{"id":202,"date":"2017-08-23T09:57:17","date_gmt":"2017-08-23T09:57:17","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=202"},"modified":"2017-08-30T09:52:14","modified_gmt":"2017-08-30T09:52:14","slug":"measuring-three-dimensional-figures","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/measuring-three-dimensional-figures\/","title":{"rendered":"Measuring Three-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\/measuring-two-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\/proportions-in-measurement\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Measuring Three-Dimensional Figures<\/h1>\n<h4>Objective<\/h4>\n<p>In this lesson, you will focus on determining the surface area and volume of three-dimensional figures. You will also\u00a0explore how changing dimensions of a figure changes the size of the surface area and the volume.<\/p>\n<h4>Previously Covered:<\/h4>\n<ul>\n<li>We looked at connections between formulas for the area and perimeter of two-dimensional figures, such as circles\u00a0and polygons.<\/li>\n<\/ul>\n<section>\n<h3>How do you determine surface area of a figure?<\/h3>\n<p>To determine the surface area of a figure, first think about\u00a0disassembling the figure into separate faces. For instance, think\u00a0about a right rectangular parallelepiped with dimensions of 6\u00a0centimeters by 8 centimeters by 20 centimeters.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/Math-Mod-5.2-Art-001.gif\" alt=\" Right rectangular parallelepiped\" width=\"244\" height=\"115\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>There are six faces.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/Math-Mod-5.2-Art-002--Faces.gif\" alt=\"Faces\" width=\"377\" height=\"113\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The surface area of the figure is the sum of the areas of the\u00a0six faces.<\/p>\n<p align=\"CENTER\"><em>A<\/em> = 2(6)(8) + 2(6)(20) + 2(8)(20) = 656<br \/>\nsquare centimeters<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the surface area of a cylinder that is 15 feet in\u00a0diameter and 8 feet high?<\/p>\n<ol>\n<li>120 square feet<\/li>\n<li>64 + 30<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p2_clip_image003.gif\" width=\"15\" height=\"15\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0square feet<\/li>\n<li>176.25<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p2_clip_image006.gif\" width=\"15\" height=\"15\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0square feet<\/li>\n<li>232.5<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p2_clip_image009.gif\" width=\"15\" height=\"15\" name=\"graphics7\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0square feet<\/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 D. The cylinder can be deconstructed\u00a0into three faces- two circles and one rectangle.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/Math-Mod-5.2-Art-002.gif\" alt=\"cylinder deconstructed\" width=\"541\" height=\"181\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The surface area is the sum of the three faces,\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p2_clip_image012.gif\" width=\"179\" height=\"21\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<h3><strong>How do you determine volume?<\/strong><\/h3>\n<p>There are a number of volume formulas. In this section, we will\u00a0explore two basic categories of volume formulas: non-pointed and\u00a0pointed figures.<\/p>\n<p>First, let\u2019s review some formulas for non-pointed solids:<\/p>\n<ul>\n<li>Volume of a rectangular solid:\u00a0<em>lwh<\/em>, where <em>lw<\/em> is the area of the base, and <em>h\u00a0<\/em>is the height of the solid<\/li>\n<li>Volume of a cylinder:\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p3_clip_image003.gif\" width=\"36\" height=\"21\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0where\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p3_clip_image006.gif\" width=\"28\" height=\"21\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is the area of the base, and <em>h<\/em> is the height of the solid<\/li>\n<li>Volume of a solid with trapezoidal bases and rectangular\u00a0sides:\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p3_clip_image009.gif\" width=\"88\" height=\"41\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0where\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p3_clip_image012.gif\" width=\"80\" height=\"41\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is the area of the trapezoid, and <em>h<\/em> is the height of the\u00a0solid.<\/li>\n<\/ul>\n<p>These first three formulas refer to non-pointed figures. Notice\u00a0that each figure in this category has a volume that can be\u00a0calculated by multiplying the area of its base by its height.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/Math%20Mod%205.2%20Art%20003.JPG\" alt=\"Rectangular Solid\" width=\"300\" height=\"284\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>A rectangular solid is also called a rectangular\u00a0parallelepiped.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/Math%20Mod%205.2%20Art%20004.JPG\" alt=\"Trapezoidal base solid\" width=\"300\" height=\"217\" name=\"graphics8\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>A solid with trapezoidal bases and rectangular sides is also called a trapezoidal parallelepiped.<\/p>\n<p>Next, let\u2019s consider solids that are pointed at the top.<\/p>\n<ul>\n<li>Volume of a square pyramid:\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p3_clip_image015.gif\" width=\"36\" height=\"44\" name=\"graphics9\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0where <em>s<\/em> is the length of a side on the square base, <em>s<\/em><sup>2\u00a0<\/sup>is the area of the square base, and <em>h<\/em> is the height of the\u00a0pyramid<\/li>\n<li>Volume of a triangular pyramid:\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p3_clip_image018.gif\" width=\"72\" height=\"45\" name=\"graphics10\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0where\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p3_clip_image021.gif\" width=\"49\" height=\"45\" name=\"graphics11\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0is the area of the triangular base, and <em>h<\/em> is the height of\u00a0the pyramid<\/li>\n<li>Volume of a cone:\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p3_clip_image024.gif\" width=\"48\" height=\"41\" name=\"graphics12\" align=\"ABSMIDDLE\" border=\"0\" \/>,\u00a0where\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p3_clip_image027.gif\" width=\"27\" height=\"21\" name=\"graphics13\" align=\"absmiddle\" border=\"0\" \/>\u00a0is the area of the circular base, and <em><span style=\"text-decoration: none;\">h\u00a0<\/span><\/em>is the height of the cone<\/li>\n<\/ul>\n<p>These three formulas refer to pointed figures. Notice that the\u00a0volume of each pointed figure is\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p3_clip_image030.gif\" width=\"15\" height=\"41\" name=\"graphics14\" align=\"ABSMIDDLE\" border=\"0\" \/>\u00a0of the area of the base multiplied by the height.<\/p>\n<p class=\"notebox_text\" align=\"left\">To find the volume of a non-pointed solid, multiply\u00a0the area of the base by the height of the solid, measured at a\u00a0right angle to the base.<\/p>\n<p class=\"notebox_text\" align=\"left\">To find the volume of a pointed solid, multiply the\u00a0area of the base by the height of the solid, measured at a right\u00a0angle to the base, and divide by 3.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>The solid below has a right triangular base. What is the volume?<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/Math%20Mod%205.2%20Art%20005.JPG\" alt=\"Solid with triangular base\" width=\"181\" height=\"135\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<ol>\n<li>24 cubic inches<\/li>\n<li>24 square inches<\/li>\n<li>48 cubic inches<\/li>\n<li>48 square inches<\/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 volume is the area of the\u00a0triangular base multiplied by the height of the solid:\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p4_clip_image007.gif\" width=\"124\" height=\"45\" name=\"graphics4\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0Volume is always measured in cubic units.<\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>What is the approximate surface area of the solid in the\u00a0question above?<\/p>\n<ol>\n<li>49 square inches<\/li>\n<li>52 square inches<\/li>\n<li>55 square inches<\/li>\n<li>57 square inches<\/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 D.Surface area is measured in square\u00a0units. The solid has 5 faces. The bases are two isosceles right\u00a0triangles, each with 2 sides of length 4 inches and a third side\u00a0of length\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p4_clip_image014.gif\" width=\"33\" height=\"25\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/>.\u00a0There are two 3 by 4 rectangles and one rectangle that is 3 by\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p4_clip_image017.gif\" width=\"33\" height=\"25\" name=\"graphics6\" align=\"ABSMIDDLE\" border=\"0\" \/>.<\/p>\n<p>The total surface area is<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p4_clip_image020.gif\" width=\"469\" height=\"51\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/>.<\/p>\n<\/div>\n<\/section>\n<h3>What happens when the dimensions of a figure are changed?<\/h3>\n<ul>\n<li>When a\u00a0two-dimensional figure is changed in size by a factor of <em><span style=\"text-decoration: none;\">n<\/span><\/em>,\u00a0the area is changed by a factor of <em>n<\/em><sup>2<\/sup>.\u00a0When a three-dimensional figure is changed in\u00a0size by a factor of <em>n<\/em><u>,<\/u> the area is changed\u00a0by a factor of <em>n<\/em><sup>3<\/sup>.<\/li>\n<\/ul>\n<p>Take the figure below, for example. The orange rectangle is 4\u00a0by 6 and has an area of 24 square units.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/Math%20Mod%205.2%20Art%20006.JPG\" alt=\"fiigures made up of unit squares\" width=\"300\" height=\"334\" name=\"graphics3\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The yellow rectangle has the dimensions of the orange rectangle\u00a0multiplied by a factor of three, which results in dimensions of 12\u00a0by 18. The area of this rectangle is 216 square units, which is\u00a0the area of the orange rectangle multiplied by the square of the\u00a0factor of three, or nine.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p5_clip_image003.gif\" width=\"85\" height=\"25\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>The blue rectangle has the dimensions of the orange rectangle\u00a0multiplied by a factor of one-half, which results in dimensions of\u00a02 by 3. The area is 6 square units, which is the area of the\u00a0orange rectangle multiplied by the square of the factor of\u00a0one-half, or one-fourth.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p5_clip_image006.gif\" width=\"81\" height=\"53\" name=\"graphics5\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>Let\u2019s say that these rectangles are actually solids. If\u00a0the orange solid were 4 by 6 by 2, it would have a volume of 48\u00a0cubic units.<\/p>\n<p>If the dimensions of the yellow rectangular solid differ from\u00a0the dimensions of the orange solid by a factor of three, its\u00a0dimensions would be 12 by 18 by 6. The volume of the yellow\u00a0rectangular solid would be 1,296 cubic units, which is the volume\u00a0of the orange rectangular solid multiplied by the cube of the\u00a0factor of three.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p5_clip_image009.gif\" width=\"95\" height=\"25\" name=\"graphics6\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<p>If the dimensions of the blue rectangular solid differ from the\u00a0dimensions of the orange solid by a factor of one half, the\u00a0dimensions would be 2 by 3 by 1. The volume of the blue\u00a0rectangular solid would be 6 cubic units, which is the volume of\u00a0the orange rectangular solid multiplied by the cube of the factor\u00a0of one-half.<\/p>\n<p align=\"CENTER\"><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p5_clip_image012.gif\" width=\"80\" height=\"53\" name=\"graphics7\" align=\"BOTTOM\" border=\"0\" \/><\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>A statue of a horse has a volume of 12 cubic meters. The\u00a0artist\u2019s model had dimensions of one-tenth of the final\u00a0statue. What was the volume of the artist\u2019s model?<\/p>\n<ol>\n<li>0.012 cubic meters<\/li>\n<li>0.12 cubic meters<\/li>\n<li>0.2 cubic meters<\/li>\n<li>1,200 cubic 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 A. The dimensions of the model were\u00a0one-tenth the dimensions of the statue, so the volume of the\u00a0model would be\u00a0<img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p6_clip_image005.gif\" width=\"117\" height=\"53\" name=\"graphics3\" align=\"ABSMIDDLE\" border=\"0\" \/>of\u00a0the original.<\/p>\n<p align=\"CENTER\"><strong><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p6_clip_image008.gif\" width=\"128\" height=\"49\" name=\"graphics4\" align=\"BOTTOM\" border=\"0\" \/><\/strong><\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>How many square inches are in a square foot?<\/p>\n<ol>\n<li>12<\/li>\n<li>24<\/li>\n<li>48<\/li>\n<li>144<\/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 D. A square foot is 12 inches by 12\u00a0inches; it has an area of 144 square inches.<\/p>\n<\/div>\n<\/section>\n<section class=\"question\">\n<h4>Question<\/h4>\n<div>\n<p>A copy machine increases the area of a figure to 200% of the\u00a0original. If the original height of the figure is 6 units, what is\u00a0the height of the final figure?<\/p>\n<ol>\n<li>6 units<\/li>\n<li><img loading=\"lazy\" decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/Images\/math\/5\/images\/s3_p6_clip_image016.gif\" width=\"33\" height=\"23\" name=\"graphics5\" align=\"ABSMIDDLE\" border=\"0\" \/><br \/>\nunits<\/li>\n<li>24 units<\/li>\n<li>48 units<\/li>\n<li>48 units<\/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 area was doubled, so the lengths\u00a0of the sides must have been increased by a factor of the square\u00a0root of two.<\/p>\n<\/div>\n<\/section>\n<h3>Review of New Vocabulary and Concepts<\/h3>\n<ul>\n<li>The <em><strong>volume\u00a0<\/strong><\/em>of a solid is measured in cubic units.<\/li>\n<li>The <em><strong>surface\u00a0<\/strong><\/em>area of a solid is measured in square units.<\/li>\n<li>The volume of a non-pointed\u00a0solid is found by multiplying the area of the base by the height\u00a0of the solid.<\/li>\n<li>The volume of a pointed solid\u00a0is found by multiplying the area of the base by the height of the\u00a0solid and dividing by three.<\/li>\n<li>The surface area of a solid is\u00a0found by finding the sum of the areas of all of the surfaces of\u00a0the solid.<\/li>\n<li>When the dimensions of a figure are increased by a factor\u00a0of <em>n<\/em>, any area is increased by a factor of <em>n<\/em><sup>2<\/sup>,\u00a0and any volume is increased by a factor of <em>n<\/em><sup>3<\/sup>.<\/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\/measuring-two-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\/proportions-in-measurement\">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 Three-Dimensional Figures Objective In this lesson, you will focus on determining the surface area and volume of three-dimensional figures. You will also\u00a0explore how changing dimensions of a figure changes the size of the surface area and the volume. Previously Covered: We looked at connections between formulas for the [&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-202","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/202","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=202"}],"version-history":[{"count":5,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/202\/revisions"}],"predecessor-version":[{"id":556,"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/pages\/202\/revisions\/556"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/mathematics\/wp-json\/wp\/v2\/media?parent=202"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}