{"id":201,"date":"2017-08-23T09:56:43","date_gmt":"2017-08-23T09:56:43","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=201"},"modified":"2017-09-18T19:00:22","modified_gmt":"2017-09-18T19:00:22","slug":"measuring-two-dimensional-figures","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/measuring-two-dimensional-figures\/","title":{"rendered":"Measuring Two-Dimensional Figures"},"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\/about-measurement\">\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\/measuring-three-dimensional-figures\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Measuring Two-Dimensional Figures<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson you will reflect on connections between formulas for the area and perimeter of\u00a0two-dimensional figures, such as circles and polygons.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>A <em><strong>polygon <\/strong><\/em>is\u00a0a closed figure with sides made up of line segments.<\/li>\n<li>A <strong><em>regular polygon\u00a0<\/em><\/strong>is a convex polygon in which all sides are equal in length and\u00a0angles are equal in measure.<\/li>\n<li>A <strong><em>quadrilateral\u00a0<\/em><\/strong>is a four-sided polygon.<\/li>\n<li>A <strong><em>parallelogram\u00a0<\/em><\/strong>is a quadrilateral which has exactly two pairs of opposite\u00a0parallel sides.<\/li>\n<li>A <em><strong>trapezoid <\/strong><\/em>is\u00a0a quadrilateral with exactly one pair of parallel sides. The <em><strong>perimeter<\/strong><\/em> of a figure is the\u00a0distance around the edge of a figure.<\/li>\n<\/ul>\n<section>\n<h3>How do you keep all the different area formulas straight?<\/h3>\n<p>The formulas for areas of polygons can be examined by their\u00a0similarities.<\/p>\n<p>The area of a rectangle can be found by determining the number\u00a0of unit squares that will cover it. By unit squares, we mean\u00a0squares that each have an area of 1 square unit.<\/p>\n<p>The number of unit squares that cover a rectangle can be\u00a0determined by multiplying the length of the base <em>b <\/em>by the\u00a0height <em>h<\/em> of the rectangle. The area of a rectangle is <em>A\u00a0<\/em>= <em>bh<\/em>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/Math%20Mod%205.1%20Art%20001.JPG\" alt=\"Area of a rectangle\" width=\"189\" height=\"152\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>With a height of four units and base of five units, this\u00a0rectangle has an area of 20 square units.<\/p>\n<p>Because the area of a parallelogram can be easily rearranged to\u00a0form a rectangle, the area formula is the same, <em>A<\/em> = <em>bh<\/em>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/Math%20Mod%205.1%20Art%20002.JPG\" alt=\"Area of a parallelogram\" width=\"269\" height=\"74\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The height of a parallelogram is the verticalheight, measured\u00a0at a 90\u00b0 angle to the base.<\/p>\n<p>Because any pair of congruent triangles can be arranged into a\u00a0parallelogram, the area of a triangle is half the area of a\u00a0parallelogram,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s2_p2_clip_image003.gif\" width=\"61\" height=\"41\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0where <em><span style=\"text-decoration: none;\">h\u00a0<\/span><\/em><span style=\"text-decoration: none;\">is th<\/span>e altitude of the triangle.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/Math%20Mod%205.1%20Art%20003.JPG\" alt=\"Area of a triangle\" width=\"165\" height=\"83\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Because any pair of congruent trapezoids can be arranged into a\u00a0parallelogram, the area of a trapezoid is also based on the area\u00a0of a parallelogram,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s2_p2_clip_image006.gif\" width=\"107\" height=\"41\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0where <em>b<\/em><sub>1<\/sub> and <em>b<\/em><sub>2<\/sub> are the\u00a0lengths of the two parallel sides, and <em>h <\/em>is the vertical\u00a0distance between those two lengths.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/Math%20Mod%205.1%20Art%20004.JPG\" alt=\"Area of a Trapezoid\" width=\"194\" height=\"61\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Any regular polygon can be subdivided into congruent triangles.\u00a0The height of those triangles is called the <abbr title=\"a line segment from the center of a polygon perpendicular to a side of the polygon \"><span style=\"text-decoration: none;\">apothem<\/span><\/abbr><strong><span style=\"text-decoration: none;\">.\u00a0<\/span><\/strong><span style=\"text-decoration: none;\">The area of a regular polygon can be found using the formula\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s2_p2_clip_image009.gif\" width=\"52\" height=\"41\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0where <em>p<\/em> is the perimeter of the polygon<\/span><em><span style=\"text-decoration: none;\">,\u00a0<\/span><\/em><span style=\"text-decoration: none;\">and <em>a<\/em> is\u00a0the length of the apothem. <\/span><\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/Math%20Mod%205.1%20Art%20005.JPG\" alt=\"Area of a polygon\" width=\"150\" height=\"141\" name=\"graphics10\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<h3>What about circles?<\/h3>\n<p>The distance around the edge of a circle is called the\u00a0<abbr title=\"the distance around a circle; MathType, where r is the length of the radius of the circle \">circumference<\/abbr>. That distance is found using the formula\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s2_p3_clip_image003.gif\" width=\"56\" height=\"19\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0where <em>r<\/em> is the radius of the circle.<\/p>\n<p>A circle can be decomposed into wedges. If the wedges are\u00a0infinitely thin, they can be rearranged into a parallelogram with\u00a0height of <em>r<\/em> and a base of half of the circumference, or\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s2_p3_clip_image006.gif\" width=\"45\" height=\"18\" name=\"graphics4\" align=\"TEXTTOP\" border=\"0\" \/>;\u00a0thus, the area is <img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s2_p3_clip_image009.gif\" width=\"122\" height=\"24\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/Math%20Mod%205.1%20Art%20006.JPG\" alt=\" Circle broken into wedge\" width=\"268\" height=\"112\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<h3>Review of New Vocabulary and Concepts<\/h3>\n<ul>\n<li>The area of a rectangle is <em>A\u00a0<\/em>= <em>bh<\/em>.<\/li>\n<li>The area of a parallelogram is\u00a0<em>A <\/em>= <em>bh<\/em>.<\/li>\n<li>The area of a triangle is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s2_p4_clip_image003.gif\" width=\"61\" height=\"41\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<li>The area of a trapezoid is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s2_p4_clip_image006.gif\" width=\"105\" height=\"41\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<li>The area of a circle is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s2_p4_clip_image009.gif\" width=\"53\" height=\"21\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<li>The perimeter of a figure is\u00a0the distance around that figure.<\/li>\n<li>The perimeter of a circle is called the circumference and\u00a0can be found by using the formula\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s2_p4_clip_image012.gif\" width=\"56\" height=\"19\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/li>\n<\/ul>\n<\/section>\n<p><!-- CONTENT ENDS HERE --><\/p>\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/mathematics\/about-measurement\">\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\/measuring-three-dimensional-figures\">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 Measuring Two-Dimensional Figures Objective In this lesson you will reflect on connections between formulas for the area and perimeter of\u00a0two-dimensional figures, such as circles and polygons. Previously Covered: A polygon is\u00a0a closed figure with sides made up of line segments. A regular polygon\u00a0is a convex polygon in which all [&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-201","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/201","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=201"}],"version-history":[{"count":6,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/201\/revisions"}],"predecessor-version":[{"id":554,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/201\/revisions\/554"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=201"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}