{"id":267,"date":"2017-08-23T04:38:45","date_gmt":"2017-08-23T04:38:45","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/elementary-education\/?page_id=267"},"modified":"2021-04-30T05:25:10","modified_gmt":"2021-04-30T05:25:10","slug":"measuring-other-solids","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/elementary-education\/measuring-other-solids\/","title":{"rendered":"Measuring Other Solids"},"content":{"rendered":"<div class=\"twelve columns\" style=\"margin-top: 10%;\">\n<div class=\"advance\"><a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/measurements-involving-three-dimensional-figures\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/geometry-measurement\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/the-effects-on-measures-of-changes-in-dimensions\">Next Lesson \u27a1<\/a><\/div>\n<p><!-- CONTENT BEGINS HERE --><\/p>\n<h1 id=\"title\">Measuring Other Solids<\/h1>\n<h4>Objective<\/h4>\n<p>Now we&#8217;ll look at some other solids and how to calculate their volumes, surface areas, and other characteristics. We&#8217;ll also present some more questions and give you a chance to practice these skills.<\/p>\n<h4>Previously Covered<\/h4>\n<ul>\n<li>A cylinder is a solid whose bases are circles.<\/li>\n<li>A cone is a solid with one circular base and one vertex.<\/li>\n<li>A sphere is the set of all points in space that are equidistant from a given point called the center.<\/li>\n<\/ul>\n<section>\n<h3>Cylinders<\/h3>\n<p>Recall that the surface area of a polyhedron is the sum of the areas of its polygonal faces. The surface area of a cylinder is the sum of the areas of its faces, as well. Clearly, the two bases are circles, but the other \u201cface\u201d is harder to see. This might help:<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/cylinderz.gif\" alt=\"Cylinder\" \/><\/center>Since the radius of each circular base is r, then each circular base has area A = \u03c0r<sup>2<\/sup>. When the side of the cylinder is unwrapped, it becomes a rectangle with height h and width equal to the circumference of the circle, or C = 2\u03c0r. Thus, the area of the cylinder is A = 2\u03c0rh<\/p>\n<p>So, the surface area of a cylinder is given by the formula A = 2\u03c0r<sup>2<\/sup> + 2\u03c0rh, where r is the radius of the circular bases and h is the height.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>What is the surface area of a cylinder with a base of radius 2 ft and a height of 3 ft? Give your answer in terms of \u03c0 and round your answer to the nearest square foot.<\/p>\n<ol>\n<li>8\u03c0 ft<sup>2<\/sup><\/li>\n<li>12\u03c0 ft<sup>2<\/sup><\/li>\n<li>16\u03c0 ft<sup>2<\/sup><\/li>\n<li>20\u03c0 ft<sup>2<\/sup><\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">Choice D is the correct answer. The surface area of this cylinder is equal to A = 2\u03c0r<sup>2<\/sup> + 2\u03c0rh, where r = 2 ft and h = 3 ft. The surface area is:<\/p>\n<p class=\"q-reveal center\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3a_022.gif\" alt=\"Solution equation\" width=\"199\" height=\"105\" \/><\/p>\n<\/section>\n<h3>Cones<\/h3>\n<p>Let\u2019s start with a diagram of a cone so that you can get some sense of where the formula for its surface area comes from.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/cones.jpg\" alt=\"Cones\" \/><\/center>The cone has a circular base with area A = \u03c0r<sup>2<\/sup>. It also has a \u201cface\u201d whose area is a bit difficult to interpret. It\u2019s actually a sector, or fraction of a circle, that depends on the circumference of the circle, 2\u03c0r, and the slant height, l.<\/p>\n<p>The surface area of a cone is given by the formula, A = \u03c0r<sup>2<\/sup> + 2\u03c0rl, where r is the radius of the base and l is the slant height.<\/p>\n<h3>Spheres<\/h3>\n<p>Recall that a sphere is the set of all points in space that are equidistant from a given point. That equal distance is the radius of the circle. This makes things simpler, because everything about a sphere depends on its radius. And so the formula for its surface area depends only on its radius.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/spherez.gif\" alt=\"Sphere\" \/><\/center>The surface area of a sphere is given by the formula A = 4\u03c0r<sup>2<\/sup>, where r is the radius of the sphere.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>What is the surface area of sphere with diameter 8 yd? Round your answer to the nearest square yard.<\/p>\n<ol>\n<li>50 yd<sup>2<\/sup><\/li>\n<li>101 yd<sup>2<\/sup><\/li>\n<li>201 yd<sup>2<\/sup><\/li>\n<li>804 yd<sup>2<\/sup><\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">Choice C is the correct answer. The surface area of the sphere is given by the formula A = 4\u03c0r<sup>2<\/sup>, where r = 4 yd. If you picked choice D, you may have used 8 yd for the value of the radius rather than the value of the diameter. Remember, everything about a sphere depends on its radius, so the most common way to mix up a circle or sphere problem is to give the diameter instead of the radius. Please be careful. The sphere has surface area equal to:<\/p>\n<p class=\"q-reveal center\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3a_029.gif\" alt=\"Solution formula\" width=\"120\" height=\"125\" \/><\/p>\n<\/section>\n<p>Got your fill of formulas yet? The surface area formulas for polyhedra are quite intuitive; the surface area formula for a cone is not. Or is it?<\/p>\n<h3>Volume<\/h3>\n<h4>Cylinders<\/h4>\n<p>Recall that the volume of a prism is V = Bh, where B is the area of the base and h is the height. The same is true for cylinders, except now the base is a circle. So B = \u03c0r<sup>2<\/sup>.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/cylinder1.jpg\" alt=\"Volume of a cylinder\" \/><\/center>The volume of a cylinder is given by the formula V = \u03c0r<sup>2<\/sup>h, where r is the radius of a circular base and h is the height.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>What is the volume of a cylinder of height 10 in. and radius 15 in.? Give your answer in terms of \u03c0 and round to the nearest cubic inch.<\/p>\n<ol>\n<li>1500\u03c0 in<sup>3<\/sup><\/li>\n<li>2250\u03c0 in<sup>3<\/sup><\/li>\n<li>4714\u03c0 in<sup>3<\/sup><\/li>\n<li>7069\u03c0 in<sup>3<\/sup><\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">Choice B is the correct answer. The volume of the cylinder is given by the formula V = \u03c0r<sup>2<\/sup>h, where r = 15 in. and h = 10 in. The volume of the cylinder is<\/p>\n<p class=\"q-reveal center\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3a_036.gif\" alt=\"Solution formula\" width=\"153\" height=\"105\" \/><\/p>\n<\/section>\n<h4>Cones<\/h4>\n<p>Recall that the volume of a pyramid is one third of the volume of a prism with the same base and height. The same relationship is true for cylinders and cones.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/cone.jpg\" alt=\"Volume of a cone\" \/><\/center>The volume of a cone is given by the formula <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3a_037.gif\" alt=\"Formula for volume of a cone\" \/>\u00a0Where r is the radius of the circular base and h is the height.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>What is the volume of a cone whose height and radius each equal 1 yd? Give your answer in terms of and round to the nearest cubic yard.<\/p>\n<ol>\n<li><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3a_038.gif\" alt=\"\u03c0\/3\" \/> yd<sup>3<\/sup><\/li>\n<li><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3a_039.gif\" alt=\"2\u03c0\/3\" \/> yd<sup>3<\/sup><\/li>\n<li>\u03c0 yd<sup>3<\/sup><\/li>\n<li>3\u03c0 yd<sup>3<\/sup><\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">Choice A is the correct answer. The volume of the cone is given by the formula <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3a_037.gif\" alt=\"Formula for volume of a cone\" \/>, where r = 1 yd and h = 1 yd. The volume of the cone is<\/p>\n<p class=\"q-reveal center\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3a_041.gif\" alt=\"Solution formula\" width=\"150\" height=\"147\" \/><\/p>\n<\/section>\n<h4>Spheres<\/h4>\n<p>Recall that everything about a sphere depends on its radius. Also recall that volume is measured in cubic units. These two facts take us some of the way to the formula for the volume of a sphere.<\/p>\n<p><center><img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3bsphere.jpg\" alt=\"Volume of a sphere\" \/><\/center>The volume of a sphere is given by the formula <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3a_045.gif\" alt=\"Formula for volume of a sphere\" \/>, where r is the radius of the sphere.<\/p>\n<p>Let&#8217;s try a practice question and then we&#8217;ll look at some other examples of these formulas.<\/p>\n<section class=\"question\">\n<h4>Question<\/h4>\n<p>What is the volume of a sphere of radius 3 cm? Round your answer to the nearest cubic centimeter.<\/p>\n<ol>\n<li>12\u03c0 cm<sup>3<\/sup><\/li>\n<li>24\u03c0 cm<sup>3<\/sup><\/li>\n<li>36\u03c0 cm<sup>3<\/sup><\/li>\n<li>108\u03c0 cm<sup>3<\/sup><\/li>\n<\/ol>\n<p><a class=\"q-answer button button-primary\">Reveal Answer<\/a><\/p>\n<p class=\"q-reveal\">Choice C is the correct answer. The volume of the sphere is given by the formula <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3a_045.gif\" alt=\"Formula for volume of a sphere\" \/>. Plug 3 in for the radius and solve.<\/p>\n<\/section>\n<p><object class=\"old_animation\" width=\"744\" height=\"445\" classid=\"clsid:D27CDB6E-AE6D-11cf-96B8-444553540000\"  codebase=\"http:\/\/download.macromedia.com\/pub\/shockwave\/cabs\/flash\/swflash.cab#version=6,0,29,0\"><param name=\"movie\" value=\"flashPlayer.swf?sURL=fish.swf\" \/><param name=\"quality\" value=\"high\" \/><embed src=\"http:\/\/americanboard.org\/Subjects\/Images\/physics\/flash\/flashPlayer.swf?sURL=http:\/\/americanboard.org\/Subjects\/Images\/physics\/flash\/fish.swf\" quality=\"high\" pluginspage=\"http:\/\/www.macromedia.com\/go\/getflashplayer\" type=\"application\/x-shockwave-flash\" height=\"445\" width=\"744\" \/><\/object><\/p>\n<div class=\"slider-container new_animation\">\n<iframe loading=\"lazy\" src=\"https:\/\/americanboard.org\/Subjects\/animation_sliders\/fishing\/fishing.html\" referrerpolicy=\"origin\" width=\"750\" height=\"450\" frameborder=\"0\" scrolling=\"no\"><br \/>\n<\/iframe>\n<\/div>\n<h3>Review<\/h3>\n<ul>\n<li>The volume of a cylinder is given by the formula V = \u03c0r<sup>2<\/sup>h, where r is the radius of a circular base and h is the height.<\/li>\n<li>The volume of a cone is given by the formula <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3a_037.gif\" alt=\"Formula for volume of a cone\" \/>, where r is the radius of the circular base and h is the height.<\/li>\n<li>The volume of a sphere is given by the formula <img decoding=\"async\" src=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/wp-content\/uploads\/sites\/4\/2017\/08\/6.3a_045.gif\" alt=\"Formula for volume of a sphere\" \/>, where r is the radius of the sphere.<\/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\/elementary-education\/measurements-involving-three-dimensional-figures\">\u2b05 Previous Lesson<\/a>\u00a0<a class=\"button\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/geometry-measurement\">Workshop Index<\/a>\u00a0<a class=\"button button-primary\" href=\"http:\/\/americanboard.org\/Subjects\/elementary-education\/the-effects-on-measures-of-changes-in-dimensions\">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 Other Solids Objective Now we&#8217;ll look at some other solids and how to calculate their volumes, surface areas, and other characteristics. We&#8217;ll also present some more questions and give you a chance to practice these skills. Previously Covered A cylinder is a solid whose bases are circles. A [&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-267","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/267","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/comments?post=267"}],"version-history":[{"count":15,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/267\/revisions"}],"predecessor-version":[{"id":1482,"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/pages\/267\/revisions\/1482"}],"wp:attachment":[{"href":"https:\/\/americanboard.org\/Subjects\/elementary-education\/wp-json\/wp\/v2\/media?parent=267"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}