{"id":26054,"date":"2023-03-09T07:07:37","date_gmt":"2023-03-09T07:07:37","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=26054"},"modified":"2024-02-05T16:12:28","modified_gmt":"2024-02-05T16:12:28","slug":"octagon-formula-for-area-and-perimeter-with-derivation","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/octagon-formula","title":{"rendered":"Octagon Formula For Area and Perimeter With Derivation"},"content":{"rendered":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\tMath-Vocabulary\n<\/span><\/div>\n\n<div class=\"ub_table-of-contents\" data-showtext=\"show\" data-hidetext=\"hide\" data-scrolltype=\"auto\" id=\"ub_table-of-contents-657c7ab7-9e4b-463f-a779-f510f6e1c969\" data-initiallyhideonmobile=\"false\"\n                    data-initiallyshow=\"true\"><div class=\"ub_table-of-contents-extra-container\"><div class=\"ub_table-of-contents-container ub_table-of-contents-1-column \"><ul><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/octagon-formula#0-what-are-the-octagon-formulas>What Are The Octagon Formulas?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/octagon-formula#1-the-formula-for-the-area-of-an-octagon>The Formula for the Area of an Octagon<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/octagon-formula#3-formula-of-the-perimeter-of-octagon>Formula of the Perimeter of Octagon<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/octagon-formula#7-solved-examples-based-on-octagon-formulas>Solved Examples based on Octagon Formulas<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/octagon-formula#8-practice-problems-on-octagon-formulas>Practice Problems on Octagon Formulas<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/octagon-formula#9-frequently-asked-questions-about-octagon-formulas>Frequently Asked Questions about Octagon Formulas<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"0-what-are-the-octagon-formulas\">What Are The Octagon Formulas?<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">The octagon is an 8-sided polygon. An octagon is referred to as a regular octagon if all of its sides have equal lengths and angles are of equal measures. A regular octagon has each interior angle measuring $135^{\\circ}$ and each exterior angle measures $45^{\\circ}$.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"383\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/a-regular-octagon-with-each-interior-angle-of-135\u00b0.webp\" alt=\"A regular octagon with each interior angle of 135\u00b0\" class=\"wp-image-35683\" title=\"A regular octagon with each interior angle of 135\u00b0\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/a-regular-octagon-with-each-interior-angle-of-135\u00b0.webp 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/a-regular-octagon-with-each-interior-angle-of-135\u00b0-300x185.webp 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\">We will discuss octagon formulas such as the area of an octagon, perimeter of an octagon.<\/p>\n\n\n\n<div id=\"recommended-games-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Games<\/h4><div class=\"recommended-games-container-slides\"><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/find-volume-using-the-formula\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/geo_meas_vol_formula_pt.png\" alt=\"Find Volume using the Formula Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Find Volume using the Formula Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><\/div><p class=\"recommended-games-container-desc\"><a href=\"https:\/\/www.splashlearn.com\/games\">More Games<\/a><\/p><button class=\"scroll-right-arrow\"><\/button><\/div><script type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-games-container-id\");\n            const slidesContainer = container.querySelector(\".recommended-games-container-slides\");\n            const cards = slidesContainer.querySelectorAll(\".game-card-container-outer\");\n            const scrollRightArrow = container.querySelector(\".scroll-right-arrow\");\n\n            function adjustContainerStyles() {\n                const numCards = cards.length;\n\n                if (numCards === 1) {\n                    container.style.maxWidth = \"30%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 2) {\n                    container.style.maxWidth = \"50%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 3) {\n                    container.style.maxWidth = \"75%\";\n                    container.style.textAlign = \"center\";\n                } else {\n                    container.style.maxWidth = \"\";\n                    container.style.textAlign = \"\";\n                }\n            }\n\n            function checkScrollPosition() {\n                const maxScrollLeft = slidesContainer.scrollWidth - slidesContainer.clientWidth;\n                if ((slidesContainer.scrollLeft + 10) >= maxScrollLeft) {\n                    scrollRightArrow.style.display = \"none\"; \/\/ Hide the arrow if fully scrolled\n                } else {\n                    scrollRightArrow.style.display = \"block\"; \/\/ Show the arrow if not fully scrolled\n                }\n            }\n\n            scrollRightArrow.addEventListener(\"click\", function() {\n                const scrollAmount = 300; \/\/ Adjust based on the container's width and your needs\n                slidesContainer.scrollLeft += scrollAmount;\n                setTimeout(checkScrollPosition, 100); \/\/ Delay to allow scroll update\n            });\n\n            adjustContainerStyles();\n            checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading eplus-wrapper\" id=\"1-the-formula-for-the-area-of-an-octagon\">The Formula for the Area of an Octagon<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">Area of an octagon is the region bounded within the boundaries of an octagon. The <strong>area of an octagon <\/strong>$= 2a^{2}(\\sqrt{2} + 1)$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">where a represents the side of the octagon.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"2-derivation\">Derivation<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">To find the area of an octagon, it is divided into 8 equal isosceles triangles. The entire area of the polygon can be determined by multiplying the area of one triangle by 8.<strong>&nbsp;<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"459\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/a-regular-octagon-with-side-a-divided-into-8-isosceles-triangles.png\" alt=\"A regular octagon with side \u2018a\u2019 divided into 8 isosceles triangles\" class=\"wp-image-35684\" title=\"A regular octagon with side \u2018a\u2019 divided into 8 isosceles triangles\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/a-regular-octagon-with-side-a-divided-into-8-isosceles-triangles.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/a-regular-octagon-with-side-a-divided-into-8-isosceles-triangles-300x222.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\">One isosceles triangle is shown below where $OA = OB$.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Here, OD is the height of the triangle, and AB is the base of the triangle. Let OD bisects the angle AOB and the side AB.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Here, base $= AB = a$<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"404\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/an-isosceles-triangle-OAB.png\" alt=\"An isosceles triangle OAB\" class=\"wp-image-35686\" title=\"An isosceles triangle OAB\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/an-isosceles-triangle-OAB.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/11\/an-isosceles-triangle-OAB-300x195.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Area of octagon<\/strong> $= 8 \\times\\; Area\\; of\\; one\\; isosceles\\; triangle$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">By using the identities, we have<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$tan^{2}(\\frac{\\theta}{2}) = \\frac{1 \\;-\\; cos\\; \\theta}{1 \\;+\\; cos\\;\\theta} = \\frac{1\\;-\\; cos\\; 45^{\\circ}}{1\\;+ \\;cos\\; 45^{\\circ}} = \\frac{1-(1\\sqrt{2})}{1+(1\\sqrt{2})} = \\frac{\\sqrt{2}\\;-\\;1}{\\sqrt{2}\\;+\\;1} = (\\sqrt{2}\\;-\\;1)^{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$tan\\; (\\frac{45^{\\circ}}{2}) = \\frac{BD}{OD} = \\sqrt{2}\\;-\\;1$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Thus, $OD = \\frac{BD}{\\sqrt{2}\\;-\\;1} = \\frac{a\/2}{\\sqrt{2}\\;-\\;1} = \\frac{a(\\sqrt{2}\\;+\\;1)}{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Area of $\\Delta AOB\\; = 12 \\times AB \\times OD$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$= \\frac{1}{2} \\times a \\times \\frac{a}{2}(\\sqrt{2}\\;+\\;1)$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$= \\frac{1}{4}a^{2}\\;(\\sqrt{2}\\;+\\;1)$<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Area of octagon <\/strong>$= 8 \\times$ <strong>Area of one isosceles triangle<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">Area of octagon $= 8 \\times \\frac{1}{4}a^{2}\\;(\\sqrt{2}\\;+\\;1)$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Area of octagon $= 2a^{2}\\;(\\sqrt{2}\\;+\\;1)$<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"3-formula-of-the-perimeter-of-octagon\">Formula of the Perimeter of Octagon<\/h2>\n\n\n\n<p class=\"eplus-wrapper\"><strong>The length of an octagon\u2019s boundary is referred to as its perimeter<\/strong>. Therefore, the perimeter will equal the total length of all sides. The formula for the perimeter of a regular octagon having 8 sides each measuring \u201ca\u201d units is given by<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>So, the perimeter of a regular octagon = 8a units<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"4-octagon-formulas\">Octagon Formulas<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">Area of a regular octagon $= 2s^{2}\\;(1\\;+\\;\\sqrt{2})$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Perimeter of a regular octagon $= 8s$<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"5-properties-of-a-regular-octagon\">Properties of a Regular Octagon<\/h2>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">There are 8 sides and 8 interior angles.<\/li>\n\n\n\n<li class=\"eplus-wrapper\">A regular octagon has a total of 20 diagonals.<\/li>\n\n\n\n<li class=\"eplus-wrapper\">Each inside angle measures $135^{\\circ}$. The sum of interior angles is $1080^{\\circ}$.<\/li>\n\n\n\n<li class=\"eplus-wrapper\">Each exterior angle measures $45^{\\circ}$, so the sum of all the exterior angles is $360^{\\circ}$.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"6-conclusion\">Conclusion<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">In this article, we learned about octagon, octagonal formulas, derivation of octagon formula, and properties of a regular octagon.&nbsp; Let\u2019s solve a few examples now.<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"7-solved-examples-based-on-octagon-formulas\">Solved Examples based on Octagon Formulas<\/h2>\n\n\n\n<p class=\"eplus-wrapper\"><strong>1. If the length of an octagon\u2019s side is 14 inches, calculate its area.<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">The side\u2019s length, \u201cs,\u201d is 14 inches.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Using the octagon\u2019s surface area formula,<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$A = 2s^{2}\\;(\\sqrt{2}\\;+\\;1)$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$A = 2 \\times14^{2}\\;(\\sqrt{2}\\;+\\;1)$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$A = 946.37$ square inches<\/p>\n\n\n\n<p class=\"eplus-wrapper\">As a result, the octagon has a surface area of 946.37 square inches.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>2. The area of an octagon is 25.54 square units. Find the length of the side.<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">The octagon&#8217;s area is 25.54 square units.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$A = 2s^{2}(1\\;+\\;\\sqrt{2})$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$25.54 = 2 \\times s^{2}\\;(1\\;+\\;\\sqrt{2})$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$s = 2.3$ units<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The octagon\u2019s side length is 2.3 units.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>3. If a normal octagon\u2019s side is 5 units long, calculate its area and perimeter.<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">An ordinary octagon has equal-length sides.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The side length in this case is 5 units, or \u201cs.\u201d<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Area of a regular octagon $= 2s^{2}\\;(1 \\;+\\; \\sqrt{2}) = 2 (5)^{2} (1 \\;+\\; \\sqrt{2}) = 50 (1 \\;+\\; 2) = 120.71$ square units&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Perimeter of a regular octagon $= 8s = 40$ units<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"8-practice-problems-on-octagon-formulas\">Practice Problems on Octagon Formulas<\/h2>\n\n\n\n<p class=\"eplus-wrapper\"><div class=\"spq_wrapper\"><h2 style=\"display:none;\">Octagon Formula For Area and Perimeter With Derivation<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">Determine the perimeter of a regular octagon with side 2 inches.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">15<\/div><div class=\"spq_answer_block\" data-value=\"1\">16<\/div><div class=\"spq_answer_block\" data-value=\"2\">17<\/div><div class=\"spq_answer_block\" data-value=\"3\">18<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: 16<br\/>Solution: Side of the octagon $= 2$ inches<br>\r\nThe perimeter of a regular octagon, with side length \u2018a\u2019, is given by the following formula,<br>\r\nThe perimeter $= 8a = 8 \\times 2 = 16$ inches<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"3\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">An octagon has a 40-inch perimeter. Find the area.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">5 sq. in<\/div><div class=\"spq_answer_block\" data-value=\"1\">125 sq. in<\/div><div class=\"spq_answer_block\" data-value=\"2\">60 sq. in<\/div><div class=\"spq_answer_block\" data-value=\"3\">120 sq. in<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: 120 sq. in<br\/>Octagon\u2019s perimeter $= 8a$<br>\r\n32 inches $= 8a$<br>\r\n$a = \\frac{40}{8} = 5$ inches<br>\r\nOctagon Area $= 2 \\;a^{2}\\; (1\\;+\\;\\sqrt{2}) = 2 \\times 5^{2}\\; (1\\;+\\;\\sqrt{2})\\;\\simeq 120\\; inches^{2}$<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">Each interior angle of a regular octagon measures _____ degrees.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">85<\/div><div class=\"spq_answer_block\" data-value=\"1\">135<\/div><div class=\"spq_answer_block\" data-value=\"2\">145<\/div><div class=\"spq_answer_block\" data-value=\"3\">67<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: 135<br\/>Each interior angle of a regular octagon measures 135 degrees.<\/span><\/div><\/div><\/div><\/div>  <script type=\"application\/ld+json\">{\n        \"@context\": \"https:\/\/schema.org\/\", \n        \"@type\": \"Quiz\", \n        \"typicalAgeRange\": \"3-11\",\n        \"educationalLevel\":  \"beginner\",\n        \"assesses\" : \"Attend this quiz & Test your knowledge.\",\n        \"educationalAlignment\": [\n              {\n                \"@type\": \"AlignmentObject\",\n                \"alignmentType\": \"educationalSubject\",\n                \"targetName\": \"Math\"\n              }] ,\n        \"name\": \"Octagon Formula For Area and Perimeter With Derivation\",        \n        \"about\": {\n                \"@type\": \"Thing\",\n                \"name\": \"Octagon Formula For Area and Perimeter With Derivation\"\n        },  \n        \"hasPart\": [{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Determine the perimeter of a regular octagon with side 2 inches.\",\n                    \"text\": \"Determine the perimeter of a regular octagon with side 2 inches.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"Solution: Side of the octagon $$= 2$$ inches<br>\r\nThe perimeter of a regular octagon, with side length \u2018a\u2019, is given by the following formula,<br>\r\nThe perimeter $$= 8a = 8 \\\\times 2 = 16$$ inches\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"15\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Solution: Side of the octagon $$= 2$$ inches<br>\r\nThe perimeter of a regular octagon, with side length \u2018a\u2019, is given by the following formula,<br>\r\nThe perimeter $$= 8a = 8 \\\\times 2 = 16$$ inches\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"17\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Solution: Side of the octagon $$= 2$$ inches<br>\r\nThe perimeter of a regular octagon, with side length \u2018a\u2019, is given by the following formula,<br>\r\nThe perimeter $$= 8a = 8 \\\\times 2 = 16$$ inches\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"18\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Solution: Side of the octagon $$= 2$$ inches<br>\r\nThe perimeter of a regular octagon, with side length \u2018a\u2019, is given by the following formula,<br>\r\nThe perimeter $$= 8a = 8 \\\\times 2 = 16$$ inches\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"16\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"Solution: Side of the octagon $$= 2$$ inches<br>\r\nThe perimeter of a regular octagon, with side length \u2018a\u2019, is given by the following formula,<br>\r\nThe perimeter $$= 8a = 8 \\\\times 2 = 16$$ inches\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"Solution: Side of the octagon $$= 2$$ inches<br>\r\nThe perimeter of a regular octagon, with side length \u2018a\u2019, is given by the following formula,<br>\r\nThe perimeter $$= 8a = 8 \\\\times 2 = 16$$ inches\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"An octagon has a 40-inch perimeter. Find the area.\",\n                    \"text\": \"An octagon has a 40-inch perimeter. Find the area.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"Octagon\u2019s perimeter $$= 8a$$<br>\r\n32 inches $$= 8a$$<br>\r\n$$a = \\\\frac{40}{8} = 5$$ inches<br>\r\nOctagon Area $$= 2 \\\\;a^{2}\\\\; (1\\\\;+\\\\;\\\\sqrt{2}) = 2 \\\\times 5^{2}\\\\; (1\\\\;+\\\\;\\\\sqrt{2})\\\\;\\\\simeq 120\\\\; inches^{2}$$\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"5 sq. in\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Octagon\u2019s perimeter $$= 8a$$<br>\r\n32 inches $$= 8a$$<br>\r\n$$a = \\\\frac{40}{8} = 5$$ inches<br>\r\nOctagon Area $$= 2 \\\\;a^{2}\\\\; (1\\\\;+\\\\;\\\\sqrt{2}) = 2 \\\\times 5^{2}\\\\; (1\\\\;+\\\\;\\\\sqrt{2})\\\\;\\\\simeq 120\\\\; inches^{2}$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"125 sq. in\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Octagon\u2019s perimeter $$= 8a$$<br>\r\n32 inches $$= 8a$$<br>\r\n$$a = \\\\frac{40}{8} = 5$$ inches<br>\r\nOctagon Area $$= 2 \\\\;a^{2}\\\\; (1\\\\;+\\\\;\\\\sqrt{2}) = 2 \\\\times 5^{2}\\\\; (1\\\\;+\\\\;\\\\sqrt{2})\\\\;\\\\simeq 120\\\\; inches^{2}$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"60 sq. in\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Octagon\u2019s perimeter $$= 8a$$<br>\r\n32 inches $$= 8a$$<br>\r\n$$a = \\\\frac{40}{8} = 5$$ inches<br>\r\nOctagon Area $$= 2 \\\\;a^{2}\\\\; (1\\\\;+\\\\;\\\\sqrt{2}) = 2 \\\\times 5^{2}\\\\; (1\\\\;+\\\\;\\\\sqrt{2})\\\\;\\\\simeq 120\\\\; inches^{2}$$\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 3,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"120 sq. in\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"Octagon\u2019s perimeter $$= 8a$$<br>\r\n32 inches $$= 8a$$<br>\r\n$$a = \\\\frac{40}{8} = 5$$ inches<br>\r\nOctagon Area $$= 2 \\\\;a^{2}\\\\; (1\\\\;+\\\\;\\\\sqrt{2}) = 2 \\\\times 5^{2}\\\\; (1\\\\;+\\\\;\\\\sqrt{2})\\\\;\\\\simeq 120\\\\; inches^{2}$$\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"Octagon\u2019s perimeter $$= 8a$$<br>\r\n32 inches $$= 8a$$<br>\r\n$$a = \\\\frac{40}{8} = 5$$ inches<br>\r\nOctagon Area $$= 2 \\\\;a^{2}\\\\; (1\\\\;+\\\\;\\\\sqrt{2}) = 2 \\\\times 5^{2}\\\\; (1\\\\;+\\\\;\\\\sqrt{2})\\\\;\\\\simeq 120\\\\; inches^{2}$$\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Each interior angle of a regular octagon measures _____ degrees.\",\n                    \"text\": \"Each interior angle of a regular octagon measures _____ degrees.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"Each interior angle of a regular octagon measures 135 degrees.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"85\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Each interior angle of a regular octagon measures 135 degrees.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"145\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Each interior angle of a regular octagon measures 135 degrees.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"67\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Each interior angle of a regular octagon measures 135 degrees.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"135\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"Each interior angle of a regular octagon measures 135 degrees.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"Each interior angle of a regular octagon measures 135 degrees.\"\n                      }\n                    } \n\n                    }]}<\/script><\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"9-frequently-asked-questions-about-octagon-formulas\">Frequently Asked Questions about Octagon Formulas<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-cdee7e44-5393-44ab-af14-f6867fb566c1\" data-mobilecollapse=\"true\" data-desktopcollapse=\"true\">\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-0-cdee7e44-5393-44ab-af14-f6867fb566c1\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-cdee7e44-5393-44ab-af14-f6867fb566c1\"><strong>What are the exterior angles of a regular octagon?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-0-cdee7e44-5393-44ab-af14-f6867fb566c1\">\n\n<p class=\"eplus-wrapper\">The exterior angles are the angles that form between a side and the extension of the side adjacent to it. A regular octagon\u2019s interior angles are all $135^{\\circ}$, so the exterior angle is measured as the angle supplement to it,&nbsp; $180^{\\circ} \\;-\\; 135^{\\circ} = 45^{\\circ}$.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-1-cdee7e44-5393-44ab-af14-f6867fb566c1\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-cdee7e44-5393-44ab-af14-f6867fb566c1\"><strong>How many diagonals does an octagon have?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-1-cdee7e44-5393-44ab-af14-f6867fb566c1\">\n\n<p class=\"eplus-wrapper\">The number of diagonals of a polygon of n sides is given by $\\frac{n(n-3)}{2}$, substituting the value $n = 8$, we have 20 diagonals.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-2-cdee7e44-5393-44ab-af14-f6867fb566c1\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-cdee7e44-5393-44ab-af14-f6867fb566c1\"><strong>Can we derive a formula for finding the area of an irregular octagon?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-2-cdee7e44-5393-44ab-af14-f6867fb566c1\">\n\n<p class=\"eplus-wrapper\">Since an irregular polygon (with more than 3 sides, heron\u2019s formula for a scalene triangle) has no fixed dimensions, we cannot find a direct formula for its area. We can divide the octagon into separate triangles and find the sum of the areas of those triangles.<\/p>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>What Are The Octagon Formulas? The octagon is an 8-sided polygon. An octagon is referred to as a regular octagon if all of its sides have equal lengths and angles are of equal measures. A regular octagon has each interior angle measuring $135^{\\circ}$ and each exterior angle measures $45^{\\circ}$.&nbsp; We will discuss octagon formulas such &#8230; <a title=\"Octagon Formula For Area and Perimeter With Derivation\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/octagon-formula\" aria-label=\"More on Octagon Formula For Area and Perimeter With Derivation\">Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-26054","post","type-post","status-publish","format-standard","hentry","category-all"],"featured_image_src":null,"author_info":{"display_name":"Mithun Jhawar","author_link":"https:\/\/www.splashlearn.com\/math-vocabulary\/author\/mithun-jhawarsplashlearn-com\/"},"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/26054","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/comments?post=26054"}],"version-history":[{"count":21,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/26054\/revisions"}],"predecessor-version":[{"id":39961,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/26054\/revisions\/39961"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=26054"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=26054"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=26054"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}