{"id":203,"date":"2017-08-23T09:57:54","date_gmt":"2017-08-23T09:57:54","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=203"},"modified":"2017-08-30T09:56:29","modified_gmt":"2017-08-30T09:56:29","slug":"proportions-in-measurement","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/proportions-in-measurement\/","title":{"rendered":"Proportions in Measurement"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/measuring-three-dimensional-figures\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/measurement-and-linear-algebra\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/matrices-operations-and-inverses\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Proportions in Measurement<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will use proportions to compute arc lengths and areas of sectors of circles. You will also apply\u00a0proportions to interpret scale drawings and maps.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>The area of a circle is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s4_p1_clip_image003.gif\" width=\"27\" height=\"21\" name=\"graphics2\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0where <em><span style=\"text-decoration: none;\">r<\/span><\/em> is\u00a0the radius of the circle.<\/li>\n<li>A <em><strong>sector<\/strong><\/em> is\u00a0a portion of a circle with boundaries consisting of an arc on the\u00a0circle\u2019s circumference and two radii from the center of the\u00a0circle to the endpoints of the arc.<\/li>\n<li>The <strong><em>circumference\u00a0<\/em><\/strong>of a circle is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s4_p1_clip_image006.gif\" width=\"31\" height=\"17\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0where <em>r<\/em> is the radius of the circle.<\/li>\n<li>The <em><strong>central angle<\/strong><\/em> of a circle is\u00a0360 degrees, or\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s4_p1_clip_image009.gif\" width=\"24\" height=\"17\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0radians.<\/li>\n<\/ul>\n<section>\n<h3>How do you find the area of a sector of a circle?<\/h3>\n<p>A sector is a portion of a circle. The area of a sector can be\u00a0found by setting up a proportion using the angle measure of the\u00a0sector, the central angle of the circle, and the area of the\u00a0entire circle.<\/p>\n<p>Let\u2019s try an example. Find the area of a sector with an\u00a0angle of 20\u02da in a circle with radius 15 meters.<\/p>\n<p>The entire area of the circle would be\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/s4_p2_html_m1a1680f5.gif\" width=\"144\" height=\"29\" name=\"graphics3\" align=\"absmiddle\" border=\"0\" \/>\u00a0square meters.<\/p>\n<p>The area of the sector is proportionate to the ratio of the\u00a0angle of the sector over the central angle of the circle.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/s4_p2_html_2fdb297c.gif\" width=\"153\" height=\"88\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Thus, the area of the sector would equal\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/s4_p2_html_m4a386e0d.gif\" width=\"41\" height=\"19\" name=\"graphics20\" align=\"BOTTOM\" border=\"0\" \/>\u00a0square meters.<\/p>\n<h3>How do you find arc lengths?<\/h3>\n<p>Finding an arc length is similar to\u00a0finding the area of a sector. The arc length is a part of the\u00a0circumference of a circle, so its length is proportionate in a way\u00a0similar to the way a sector\u2019s area is proportionate to its\u00a0angle.<\/p>\n<p>Find the arc length of the sector of the circle described in\u00a0the previous example.<\/p>\n<p>The circle has a circumference of\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s4_p2_clip_image009.gif\" width=\"135\" height=\"21\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>The arc length can be found using a proportion.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s4_p2_clip_image012.gif\" width=\"128\" height=\"85\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the area of a sector of a circle with radius 12\u00a0centimeters, with measure between radii\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s4_p2_clip_image015.gif\" width=\"28\" height=\"41\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0radians?<\/p>\n<ol>\n<li>9 square centimeters<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s4_p2_clip_image018.gif\" width=\"31\" height=\"17\" name=\"graphics8\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0square centimeters<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s4_p2_clip_image021.gif\" width=\"37\" height=\"17\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0square centimeters<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s4_p2_clip_image024.gif\" width=\"43\" height=\"23\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0square centimeters<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is B. The ratio of the areas is equal to\u00a0the ratio of the angles.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s4_p2_clip_image027.gif\" width=\"92\" height=\"83\" name=\"graphics11\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the arc length of a\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s4_p2_clip_image030.gif\" width=\"28\" height=\"41\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/>-radian\u00a0sector of a circle with radius 12 centimeters?<\/p>\n<ol>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s4_p2_clip_image033.gif\" width=\"23\" height=\"17\" name=\"graphics13\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0centimeters<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s4_p2_clip_image036.gif\" width=\"29\" height=\"17\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0centimeters<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s4_p2_clip_image039.gif\" width=\"31\" height=\"17\" name=\"graphics15\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0centimeters<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s4_p2_clip_image042.gif\" width=\"37\" height=\"17\" name=\"graphics16\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0centimeters<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The correct choice is A. The ratio of the arc lengths is equal\u00a0to the ratio of the angles.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s4_p2_clip_image045.gif\" width=\"95\" height=\"83\" name=\"graphics17\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<\/div>\n<\/section>\n<h3>How do proportions apply to scale drawings?<\/h3>\n<p>Maps\u00a0of states or drawings of buildings are not drawn life sized because it\u00a0would never fit on the paper. Instead a <abbr title=\"a proportional drawing of an object. If the scale is 0.1, then the dimensions of the drawing are one-tenth of the original. If the scale is 4, then the dimensions of the drawing are four times the original.\">scale drawing<\/abbr> is made. The scale is\u00a0usually identified on the map or drawing.<\/p>\n<p>Let\u2019s\u00a0try an example. Find the length of the actual car if a <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s4_p3_clip_image003.gif\" width=\"23\" height=\"43\" name=\"graphics3\" align=\"middle\" border=\"0\" \/>\u00a0scale model is 12 inches long.<\/p>\n<p>The\u00a0ratio of the length of the model to the actual car is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s4_p3_clip_image006.gif\" width=\"23\" height=\"43\" name=\"graphics4\" align=\"middle\" border=\"0\" \/>.<\/p>\n<p align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s4_p3_clip_image009.gif\" width=\"264\" height=\"43\" name=\"graphics5\" align=\"bottom\" border=\"0\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>The\u00a0Washington Monument is 555 feet tall. If a scale model is made that is\u00a030 inches tall, which of the following represents the scale of the\u00a0model?<\/p>\n<ol>\n<li>1:18.5<\/li>\n<li>1:22.2<\/li>\n<li>1:185<\/li>\n<li>1:222<\/li>\n<\/ol>\n<\/div>\n<p><a class=\"button button-primary q-answer\"> Reveal Answer <\/a><\/p>\n<div class=\"q-reveal\">\n<p>The\u00a0correct choice is D. Convert 555 ft to inches: (555)(12)=6,660 in. The\u00a0scale of the model is 30 in\/6,660 in = 1\/222.<\/p>\n<\/div>\n<\/section>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/measuring-three-dimensional-figures\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/measurement-and-linear-algebra\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/matrices-operations-and-inverses\">Next Lesson \u27a1<\/a><\/div>\n<p><a class=\"backtotop\" href=\"#title\">Back to Top<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\u2b05 Previous Lesson\u00a0Workshop Index\u00a0Next Lesson \u27a1 Proportions in Measurement Objective In this lesson, you will use proportions to compute arc lengths and areas of sectors of circles. You will also apply\u00a0proportions to interpret scale drawings and maps. Previously Covered: The area of a circle is\u00a0,\u00a0where r is\u00a0the radius of the circle. A sector is\u00a0a portion [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-203","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/203","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/comments?post=203"}],"version-history":[{"count":5,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/203\/revisions"}],"predecessor-version":[{"id":558,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/203\/revisions\/558"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=203"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}