[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-207766-en":3,"doc-seo-207766-105":30,"detail-sidebar-cat-0-en-105":89},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},207766,5909887254083,"\tWilliam","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",4,"Exam","Circles - Key Terms, Formulas and Circle Theorems","Covers the fundamentals of circles as 2D geometry, defining key terms such as circumference, chord, diameter, radius, tangent, arc, sector, and segment. Provides essential formula relationships for circumference, area of circles, arc length, area of sectors, and dimensions of curved solids including cylinders and cones, plus sphere volume and surface area. Includes key circle theorems on tangent-radius perpendicularity, equal tangents, alternate segment angles, semicircle right angles, cyclic quadrilaterals, and angle relationships at the centre and circumference, with worked example-style angle calculations.","CIRCLES  \nA circle is a 2D shape where all of the points that form the circle are an equal distance from the centre .π ‘Pi” is ratio of the circumference of every circle to its diameter π = 3.1415926535897932...  \nKey terms  \nCircumference   \nThe distance around the outside of the circle.  \nChord  \nA straight line that goes from one side of the circle to the other without passing through the centre.  \n~~ ~~ Diameter  \nA straight line that goes from one side of the circleto the other and passes through the centre.  \nRadius  \nA straight line that goes from the centre Tangent to the circumference of the circle. A straight line that touches the circle at one point.  \nArc  \nA part of the circle’s circumference.  \nSector  \nThe area enclosed by two radii and an arc. It looks like a slice of pizza!  \nSegment  \nThe area enclosed by a chord and an arc.  \nCircumference  \nC = πd or C = 2πr C = πd  \n5cm = π x 5  \n= 5π  \n= 15.7 cm to 1 d.p.  \nArea  \n10cm  \nA = πr2 = π x 102  \n= 100π  \n= 314.2 cm2 to 1 d.p.  \nTake care when working with semi circles or compound shapes involving circles.  \nLength of an arc  \nLength of arc = 3~~θ~~60 × r  \nθ 2.5cm  \nLength of arc = 3~~θ~~60 ×= 120360 × π × 5 = 5.2cm to 1 d.p.  \nπd  \n120°  \n2.5cm  \nπd  \nArea of a sector  \nArea of sector =  \n3~~θ~~60 × πr2  \n10cm  \nArea ofsector = 3~~θ~~60 × 2  \nπr  \n= 80360 × π × 102  \n= 69.8 cm2 to 1 d.p.  \nVolume of a cylinder  \nV = πr2h  \nr = 3cm  \n9m  \n= π x 32 x 9 = 81π  \n= 254.5 m3 to 1 d.p.  \nSurface area of a cylinder  \nSA = 2πr2 + 2πrh  \n\n| \u003Cbr>2πr\u003Cbr>| \u003Cbr>h\u003Cbr>2πrh |\n| --- | --- |\n\nVolume of a cone  \nV =  πr2h  \n=  π x 42 x 6 = 32π  \n= 100.5 cm3  \nto 1 d.p.  \nSurface area of a cone  \nSA = πr2 + πrl  \nπrl  \nπr2  \nh l   \nr  \n* l2 = h2 + r2 l = h2 + r2  \nCircle Theorems  \nThe tangent is perpendicular [i.e. at 90](i.e. at 90)° to the radius.  \nThe lengths of two tangents from a point are always equal.  \nThe angle between a tangent and a chord is equal to the angle in the alternate segment.  \nb  \na b  \na  \nThe angle in a semicircle is a right-angle i.e. 90°.  \ny + 39 + 90 = 180  \ny + 129 = 180  \ny = 180 − 129 = 51°  \n39°  \ny  \nOpposite angles in a cyclic quadrilateral add to 180°.  \na  \nb  \nc  \nd  \na + d = 180 ° b + c = 180 °  \nt + 98 = 180 t = 180 − 98 = 82°  \nt  \n98°  \nThe angle at the centre is double the angle atthe circumference.  \na  \n2a  \nz = 2 x 56 = 112°  \n56°  \nz  \nAngles in the same segment are equal.  \na  \na  \np = 61°  \nb  \n61°  \np  \nb  \nQuestions involving more than one theorem  \nExample  \nFind the angles marked p and q. You must give a reason for each of your answers.  \np  \n118°  \nq  \n* Diagram not drawn to scale  \np = 118 ÷ 2  \np = 59°  \nThis is because the angle at the centre of the circle is double the angle at the circumference.  \nq + p = 180°  \nq + 59 = 180  \nq = 180 – 59  \nq = 121°  \nThis is because opposite angles in a cyclic quadrilateral add to 180° .  \nCheck that you:  \n• recognise 2D shapes and their properties  \n• understand the difference between perimeter, area and volume  \n• can substitute values into a formula  \n• can find missing angles on a straight line, at a point and in a triangle.  \nVolume of a Sphere  \nV = 43~~ ~~ πr3  \n12cm  \n= 43~~ ~~ π × 123  \n= 2304π  \n= 7238.2cm3 to 1 d.p.  \nSurface area of a sphere SA = 4πr2  \nArea of a segment  \nr θ  \nr  \nArea of segment = area of sector – area of triangle  \n=  \n θ   \n360  \n× πr2 –  \n1 2  \nr2Sinθ  \n*  \n1 2  \nabSinC  \nC  \nb a  \nA c B  \nArea of segment = 3~~θ~~60 × πr2 – 12 r2Sinθ= 72360 × π × 102 – 12 × 102 × Sin72 = 15.3cm2 to 1 d.p.  \n72°  \n10cm  \n10cm  \n6m  \nπr2 πr2","cbCaiaHYmdNqBYg4","https://ap.wps.com/l/cbCaiaHYmdNqBYg4","pdf",878624,2,1,"English","en",105,"# Key terms\n## Geometric definitions\n# Circle measurements and formulas\n## Circumference, area, arcs, sectors\n## Cylinder, cone, and sphere\n# Circle theorems\n## Tangents and angles\n## Cyclic quadrilaterals and segment theorems\n# Worked example\n## Finding missing angles","[{\"question\":\"What is the definition of circumference and how is it calculated?\",\"answer\":\"Circumference is the distance around the outside of the circle. Use C = πd or C = 2πr.\"},{\"question\":\"How do you find the area of a sector?\",\"answer\":\"Use the sector formula Area of sector = (θ/360) × πr², substituting θ and r to get the required value.\"},{\"question\":\"What is the tangent-radius angle rule in circle theorems?\",\"answer\":\"The tangent is perpendicular to the radius at the point of tangency, meaning the angle is 90°.\"}]","Circles - Key Terms, Formulas and Circle Theorems | PDF",1788601082,3,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":84,"head_meta":86,"extra_data":88,"updated_unix":28},"circles-key-terms-formulas-and-circle-theorems","",{"@graph":36,"@context":83},[37,51,66],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,49],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":29},"https://docshare.wps.com/document/exam/",{"item":50,"name":13,"@type":43,"position":11},"https://docshare.wps.com/document/circles-key-terms-formulas-and-circle-theorems/207766/",{"url":50,"name":13,"@type":52,"author":53,"headline":13,"publisher":55,"fileFormat":58,"inLanguage":23,"description":14,"dateModified":59,"datePublished":60,"encodingFormat":58,"isAccessibleForFree":61,"interactionStatistic":62},"DigitalDocument",{"name":9,"@type":54},"Person",{"url":41,"name":56,"@type":57},"DocShare","Organization","application/pdf","2026-09-08","2026-09-05",true,{"@type":63,"interactionType":64,"userInteractionCount":20},"InteractionCounter",{"@type":65},"ViewAction",{"@type":67,"mainEntity":68},"FAQPage",[69,75,79],{"name":70,"@type":71,"acceptedAnswer":72},"What is the definition of circumference and how is it calculated?","Question",{"text":73,"@type":74},"Circumference is the distance around the outside of the circle. Use C = πd or C = 2πr.","Answer",{"name":76,"@type":71,"acceptedAnswer":77},"How do you find the area of a sector?",{"text":78,"@type":74},"Use the sector formula Area of sector = (θ/360) × πr², substituting θ and r to get the required value.",{"name":80,"@type":71,"acceptedAnswer":81},"What is the tangent-radius angle rule in circle theorems?",{"text":82,"@type":74},"The tangent is perpendicular to the radius at the point of tangency, meaning the angle is 90°.","https://schema.org",{"og:url":50,"og:type":85,"og:title":13,"og:site_name":56,"og:description":14},"article",{"robots":87,"canonical":50},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":90},[91,95,99,102,107,112,117,122,127,130,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":92,"show_sort_weight":93,"slug":94},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":96,"show_sort_weight":97,"slug":98},"Literature",80,"literature",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":100,"slug":101},70,"exam",{"id":103,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},5,"Comic",60,"comic",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},6,"Technology",50,"technology",{"id":113,"doc_module":4,"doc_module_name":46,"category_name":114,"show_sort_weight":115,"slug":116},7,"Healthcare",40,"healthcare",{"id":118,"doc_module":4,"doc_module_name":46,"category_name":119,"show_sort_weight":120,"slug":121},8,"Research & Report",30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":103,"slug":137},19,"General","general"]