[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-297384-105":59,"doc-detail-297384-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","118-surface-areas-and-volumes-of-spheres","11.8 - Surface Areas and Volumes of Spheres","","Lesson 11.8 explains how to compute the surface area and volume of a sphere. It defines key terms such as center, radius, chord, diameter, great circle, and hemispheres, then derives the surface area formula S = 4πr² and uses a baseball covering analogy to justify it. The volume section uses a hemisphere, circumscribed cylinder, and cone relationship with Cavalieri’s Principle to obtain V = (4/3)πr³, reinforced by worked examples and unit-based problem solving.",{"@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/118-surface-areas-and-volumes-of-spheres/297384/",{"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/118-surface-areas-and-volumes-of-spheres/297384.png","ImageObject",300,407,{"name":92,"@type":93},"Olivia Brown","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-20","2026-09-18",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",11,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"How do you find the surface area of a sphere?","Question",{"text":113,"@type":114},"Use the formula S = 4πr², where r is the sphere’s radius. Substitute the radius and simplify, then approximate with a calculator if needed.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What key definitions are used in this lesson?",{"text":118,"@type":114},"The lesson defines the center, radius, chord, diameter, great circle, and hemispheres. It also notes that the diameter equals twice the radius and a plane through the center creates a great circle.",{"name":120,"@type":111,"acceptedAnswer":121},"How do you find the volume of a sphere?",{"text":122,"@type":114},"Use V = (4/3)πr³ (written in the lesson as 2·Vhemisphere). The derivation relates a sphere to a hemisphere and a cylinder with a cone removed, using Cavalieri’s Principle.","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},297384,1789744176,{"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},16904993612988,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","11.8  \nSurface Areas and Volumes of Spheres  \nEssential Question How can you find the surface area and the  \nvolume of a sphere?  \nUSING TOOLS STRATEGICALLY  \nFinding the Surface Area of a Sphere Work with a partner. Remove the covering from a baseball or softball.  \nYou will end up with two “figure 8” pieces of material, as shown above. From the amount of material it takes to cover the ball, what would you estimate the surface area S of the ball to be? Express your answer in terms of the radius r of the ball.  \nS =  Surface area of a sphere  \nUse the Internet or some other resource to confirm that the formula you wrote for the surface area of a sphere is correct.  \n Finding the Volume of a Sphere  \nWork with a partner. A cylinder is circumscribed about a r  \nsphere, as shown. Write a formula for the volume V of the cylinder in terms of the radius r.  \nV =  \nVolume of cylinder  \nWhen half of the sphere (a hemisphere) is filled with sand and poured into the cylinder, it takes three hemispheres to fill the cylinder. Use this information to write a formula for the volume V of a sphere in terms of the radius r.  \nV =  \nVolume of a sphere  \nr 2r  \nCommunicate Your Answer  \n3. How can you find the surface area and the volume of a sphere?  \n4. Use the results of Explorations 1 and 2 to find the surface area and the volume of a sphere with a radius of (a) 3 inches and (b) 2 centimeters.  \nSection 11.8 Surface Areas and Volumes of Spheres 647  \n11.8 Lesson  \nWhat You Will Learn  \n Find surface areas of spheres.  \n Find volumes of spheres.  \nFinding Surface Areas of Spheres  \nA sphere is the set of all points in space equidistant from a given point. This point is called the center of the sphere. A radius of a sphere is a segment from the center toa point on the sphere. A chord of a sphere is a segment whose endpoints are on the sphere. A diameter of a sphere is a chord that contains the center.  \nC center  \nradius  \ndiameter  \nAs with circles, the terms radius and diameter also represent distances, and the diameter is twice the radius.  \nIf a plane intersects a sphere, then the intersection is either a single point or a circle. If the plane contains the center of the sphere, then the intersection is a great circle of the sphere. The circumference of a great circle is the circumference of the sphere. Every great circle of a sphere separates the sphere into two congruent halves called hemispheres.  \nCore Concept  \ngreat circle  \nhemispheres  \nSurface Area of a Sphere  \nThe surface area S of a sphere is S = 4 πr2  \nwhere r is the radius of the sphere.  \nTo understand the formula for the surface area of a sphere, think of a baseball. The surface area of a baseball is sewn from two congruent shapes, each of which resembles two joined circles.  \nSo, the entire covering of the baseball consists of four circles, each with radius r. The area A of a circle with radius r is A = πr2. So, the area of the covering can be approximated by 4 πr2. This is the formula for the surface area of a sphere.  \n648 Chapter 11 Circumference, Area, and Volume  \nCOMMON ERROR  \nFinding the Surface Areas of Spheres  \nFind the surface area of each sphere.  \na.  \n8 in.  \nSOLUTION  \nb.  \nC = 12π ft  \na. S = 4 πr2 Formula for surface area of a sphere  \n= 4 π(8)2 Substitute 8 for r.  \n= 256 π Simplify.  \n≈ 804.25 Use a calculator.  \n The surface area is 256 π, or about 804.25 square inches.  \nb. The circumference of the sphere is 12 π, so the radius of the sphere is  = 6 feet. S = 4 πr2 Formula for surface area of a sphere  \n= 4 π(6)2 Substitute 6 for r.  \n= 144 π Simplify.  \n≈ 452.39 Use a calculator.  \n The surface area is 144 π, or about 452.39 square feet.  \nFinding the Diameter of a Sphere  \nFind the diameter of the sphere.  \nSOLUTION  \nS = 4 πr2 Formula for surface area of a sphere 20.25 π = 4 πr2 Substitute 20.25 π for S.  \n5.0625 = r2 Divide each side by 4 π.  \n2.25 = r Find the positive square root.  The diameter is 2r = 2 • 2.25 = 4.5 centimeters.  \nS = 20.25π cm","cbCaiqGbsrlhfS5S","https://ap.wps.com/l/cbCaiqGbsrlhfS5S","pdf",3736354,"English","# Section 11.8 Surface Areas and Volumes of Spheres\n## What You Will Learn\n## Finding Surface Areas of Spheres\n## Finding the Diameter of a Sphere\n## Finding Volumes of Spheres","[{\"question\":\"How do you find the surface area of a sphere?\",\"answer\":\"Use the formula S = 4πr², where r is the sphere’s radius. Substitute the radius and simplify, then approximate with a calculator if needed.\"},{\"question\":\"What key definitions are used in this lesson?\",\"answer\":\"The lesson defines the center, radius, chord, diameter, great circle, and hemispheres. It also notes that the diameter equals twice the radius and a plane through the center creates a great circle.\"},{\"question\":\"How do you find the volume of a sphere?\",\"answer\":\"Use V = (4/3)πr³ (written in the lesson as 2·Vhemisphere). The derivation relates a sphere to a hemisphere and a cylinder with a cone removed, using Cavalieri’s Principle.\"}]","11.8 - Surface Areas and Volumes of Spheres | PDF"]