[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-267097-105":59,"doc-detail-267097-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","act-math-drills-geometry-circles-angles-lengths","ACT Math Drills -Geometry -Circles -Angles &Lengths","","Geometry practice set focused on circles, angles, arcs, tangents, and chord-related measurements. Problems use given central angles, radii, diameters, intersecting chords, and tangent conditions to determine angle measures, minor arc lengths, and arc circumferential distances. Includes arc length and circumference/area applications such as circle graphs, rectangles and squares inscribed in circles, and cylinder-related surface area and volume using πr² and 2πrh formulas. Multiple-choice answer options support timed drill practice.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/act-math-drills-geometry-circles-angles-lengths/267097/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/act-math-drills-geometry-circles-angles-lengths/267097.png","ImageObject",300,407,{"name":92,"@type":93},"Đào","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-04","2026-09-14",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",18,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"How do you find an inscribed or central-angle-related arc measure in these circle problems?","Question",{"text":113,"@type":114},"Use the relationship between central angles and arcs: the measure of an arc equals the central angle that intercepts it. Then convert arc measures to the requested angle or arc length form when needed.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What information is required to determine a minor arc’s degree measure when chords intersect?",{"text":118,"@type":114},"You need the measure relating to the circle geometry shown (such as a given angle like ∠PST) and the fact that P is the center. With the center identified, the angle can determine the intercepted arc.",{"name":120,"@type":111,"acceptedAnswer":121},"Which formulas are used for cylinder questions in this set?",{"text":122,"@type":114},"Right-cylinder volume uses πr²h, and lateral surface area uses 2πrh. These appear directly in the notes alongside the practice problems.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},267097,1789414766,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":39,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":130,"read_time":46},1374402968488,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","10.In the figure below,G is the center of the circle,LK isa diameter,H lies on the circle,Jlies outside the circleon Lk,and jM is tangent to the circle at M.Which ofthe following angles or minor arcs has the greatestdegree measure?  \nG.M  \nH.∠JMG  \nJ.∠LHK  \nK.∠MJL  \n38.Points A and B lie on the circle below,where centralangle ∠ACB measures 110°.What is the measure of∠ABC?  \nB  \nF.35°  \nG.40°  \nH.45°  \nJ.55°  \nK.Cannot be determined from the given information  \n48.In the circle shown below,chords TR and QS intersectat P,which is the center of the circle,and the measureof ∠PST is 30°.What is the degree measure of minorarc RS?  \nF.30°G.45°  \nH.60°  \nJ.90°  \nK.Cannot be determined from the given information  \n13.In the figure above,lines m and n are tangent to the circle at points Pand Q,respectively,and Cis the center of the circle.Ifx=130,what isy?  \nACT Math Drills -Geometry  \nGeometry -Circles -Angles &Lengths  \n47.A circle with radius 10cm is divided into 3 congruentarcs.What is the length,in centimeters,of each of the3 arcs?  \nA.  \nB.  \nC.10π  \nD.  \nE.20π  \n53.Douglas wants to draw a circle graph showing thefavorite colors of his friends.When he polled hisfriends asking each their favorite color,25%of hisfriends said red;30%of his friends said blue;20%ofhis friends said green;10%of his friends said purple;and the remaining friends said colors other than red,blue,green,and purple.The colors other than red,blue,green,and purple willbe grouped together in anOther sector.What will be the degree measure of theOther sector?  \nA.108°  \nB.  54°  \nC.27°D.  15°  \nE.  10°  \nIn the figure shown below,ABCD is a rectangle,EFGH is asquare,and CD is the diameter of a semicircle.Point K isthe midpoint of CD.Point Jis the midpoint of both AB andEF.Points E and F lie on AB.The 3 given lengths are inmeters.  \n35.What is the length,in meters,of arc CD?  \nA.2.5π  \nC.6.25πD.10π  \nE.25π  \n25.A chord 24 inches long is 5 inches from the center of acircle,as shown below.What is the radius of thecircle,to the nearest tenth of an inch?  \nA.29.0  \nB.24.5  \nC.16.9  \nD.13.0  \nE.10.9  \n16.Four points,A,B,C,and D,lie on a circle having acircumference of 17 units.B is 5 units counterclockwisefrom A.C is 3 units clockwise from A.D is 11 unitsclockwise from A and 6 units counterclockwise from A.What is the order of the points,starting with A andgoing clockwise around the circle?  \nF.A,B,C,D  \nG.A,B,D,C  \nH.A,C,B,D  \nJ.A,C,D,B  \nK.A,D,C,B  \n46.The ratio of the radii of two circles is 4:9.What is theratio of their circumferences?  \nF.  2:3  \nG.4:9  \nH.16:81  \nJ.  4:8π  \nK.  9:18π  \n43.Points M and N are the endpoints of the diameter of acircle with center at O,as shown below.Point P is onthe circle,and ∠MOP measures 60°.The shortest dis-tance along the circle from M to Pis what percent ofthe distance along the circle from M to N?  \nA.75%  \nB.60%  \nC.50%  \nD.  \nE.  \n47.A circle with radius 10 cm is divided into 3 congruentarcs.What is the length,in centimeters,of each of the3 arcs?  \nA.  \nB.  \nC.10π  \nD.  \nE.20π  \n57.As shown in the figure below,a clock has a minutehand that measures 5 cm from ts tip to the center ofthe clock.To the nearest centimeter,what is thedistance traveled by the tip of the minute hand between  \n2:10 p.m.and 5:30 p.m.?  \nA.31  \nB.  68  \nC.101  \nD.105  \nE.173  \nACT Math Drills -Geometry  \nGeometry -Circles -Circumference &Area  \nNote:Figure not drawn to scale.  \n13.In the figure above,the smaller circles bothhave a radius of 3 and are tangentto the largercircle.If the larger circle has a radius of7,what is the area of the shaded region?  \n(A)18π  \n(B)31π  \n(C)40π  \n(D)49π  \n(E)64π  \n18.In the figure above,ABCD is a square,andAand C are the centers of the circles.IfAB=2,what is the total area of the shaded regions?(A)3π(B)6π(C)8π(D)8π-4(E)9π-4  \n30.The figure below shows 2 tangent circles such that the10-centimeter diameter of the smaller circle is equal tothe radius of the larger circle.What is the area,insquare centimeters,of the shaded reg","cbCaiqO0rcELFdht","https://ap.wps.com/l/cbCaiqO0rcELFdht","pdf",4224041,"English","# Circles - Angles & Lengths\n# Circles - Circumference & Area\n# Cylinders & Formulas","[{\"question\":\"How do you find an inscribed or central-angle-related arc measure in these circle problems?\",\"answer\":\"Use the relationship between central angles and arcs: the measure of an arc equals the central angle that intercepts it. Then convert arc measures to the requested angle or arc length form when needed.\"},{\"question\":\"What information is required to determine a minor arc’s degree measure when chords intersect?\",\"answer\":\"You need the measure relating to the circle geometry shown (such as a given angle like ∠PST) and the fact that P is the center. With the center identified, the angle can determine the intercepted arc.\"},{\"question\":\"Which formulas are used for cylinder questions in this set?\",\"answer\":\"Right-cylinder volume uses πr²h, and lateral surface area uses 2πrh. These appear directly in the notes alongside the practice problems.\"}]","ACT Math Drills -Geometry -Circles -Angles &Lengths | PDF"]