{"id":124,"date":"2024-07-21T10:33:50","date_gmt":"2024-07-21T10:33:50","guid":{"rendered":"https:\/\/www.mathros.net.ua\/en\/?p=124"},"modified":"2026-07-18T13:07:26","modified_gmt":"2026-07-18T13:07:26","slug":"rectangular-parallelepiped","status":"publish","type":"post","link":"https:\/\/www.mathros.net.ua\/en\/rectangular-parallelepiped.html","title":{"rendered":"Rectangular Parallelepiped: From Basics to Formulas, Let\u2019s Get It All!"},"content":{"rendered":"<p>So, you&#8217;ve come across the term <em>&#8220;Rectangular Parallelepiped&#8221;<\/em> and you&#8217;re wondering what it\u2019s all about, right? Well, let&#8217;s dive in and explore this three-dimensional figure that seems like an elongated cube or a box.<\/p>\n<p><img fetchpriority=\"high\" decoding=\"async\" class=\"aligncenter wp-image-10021646 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/07\/rectangular-parallelepiped1.jpg\" alt=\"rectangular parallelepiped\" width=\"600\" height=\"350\" \/><\/p>\n<p>A rectangular parallelepiped has volume and surface area that we can calculate using specific formulas. Sounds interesting? Let&#8217;s break it down!<\/p>\n<h2>Rectangular Parallelepiped: The Basics<\/h2>\n<p>A rectangular parallelepiped is a <em>3D<\/em> shape with six faces, twelve edges, and eight vertices. You might have also heard it called a cuboid because it shares some similarities with a cube. The key difference? While a cube&#8217;s faces are all squares, the faces of a rectangular parallelepiped are rectangles.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter wp-image-10021652 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/07\/rectangular-parallelepiped2.jpg\" alt=\"rectangular parallelepiped\" width=\"600\" height=\"350\" \/><\/p>\n<p>So, how can you spot one? Look for a shape where each face meets four other faces at right angles. And if you peek at the vertices, you&#8217;ll see that each one is where three edges and three faces intersect.<\/p>\n<h2>Types of Rectangular Parallelepiped: Which Is Which?<\/h2>\n<p>Did you know there are two types of rectangular parallelepipeds? Let&#8217;s break it down:<\/p>\n<ol>\n<li><strong>Regular Rectangular Parallelepiped<\/strong>: Here, all the faces are perpendicular to the bases. So, every side face is a rectangle.<\/li>\n<li><strong>Oblique Rectangular Parallelepiped<\/strong>: In this case, the faces aren&#8217;t perpendicular to the bases. Instead, they form parallelograms.<\/li>\n<\/ol>\n<p><img decoding=\"async\" class=\"aligncenter wp-image-10021655 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/07\/rectangular-parallelepiped3.jpg\" alt=\"oblique rectangular parallelepiped\" width=\"600\" height=\"350\" \/><\/p>\n<p>When people talk about a rectangular parallelepiped without any specifics, they&#8217;re usually referring to the regular kind.<\/p>\n<h2>Properties of a Rectangular Parallelepiped: What to Look For<\/h2>\n<p>Recognizing a rectangular parallelepiped is easy once you know its key properties:<\/p>\n<ul>\n<li>It has six faces, eight vertices, and twelve edges;<\/li>\n<li>In a regular rectangular parallelepiped, all faces are rectangles. In an oblique one, the faces are parallelograms;<\/li>\n<li>It has three dimensions: length, width, and height;<\/li>\n<li>Opposite faces are equal.<\/li>\n<\/ul>\n<h2>Formulas for a Rectangular Parallelepiped: The Math You Need<\/h2>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10021663 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/07\/rectangular-parallelepiped6.jpg\" alt=\"rectangular parallelepiped\" width=\"600\" height=\"350\" \/><\/p>\n<p>Want to calculate something about a rectangular parallelepiped? Here are the essential formulas:<\/p>\n<table>\n<tbody>\n<tr>\n<th>Term<\/th>\n<th>Definition<\/th>\n<th>Formula<\/th>\n<\/tr>\n<tr>\n<td>Total surface area of a rectangular parallelepiped<\/td>\n<td>This is the area occupied by its surface. The surface area is the sum of the areas of all six rectangular faces of the parallelepiped<\/td>\n<td style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-10021693\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/07\/rectangular-parallelepiped14.jpg\" alt=\"rectangular parallelepiped surface area formula\" width=\"155\" height=\"13\" \/><\/td>\n<\/tr>\n<tr>\n<td>The volume of a rectangular parallelepiped<\/td>\n<td>This is the space it occupies in a <a title=\"Three-dimensional space\" href=\"https:\/\/en.wikipedia.org\/wiki\/Three-dimensional_space\" target=\"_blank\" rel=\"nofollow noopener noreferrer\">three-dimensional space<\/a>. The volume of a rectangular parallelepiped is calculated by multiplying its length, width and height<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-10021694 aligncenter\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/07\/rectangular-parallelepiped15.jpg\" alt=\"rectangular parallelepiped volume formula\" width=\"155\" height=\"13\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Example Problems: Putting Theory into Practice<\/h2>\n<p>Let&#8217;s test what we&#8217;ve learned with some examples.<\/p>\n<h6>Example 1: What is a rectangular parallelepiped?<\/h6>\n<p>A rectangular parallelepiped is a three-dimensional figure with six rectangular faces, eight vertices, and twelve edges. Opposite faces are congruent.<\/p>\n<h6>Example 2: What is the difference between a cube and a rectangular parallelepiped?<\/h6>\n<p>A cube has six equal square faces, while a rectangular parallelepiped has six rectangular faces. In a rectangular parallelepiped, opposite faces are the same.<\/p>\n<h6>Example 3: What is the ratio of angles to faces in a rectangular parallelepiped?<\/h6>\n<p>A rectangular parallelepiped has eight vertices (corners) and six faces. The ratio is <em>8<\/em>:<em>6<\/em> or <em>4<\/em>:<em>3<\/em>.<\/p>\n<h6>Example 4: A rectangular parallelepiped is 5 cm long, 4 cm wide, and 4 cm high. What is its surface area?<\/h6>\n<p>Alright, let\u2019s break it down. We know:<\/p>\n<ul>\n<li>Length (<em>l<\/em>) = <em>5<\/em> cm;<\/li>\n<li>Width (<em>w<\/em>) = <em>4<\/em> cm;<\/li>\n<li>Height (<em>h<\/em>) = <em>4<\/em> cm.<\/li>\n<\/ul>\n<p>Now, how do we find the surface area? We use this formula: <em>TSA=2\u22c5(l\u22c5w+l\u22c5h+h\u22c5w)<\/em>. Plugging in our values, we get:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10021697 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/07\/rectangular-parallelepiped16.jpg\" alt=\"the area of the parallelepiped is 112 cm\u00b2\" width=\"400\" height=\"13\" \/><\/p>\n<p>So, the surface area of this rectangular parallelepiped is <em>112<\/em> square centimeters.<\/p>\n<h6>Example 5: a rectangular parallelepiped has a length of 7 cm, a width of 6 cm and a height of 8 cm. What is the volume of the parallelepiped?<\/h6>\n<p>Alright, let&#8217;s figure this out together. We&#8217;ve got a rectangular parallelepiped with:<\/p>\n<ul>\n<li>Length (<em>l<\/em>) = <em>7<\/em> cm;<\/li>\n<li>Width (<em>w<\/em>) = <em>6<\/em> cm;<\/li>\n<li>Height (<em>h<\/em>) = <em>8<\/em> cm.<\/li>\n<\/ul>\n<p>How do we find the volume? Easy! We use the volume formula for a rectangular parallelepiped. Plugging in our numbers, we get:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10021671 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/07\/rectangular-parallelepiped8.jpg\" alt=\"the volume of a rectangular parallelepiped is 336 cm\u00b3\" width=\"197\" height=\"12\" \/><\/p>\n<p>So, the volume of this rectangular parallelepiped is <em>336<\/em> cubic centimeters.<\/p>\n<h6>Example 6: Find the area of the faces of the rectangular parallelepiped ABCDA<sub>1<\/sub>B<sub>1<\/sub>C<sub>1<\/sub>D<sub>1<\/sub>, the sides of which are equal to 3 cm, 5 cm and 7 cm<\/h6>\n<p>Alright, let&#8217;s figure this out. We know that the sides of our rectangular parallelepiped are <em>3<\/em> cm, <em>5<\/em> cm, and <em>7<\/em> cm. How do we find the areas of its faces?<\/p>\n<p>First, remember that opposite faces of a rectangular parallelepiped are equal. So, we have:<\/p>\n<ul>\n<li><em>AA<sub>1<\/sub>B<sub>1<\/sub>B = DD<sub>1<\/sub>C<sub>1<\/sub>C<\/em>;<\/li>\n<li><em>ABCD = A<sub>1<\/sub>B<sub>1<\/sub>C<sub>1<\/sub>D<sub>1<\/sub><\/em>;<\/li>\n<li><em>AA<sub>1<\/sub>D<sub>1<\/sub>D = BB<sub>1<\/sub>C<sub>1<\/sub>C<\/em>.<\/li>\n<\/ul>\n<p>Since the faces are rectangles, we can use the formula for the area of a rectangle: <em>Area=length\u22c5width<\/em>. Let&#8217;s calculate each pair of faces:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-10021700 aligncenter\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/07\/rectangular-parallelepiped17.jpg\" alt=\"the area of the faces of the rectangular parallelepiped\" width=\"592\" height=\"16\" \/><\/p>\n<p>So, the areas of the faces <em>AA<sub>1<\/sub>B<sub>1<\/sub>B<\/em> and <em>DD<sub>1<\/sub>C<sub>1<\/sub>C<\/em> are <em>15<\/em> square centimeters, the areas of the faces <em>ABCD<\/em> and <em>A<sub>1<\/sub>B<sub>1<\/sub>C<sub>1<\/sub>D<sub>1<\/sub><\/em> are <em>35<\/em> square centimeters, and the areas of the faces <em>AA<sub>1<\/sub>D<sub>1<\/sub>D<\/em> and <em>BB<sub>1<\/sub>C<sub>1<\/sub>C<\/em> are <em>21<\/em> square centimeters.<\/p>\n<h6>Example 7: Find the angle DBD1 of the rectangular parallelepiped ABCDA<sub>1<\/sub>B<sub>1<\/sub>C<sub>1<\/sub>D<sub>1<\/sub>, for which AB = 4 cm, AD = 3 cm and AA<sub>1<\/sub> = 5 cm. Give the answer in degrees<\/h6>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10021677 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/07\/rectangular-parallelepiped11.jpg\" alt=\"rectangular parallelepiped\" width=\"600\" height=\"350\" \/><\/p>\n<p>Alright, let&#8217;s tackle this together. We need to find the angle <em>DBD<sub>1<\/sub><\/em> of our rectangular parallelepiped, given the sides <em>AB = 4<\/em> cm, <em>AD = 3<\/em> cm, and <em>AA<sub>1<\/sub> = 5<\/em> cm. How do we do this?<\/p>\n<p>First, consider the right triangle <em>ABD<\/em>. According to the Pythagorean theorem, we can find <em>BD<\/em>:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-10021678 size-full\" src=\"https:\/\/www.mathros.net.ua\/en\/wp-content\/uploads\/2024\/07\/rectangular-parallelepiped12.jpg\" alt=\"BD = 5 cm\" width=\"315\" height=\"17\" \/><\/p>\n<p>Next, look at the right triangle <em>BDD<sub>1<\/sub><\/em>. Notice that <em>BD<\/em> equals <em>AA<sub>1<\/sub><\/em>, which also equals <em>DD<sub>1<\/sub><\/em>. This means triangle <em>BDD<sub>1<\/sub><\/em> is isosceles. In an isosceles triangle, the angles at the base are equal.<\/p>\n<p>So, what\u2019s the required angle? Since <em>BDD<sub>1<\/sub><\/em> is isosceles and a right triangle, the base angles must be <em>45\u00b0<\/em> each.<\/p>\n<p>Therefore, the angle <em>DBD<sub>1<\/sub><\/em> is <em>45\u00b0<\/em>.<\/p>\n<h2>Deeper Dive: Additional Aspects of Rectangular Parallelepiped<\/h2>\n<p>Want to explore more about a rectangular parallelepiped? Here are some interesting topics:<\/p>\n<ol>\n<li><a title=\"Diagonals of a rectangular parallelepiped\" href=\"https:\/\/www.mathros.net.ua\/en\/diagonal-of-a-rectangular-parallelepiped.html\">Diagonals of a Rectangular Parallelepiped<\/a> &#8211; Formulas and examples to help you calculate the diagonals.<\/li>\n<li><a title=\"Surface area of a rectangular parallelepiped\" href=\"https:\/\/www.mathros.net.ua\/en\/surface-area-of-a-rectangular-parallelepiped.html\">Surface Area of a Rectangular Parallelepiped<\/a> &#8211; More formulas and examples for surface area calculations.<\/li>\n<li><a title=\"Volume of a rectangular parallelepiped\" href=\"https:\/\/www.mathros.net.ua\/en\/volume-of-a-parallelepiped.html\">Volume of a Rectangular Parallelepiped<\/a> &#8211; Learn more about volume calculations with detailed examples.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>So, you&#8217;ve come across the term &#8220;Rectangular Parallelepiped&#8221; and you&#8217;re wondering what it\u2019s all about, right? Well, let&#8217;s dive in<\/p>\n","protected":false},"author":1,"featured_media":125,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"template-centered.php","format":"standard","meta":{"footnotes":""},"categories":[10],"tags":[63,64,66,65],"class_list":["post-124","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-solid-geometric-shapes","tag-parallelepiped","tag-rectangular-parallelepiped","tag-rectangular-parallelepiped-shape","tag-what-is-a-rectangular-parallelepiped"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/posts\/124","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/comments?post=124"}],"version-history":[{"count":4,"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/posts\/124\/revisions"}],"predecessor-version":[{"id":232,"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/posts\/124\/revisions\/232"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/media\/125"}],"wp:attachment":[{"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/media?parent=124"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/categories?post=124"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.mathros.net.ua\/en\/wp-json\/wp\/v2\/tags?post=124"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}