[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-267095-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-267095-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","circles-cheat-sheet-dans-test-prep-sat-circle-theorems-equations","Circles Cheat Sheet - Dan’s Test Prep - SAT Circle Theorems & Equations","","SAT circle questions typically fall into two areas: circle theorems and circle equations. This cheat sheet summarizes must-know circle terms such as radius, diameter, circumference, and area, then provides essential circle equation forms, including standard form and general form with completing the square. It also reviews theorem-based concepts like arc length, sector definitions, angle measures, radians vs. degrees conversions, and the sector formulas needed to solve test problems efficiently.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/exam/","Exam",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/circles-cheat-sheet-dans-test-prep-sat-circle-theorems-equations/267095/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/circles-cheat-sheet-dans-test-prep-sat-circle-theorems-equations/267095.png","ImageObject",300,407,{"name":42,"@type":43},"Chumphorn","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-11-04","2026-09-14",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",19,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What two categories do most SAT circle questions fall into?","Question",{"text":63,"@type":64},"Most SAT circle questions deal with either circle theorems (arc lengths, area, central angles, sectors) or circle equations (circles on a coordinate plane using center and radius).","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How do I find a circle’s equation from different given information?",{"text":68,"@type":64},"Use the appropriate circle equation form, especially standard form when the center and radius are clear. If given center and an endpoint, solve for the radius first, then plug into the standard equation.",{"name":70,"@type":61,"acceptedAnswer":71},"How do radians and degrees conversions work for circle formulas?",{"text":72,"@type":64},"Convert by multiplying by 180°/π when going from radians to degrees, and by π/180 when going from degrees to radians. This keeps units consistent for arc length and sector area formulas.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},267095,1789414755,{"code":4,"msg":82,"data":83},"success",[84,88,92,95,100,105,110,115,120,123,127],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":93,"slug":94},70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":106,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":112,"show_sort_weight":113,"slug":114},8,"Research & Report",30,"research-report",{"id":116,"doc_module":4,"doc_module_name":25,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":25,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":25,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":55,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":96,"slug":129},"General","general",{"code":4,"msg":82,"data":131},{"doc_id":79,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":33,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":96,"language":139,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":12,"update_tm":80,"read_time":143},2336475401981,"https://ap-avatar.wpscdn.com/avatar/22000c94efd8d5204d?x-image-process=image/resize,m_fixed,w_180,h_180&k=1786935347598174694","Circles Cheat Sheet (Dan’s Test Prep)  \nThe circle questions you see on the SAT will typically deal with one of two categories: circle theorems and circle equations.  \n- Circle Equations: Deal with circles on a coordinate plane (center, distance, radius)  \n- Circle Theorems: Deal with arc lengths, area, central angles, and sectors  \nMust-Know Circle Terms:  \nRadius: The distance from the center of a circle to a point on its edge. Diameter: The distance from one point on the circle, through its center, to a point on the opposite end. The diameter is always twice the radius. Circumference: The distance around the circle (essentially the perimeter) . Area: The extent of the space that the circle encompasses in the 2D plane.  \nBasic Circle Equations:  \nRadius of a circle: 􀝎  \nDiameter of a circle: 􀜦 = 2􀝎  \nCircumference of a circle: 􀜥 = 2􀟨􀝎 = 􀟨􀜦  \nArea of a circle: 􀜣 = 􀟨􀝎 ଶ  \nEquation of a Circle in the (x, y)-Plane:  \n(􀝔 − ℎ) ଶ + (􀝕 − 􀝇)ଶ = 􀝎 ଶ Where (h, k) is the center of the circle  \nand r is the radius of the circle (Standard Form)  \nMake sure you are very familiar with this equation and how it is used. The SAT can test you on this in a tricky way, such as giving you the center of the circle and an endpoint, requiring you to solve for the radius to figure out the equation.  \nExample Circle Graphs (Helpful Visualizations):  \nHere are some examples of what changing your circle equation can do to its graph.  \n􀝔 ଶ + 􀝕 ଶ = 1 􀝔 ଶ + 􀝕 ଶ = 4  \nℎ = 0, 􀝇 = 0, 􀝎 = 1 ℎ = 0, 􀝇 = 0, 􀝎 = 2  \nℎ = −1, 􀝇 = −1, 􀝎 = 1 (􀝔 + 1) ଶ + (􀝕 + 1)ଶ = 1  \nℎ = 1, 􀝇 = 1, 􀝎 = 1 (􀝔 − 1) ଶ + (􀝕 − 1)ଶ = 1  \nGeneral Form and Completing the Square:  \nSometimes, you’ll see an equation of a circle in the following form:  \n􀝔 ଶ + 􀝕 ଶ + 􀜣􀝔 + 􀜤􀝕 + 􀜥 = 0 (General Form)  \nThis equation still graphs a circle, just like our previous one, but now it is a little less clear what the center and radius are. To make our lives easier, we can“complete the square” to put this equation into the Standard Form.  \nI’ll illustrate with an example to make things a little less abstract.  \nSuppose our General Form equation is 􀝔 ଶ + 􀝕 ଶ + 2􀝔 − 6􀝕 + 6 = 0 I’ll start by moving the constant over to the right side.  \n􀝔 ଶ + 􀝕 ଶ + 2􀝔 − 6􀝕 = −6  \nNext, I’ll group the 􀝔 and 􀝕 terms together.  \n􀝔 ଶ + 2􀝔 + 􀝕 ଶ − 6􀝕 = −6  \nNow for the process of completing the square. To complete the square, we take our coefficient on the 􀝔 or 􀝕 term, we cut in half, and square it. Then, add these new constants to both sides of the equation to maintain the balance.  \nSince our coefficient on the is 􀝔 term is 2, we divide by 2 (~~ ~~ଶଶ = 1) and square it to get 1. Repeat this process to complete the square for 􀝕 . Our 􀝕 term is 6􀝕 . The coefficient is therefore 6. 6 divided by 2 is 3, and 3 squared is 9.  \n􀝔 ଶ + 2􀝔 + 1 + 􀝕 ଶ − 6􀝕 + 9 = −6 + 1 + 9  \nNow you may start to notice something interesting. After completing the square for both 􀝔 and 􀝕 , we can actually factor our terms into General Form by recognizing our newly formed perfect squares. Now we see that h = -1, k = 3, r = 2 (􀝔 + 1)ଶ + (􀝕 − 3)ଶ = 4 or (􀝔 + 1)ଶ + (􀝕 − 3)ଶ = 2 ଶ  \nCircle Theorems and Must-Know Equations:  \nLet’s start by reviewing some more terms you’ll need to know.  \nArc length: The distance between two points along the curve of a circle Sector: A portion of area captured by an angle traveling outward from the center of a circle to its edge  \nThis can be really confusing, so I created a visualization to help clarify it:  \nHere are some formulas you absolutely need to know in order to solve these questions properly. The SAT does not provide these for you!  \nAngle Measure: Arc Length (s)  \nArea of a Sector (A)  \nRadians  \n􀝏 = 􀝎􀟠  \n􀜣 = 12 􀝎 ଶ 􀟠  \nDegrees 􀝏 = 3~~ ~~􀟠60° ∙ 2􀟨􀝎􀜣 = 3~~ ~~􀟠60° ∙ 􀟨􀝎ଶ  \nRemember that s is the arc length, A is the area of a sector, and 􀟠 is the angle coming outwards from the center of the circle.  \nYou should also know that you can easily convert between radians and degrees using this simple trick:  \nIf you’re going from radia","cbCaiiF8fiLbM8tA","https://ap.wps.com/l/cbCaiiF8fiLbM8tA","pdf",11566094,"English","# Circle Questions Overview\n## Circle Theorems vs. Circle Equations\n# Must-Know Circle Terms\n## Radius, Diameter, Circumference, Area\n# Basic Circle Equations\n## Standard Form\n## Circle From Center and Endpoint\n# General Form and Completing the Square\n## Convert to Standard Form\n# Circle Theorems and Essential Formulas\n## Arc Length and Sector Area\n## Radians and Degrees Conversions","[{\"question\":\"What two categories do most SAT circle questions fall into?\",\"answer\":\"Most SAT circle questions deal with either circle theorems (arc lengths, area, central angles, sectors) or circle equations (circles on a coordinate plane using center and radius).\"},{\"question\":\"How do I find a circle’s equation from different given information?\",\"answer\":\"Use the appropriate circle equation form, especially standard form when the center and radius are clear. If given center and an endpoint, solve for the radius first, then plug into the standard equation.\"},{\"question\":\"How do radians and degrees conversions work for circle formulas?\",\"answer\":\"Convert by multiplying by 180°/π when going from radians to degrees, and by π/180 when going from degrees to radians. This keeps units consistent for arc length and sector area formulas.\"}]","Circles Cheat Sheet - Dan’s Test Prep - SAT Circle Theorems & Equations | PDF",13]