.com\/gmat\/wp-content\/plugins\/magoosh-wpmathpub\/phpmathpublisher\/img\/math_993_a0fcd0fba78c526e1c66b0fc1b86c07e.png\" "\\"A")
. When you change from 16 to 36, what is the percentage change? Well, that’s more than double, so it must be a percent greater than 100%. The only answer choice greater than 100% is answer E<\/strong>.<\/details>\n\n
\n
![\"c](\"https:\/\/2aih25gkk2pi65s8wfa8kzvi-wpengine.netdna-ssl.com\/gmat\/wp-content\/plugins\/magoosh-wpmathpub\/phpmathpublisher\/img\/math_993_418b53eb66dfa9e48539506911ed15e3.png\" "\\"c")
.<\/li>\n.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\nShow answer and explanation<\/summary>\n
![\"c](\"https:\/\/2aih25gkk2pi65s8wfa8kzvi-wpengine.netdna-ssl.com\/gmat\/wp-content\/plugins\/magoosh-wpmathpub\/phpmathpublisher\/img\/math_993_2fff2748d85430c2f890de86723948f0.png\" "\\"c")
to solve for the radius, or ![\"c](\"https:\/\/2aih25gkk2pi65s8wfa8kzvi-wpengine.netdna-ssl.com\/gmat\/wp-content\/plugins\/magoosh-wpmathpub\/phpmathpublisher\/img\/math_993_603abfd5165132d946561a6b0d0d2b54.png\" "\\"c")
to solve for the diameter. Either way, you can find the diameter, so this statement by itself is sufficient.<\/p>\n
\n<\/details>\n
\nWant more GMAT practice problems involving circles? Click here!<\/a><\/p>\n