[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-1-en-105":3,"doc-seo-572664-105":53,"doc-detail-572664-en":127},{"code":4,"msg":5,"data":6},0,"success",[7,14,19,24,29,34,39,44,49],{"id":8,"doc_module":9,"doc_module_name":10,"category_name":11,"show_sort_weight":12,"slug":13},11,1,"Template","Presentations",90,"presentations",{"id":15,"doc_module":9,"doc_module_name":10,"category_name":16,"show_sort_weight":17,"slug":18},12,"Resumes",80,"resumes",{"id":20,"doc_module":9,"doc_module_name":10,"category_name":21,"show_sort_weight":22,"slug":23},14,"Invoices",70,"invoices",{"id":25,"doc_module":9,"doc_module_name":10,"category_name":26,"show_sort_weight":27,"slug":28},15,"Posters",60,"posters",{"id":30,"doc_module":9,"doc_module_name":10,"category_name":31,"show_sort_weight":32,"slug":33},16,"Social Media",50,"social-media",{"id":35,"doc_module":9,"doc_module_name":10,"category_name":36,"show_sort_weight":37,"slug":38},17,"Forms",40,"forms",{"id":40,"doc_module":9,"doc_module_name":10,"category_name":41,"show_sort_weight":42,"slug":43},18,"Letters",30,"letters",{"id":45,"doc_module":9,"doc_module_name":10,"category_name":46,"show_sort_weight":47,"slug":48},21,"Paper Templates",5,"papers-templates",{"id":50,"doc_module":9,"doc_module_name":10,"category_name":51,"show_sort_weight":4,"slug":52},158,"General","general-158",{"code":4,"msg":54,"data":55},"ok",{"site_id":56,"language":57,"slug":58,"title":59,"keywords":60,"description":61,"schema_data":62,"social_meta":120,"head_meta":122,"extra_data":124,"updated_unix":126},105,"en","harolds-geometry-circle-theorems-cheat-sheet-6-august-2026","Harold’s Geometry - Circle Theorems Cheat Sheet - 6 August 2026","","Concise geometry reference summarizing key circle theorems, relationships, and measurement rules. Covers terminology for angles, arcs, sectors, chords, and tangents; internal arc/angle properties such as central, inscribed, semi-circle right angles, same segment, and alternate segment theorems; and external configurations involving secants, tangents, and supplementary angle results. Also includes chord theorems and key circle metrics for circumference, arc length, sector area, and sphere surface area/volume.",{"@graph":63,"@context":119},[64,80,102],{"@type":65,"itemListElement":66},"BreadcrumbList",[67,71,74,77],{"item":68,"name":69,"@type":70,"position":9},"https://docshare.wps.com","Home","ListItem",{"item":72,"name":10,"@type":70,"position":73},"https://docshare.wps.com/template/",2,{"item":75,"name":51,"@type":70,"position":76},"https://docshare.wps.com/template/general/",3,{"item":78,"name":59,"@type":70,"position":79},"https://docshare.wps.com/template/harolds-geometry-circle-theorems-cheat-sheet-6-august-2026/572664/",4,{"url":78,"name":59,"@type":81,"image":82,"author":87,"headline":59,"publisher":90,"fileFormat":93,"inLanguage":57,"description":61,"dateModified":94,"datePublished":95,"encodingFormat":93,"isAccessibleForFree":96,"interactionStatistic":97},"DigitalDocument",{"url":83,"@type":84,"width":85,"height":86},"https://docshare.wps.com/thumbnails/harolds-geometry-circle-theorems-cheat-sheet-6-august-2026/572664.png","ImageObject",442,249,{"name":88,"@type":89},"Logic","Person",{"url":68,"name":91,"@type":92},"DocShare","Organization","application/pdf","2026-10-22","2026-10-13",true,{"@type":98,"interactionType":99,"userInteractionCount":101},"InteractionCounter",{"@type":100},"ViewAction",10,{"@type":103,"mainEntity":104},"FAQPage",[105,111,115],{"name":106,"@type":107,"acceptedAnswer":108},"What is the central angle relationship to an arc or chord?","Question",{"text":109,"@type":110},"A central angle equals the measure of its intercepted arc, and the angle at the center is twice the angle standing on the same chord or arc.","Answer",{"name":112,"@type":107,"acceptedAnswer":113},"How do inscribed angles relate to the arc they intercept?",{"text":114,"@type":110},"An inscribed angle that intercepts the same arc are congruent, and an inscribed angle equals half the measure of its intercepted arc. Angles inscribed in a semicircle form a right angle (90°).",{"name":116,"@type":107,"acceptedAnswer":117},"What are the key formulas for circle measurement shown here?",{"text":118,"@type":110},"The sheet lists circumference and arc length rules, plus circle area and sector area. It also provides sphere surface area and volume formulas.","https://schema.org",{"og:url":78,"og:type":121,"og:title":59,"og:site_name":91,"og:description":61},"article",{"robots":123,"canonical":78},"index,follow",{"doc_id":125,"site_id":56},572664,1792005716,{"code":4,"msg":5,"data":128},{"doc_id":125,"user_id":129,"nickname":88,"user_avatar":130,"doc_module":9,"category_id":50,"category_name":51,"doc_title":59,"doc_description":61,"doc_content":131,"file_id":132,"file_url":133,"file_type":134,"file_size":135,"view_count":101,"is_deleted":4,"is_public":9,"is_downloadable":9,"audit_status":9,"page_count":8,"language":136,"language_code":57,"site_id":56,"html_lang":57,"table_of_contents":137,"faqs":138,"seo_title":139,"seo_description":61,"update_tm":140,"read_time":79},1099513958762,"https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1784791008015729253","Harold’s Geometry – Circle Theorems Cheat Sheet  \n6 August 2026  \nTerminology  \n\n| Category | Examples |  |  |  |\n| --- | --- | --- | --- | --- |\n| Angles | |  |  | |\n|  | Degrees Radians |  |  |  |\n| Arcs & Sectors | |  | |  |\n| Lines, Chords & Tangents | | |  |  |\n\nInternal Arcs and Angles  \n\n| Configuration | Rule / Formula | Diagram |\n| --- | --- | --- |\n| Central Angle\u003Cbr>(Angle at Center) | Equal to arc\u003Cbr>􀟠 = 􀝔 °\u003Cbr>􀝉∠􀜣􀜱􀜥 = 􀝉̂􀜣􀜥 | |\n| Inscribed Angle\u003Cbr>(Angle in Same Segment) | Half the arc 1 °\u003Cbr>􀟠 = 2 􀝔 | |\n| Inscribed Quadrilateral\u003Cbr>(Opposite Angles of Cyclic Quadrilateral) | 􀝉∠􀜣 + 􀝉∠􀜥 = 180°\u003Cbr>􀝉∠􀜤 + 􀝉∠􀜦 = 180°\u003Cbr>The opposite angles of cyclic quadrilaterals are supplementary (180°) . | |\n| Two Chords (Internal Angle) | Half the sum\u003Cbr>1 ° °􀟠 = 2 (􀝔 + 􀝕 ) | |\n\n\n| Angle at Center | 2􀝔 ° 􀝒􀝏 . 􀝔 °\u003Cbr>The angle at the center is twice the angle standing on the same chord/arc. | \u003Cbr>Symmetrical Asymmetrical |\n| --- | --- | --- |\n| Angles Inscribed in a Semi-Circle | Right Angles\u003Cbr>(90°)\u003Cbr>An angle inscribed in a semicircle is a right angle (90°) . | |\n| Angles Inscribed in a Circle | Inscribed angles that intercept the same arc are congruent. | |\n| Same Segment Theorem\u003Cbr>(Two Inscribed Angles) | 􀝔 ° = 􀝔 °􀝕 ° = 􀝕 °\u003Cbr>Angles subtended by the same chord in the same segment are equal. | |\n| Alternate Segment Theorem | 􀝔 ° = 􀝔 °􀝕 ° = 􀝕 °\u003Cbr>The angle between a chord and a tangent is equal to the angle in the alternate segment. | |\n\n\n| Angles at the Center | 􀟠 = | 360°􀝊 | |\n| --- | --- | --- | --- |\n|  | The sum of the central angles of a circle is always 360° . |  |  |\n\nExternal Tangents and Secants  \n\n| Configuration | Rule / Formula | Diagram |\n| --- | --- | --- |\n| Radius Ʇ Tangent | The radius drawn to the point of tangency is perpendicular (90°) to the tangent. | |\n| Two Secants (External Angle) | Half the difference\u003Cbr>1 ° °\u003Cbr>􀟠 = 2 (􀝔 − 􀝕 )\u003Cbr>􀝉∠􀜦 = 12 (􀝉̂􀜧􀜨 − 􀝉 ̂􀜩􀜪) | |\n| Secant & Tangent (External Angle) | 􀝉∠􀜳 = 12 (􀝉̂􀜴􀜵 − 􀝉̂􀜴􀜶) | |\n\n\n| Two Tangents (External Angle) | 􀝉∠􀜮 = 12 (􀝉̂􀜯􀜲􀜰 − 􀝉̂􀜯􀜰) | |\n| --- | --- | --- |\n| Tangent and Intersected Chord Theorem | 􀝉∠1 = 12 (􀝉̂􀜣􀜥)\u003Cbr>􀝉∠2 = 12 (􀝉̂􀜣􀜦􀜥)\u003Cbr>If a tangent and a chord intersect at a point on acircle, then the measure of each angle formed is one-half the measure of its intercepted arc. | |\n| Supplementary Angles | 􀝉∠1 + 􀝉∠2 = 180°\u003Cbr>Adjacent angles that form alinear pair are supplementary. | |\n\nInternal Chords  \n\n| Configuration | Rule / Formula | Diagram |\n| --- | --- | --- |\n| Perpendicular Bisector of Chord Passes Through Center | The line from the center of a circle to the midpoint of a chord is perpendicular to the chord.\u003Cbr>A perpendicular line from the chord to the center bisects the chord. | |\n| Equal Chords Equidistant from Center | Equal chords are equal perpendicular distances from the center.\u003Cbr>Chords that are equidistant from the center are equal in length. | |\n| Equal Arcs, Equal Chords | Equal chords correspond to equal arcs and equal central angles. | |\n| Euclid’s Intersecting Chords Theorem | 􀜽 ∙ 􀜾 = 􀜿 ∙ 􀝀\u003Cbr>The intersecting angles do not have tobe right angles. | |\n\nExternal Tangents and Secants  \n\n| Configuration | Rule / Formula | Diagram |\n| --- | --- | --- |\n| Intersecting Secants Theorem | 􀜽 ∙ (􀜽 + 􀜾) = 􀜿 ∙ (􀜿 + 􀝀) | |\n| Intersecting SecantTangent Theorem | 􀜽 (􀜽 + 􀜾) = 􀜿 2 | |\n| Tangents from External Point | Tangent segments drawn from an external point are equal. | |\n\nArea and Perimeter  \n\n| Configuration | Rule / Formula | Diagram |  |  |\n| --- | --- | --- | --- | --- |\n| Radius | 􀝎 | |  |  |\n| Diameter | 􀝀 = 2􀝎 |  |  |  |\n| Circumference | 􀜥 = 2􀟨􀝎 = 􀟨􀝀 |  |  |  |\n| Arc Length | 􀝏 = 􀝎􀟠 where 􀟠 is in radians | 􀝏 = | 􀟠\u003Cbr>360° | (2􀟨􀝎) |\n|  |  | where 􀟠 is in degrees |  |  |\n| Area of Circle | 􀜣 = 􀟨􀝎2 | |  |  |\n| Area of a Sector | 􀜣 = 12 􀝎 2 􀟠\u003Cbr>􀜣 = (~~ ~~3􀟠6°0°~~ ~~ ) ∙ 􀟨􀝎2 􀝓ℎ􀝁􀝎􀝁 (~~ ~~3􀟠6°0°~~ ~~ ) = AAscectoircler | |  |  |\n| Surface Area of Sphere | 􀜵􀜣 = 4􀟨􀝎2 | |  |  |\n| Volume of Sphere | 􀜸 = 43 􀟨􀝎3 | |  |  |","cbCaiquW7Eu2jGZP","https://ap.wps.com/l/cbCaiquW7Eu2jGZP","pdf",999078,"English","# Terminology\n## Angles\n## Arcs & Sectors\n## Lines, Chords & Tangents\n# Internal Arcs and Angles\n## Central and inscribed angle theorems\n## Angles in cyclic quadrilaterals and same/alternate segments\n## Angles and central-angle sum\n# External Tangents and Secants\n## Radius–tangent and external-angle formulas\n## Tangent/cord angle and supplementary angles\n## Tangents from an external point\n# Internal Chords\n## Perpendicular bisector and equidistant chords\n## Equal arcs and chords\n## Euclid’s intersecting chords theorem\n# Area and Perimeter\n## Circle, sector, and arc measurements\n## Sphere surface area and volume","[{\"question\":\"What is the central angle relationship to an arc or chord?\",\"answer\":\"A central angle equals the measure of its intercepted arc, and the angle at the center is twice the angle standing on the same chord or arc.\"},{\"question\":\"How do inscribed angles relate to the arc they intercept?\",\"answer\":\"An inscribed angle that intercepts the same arc are congruent, and an inscribed angle equals half the measure of its intercepted arc. Angles inscribed in a semicircle form a right angle (90°).\"},{\"question\":\"What are the key formulas for circle measurement shown here?\",\"answer\":\"The sheet lists circumference and arc length rules, plus circle area and sector area. It also provides sphere surface area and volume formulas.\"}]","Harold’s Geometry - Circle Theorems Cheat Sheet - 6 August 2026 | PDF",1791928762]