1. Regular Octagon<\/strong><\/p>\n \u00a0<\/b><\/p>\n 2. Irregular Octagon<\/strong><\/p>\n \u00a0<\/b><\/p>\n The sum of all interior angles of an octagon is equal to <\/span>1080\u00b0.<\/span><\/p>\n <\/p>\n Given below is a regular octagon of side \u2018s\u2019 divided into eight equivalent triangles.<\/span><\/p>\n So, the area of the octagon is equal to 8 times the area of each triangle.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n The above figure shows one of the triangles which is a part of the octagon.<\/span><\/p>\n In <\/span>\u25b3<\/span>ABC<\/span><\/p>\n 1. AB = AC<\/span><\/p>\n 2. BC = s<\/span><\/p>\n 3. \u2220BAC = \\frac{360^{\\circ}}{8}=45^{\\circ}<\/span><\/span><\/p>\n 4. AD is a perpendicular bisector of BC.<\/span><\/p>\n \u21d2 \u2220ADC = 90\u00b0\u00a0<\/span><\/p>\n \\mathrm{BD}=\\mathrm{DC}=\\frac{s}{2}<\/span><\/span><\/span><\/p>\n 5. AD also bisects \u2220BAC.<\/span><\/p>\n\\angle B A D=\\angle D A C=\\frac{45^{\\circ}}{2}<\/span>\n <\/span><\/p>\n Now, In <\/span>\u25b3<\/span>DAC\u00a0<\/span><\/p>\n 1. \u2220ADC = 90\u00b0\\angle \\mathrm{ADC}=90^{\\circ}<\/span><\/span><\/p>\n 2. \\angle \\mathrm{DAC}=\\frac{45^{\\circ}}{2}<\/span><\/span><\/p>\n 3. \\tan \\angle \\mathrm{DAC}=\\tan \\left(\\frac{45^{\\circ}}{2}\\right)=\\frac{D C}{A D}<\/span><\/span><\/p>\n \u00a0 \u00a0 \u00a0\\Rightarrow \\quad A D=\\frac{D C}{\\tan \\frac{45^{\\circ}}{2}}<\/span><\/span><\/p>\n <\/span><\/p>\n According to trigonometry identities<\/span><\/p>\n \\tan \\frac{\\theta}{2}=\\frac{\\sin \\theta}{1+\\cos \\theta}<\/span><\/span><\/p>\n <\/span><\/p>\n Substituting the value of tan \\frac{45^{\\circ}}{2}<\/span><\/span>\u00a0and DC in the equation<\/span><\/p>\n \\mathrm{AD}=\\frac{D C}{\\tan \\frac{45}{2}}=\\frac{s \/ 2}{\\frac{1}{1+\\sqrt{2}}}=\\frac{\\sqrt{2}+1}{2} \\mathrm{~s}<\/span><\/span><\/p>\n \\text { Area of } \\triangle \\mathrm{ABC}=\\frac{1}{2} \\times A D \\times B C<\/span><\/span><\/p>\n \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 =\\frac{1}{2} \\times \\frac{\\sqrt{2}+1}{2} s \\times s <\/span><\/span><\/p>\n <\/p>\n Area of Octagon = 8 \u00d7 Area of <\/span>\u25b3<\/span>ABC\u00a0<\/span><\/p>\n \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 = 8 \\times \\frac{\\sqrt{2}+1}{4} s^{2}<\/span> <\/span><\/p>\n \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 = 2(\\sqrt{2}+1) s^{2}<\/span><\/span><\/p>\n <\/p>\n Hence the area of a regular octagon is 2(\\sqrt{2}+1) s^{2}<\/span>, <\/span>where s is the length of each side.\u00a0<\/span><\/span><\/p>\n <\/span><\/p>\n There is no defined formula for finding the area of an irregular octagon.<\/span><\/p>\n In this case, we have to divide the octagon into different polygons according to the question and then find the area of the irregular octagon by adding the area of the different polygons.<\/span><\/p>\n <\/p>\n <\/b><\/p>\n 1. The length of each side of a regular octagon is equal to 15 cm. Find the area of this octagon?<\/b><\/p>\n Solution:<\/strong><\/p>\n Length of each side = s = 15 cm<\/span><\/p>\n Using the area formula\u00a0<\/span><\/p>\n \\text { Area }= 2(\\sqrt{2}+1) s^{2} <\/span><\/span><\/p>\n Hence the area of this octagon is equal to 1086.3961 \\mathrm{~cm}^{2}<\/span><\/span><\/p>\n <\/span><\/p>\n 2. The area of octagon having equal sides is equal to 82 cm\u00b2<\/b>. Find the length of each side?<\/b><\/p>\n Solution:<\/strong><\/p>\n \\text { Area }=2(\\sqrt{2}+1) s^{2} <\/span><\/span><\/p>\n Hence the length of each side is equal to 4.12 cm.<\/span><\/p>\n [\/et_pb_text][\/et_pb_column][et_pb_column type=”2_5″ module_id=”stickysideR” admin_label=”Column R” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#fdefe0″ custom_padding=”25px|25px|25px|25px|true|true” sticky_position=”top” sticky_offset_top=”-280px” sticky_limit_top=”row” sticky_limit_bottom=”row” sticky_position_tablet=”none” sticky_position_phone=”none” sticky_position_last_edited=”on|desktop” sticky_limit_bottom_tablet=”” sticky_limit_bottom_phone=”” sticky_limit_bottom_last_edited=”on|phone” border_radii=”on|15px|15px|15px|15px” box_shadow_style=”preset3″ global_colors_info=”{}”][et_pb_image src=”https:\/\/eistudymaterial.s3.amazonaws.com\/1080×1080.png” alt=”Free Trial banner” title_text=”Mindspark Free Trial Banner” url=”https:\/\/mindspark.in\/free-trial” align=”center” module_class=”adsimg” _builder_version=”4.11.1″ _module_preset=”default” custom_padding=”||||false|false” border_radii=”on|10px|10px|10px|10px” global_colors_info=”{}” transform_styles__hover_enabled=”on|hover” transform_scale__hover_enabled=”on|hover” transform_translate__hover_enabled=”on|desktop” transform_rotate__hover_enabled=”on|desktop” transform_skew__hover_enabled=”on|desktop” transform_origin__hover_enabled=”on|desktop” transform_scale__hover=”102%|102%”][\/et_pb_image][et_pb_text admin_label=”Explore Other Topics [\/et_pb_text][et_pb_text _builder_version=”4.10.7″ _module_preset=”default” text_line_height=”2.2em” link_font_size=”16px” custom_margin=”||0px||false|false” custom_padding=”10px|15px|10px|28px|true|false” locked=”off” global_colors_info=”{}”]<\/p>\n [\/et_pb_text][et_pb_text admin_label=”Related Concepts [\/et_pb_text][et_pb_text _builder_version=”4.13.1″ _module_preset=”default” text_line_height=”2.2em” link_font_size=”16px” custom_margin=”||0px||false|false” custom_padding=”10px|15px|10px|28px|true|false” locked=”off” global_colors_info=”{}”]<\/p>\n\n
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Area of a regular octagon<\/b><\/h2>\n
![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Area-of-octagon-01-280x300.png\")
<\/span><\/p>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Area-of-octagon-02.png\")
<\/span><\/p>\n
\\tan \\frac{45^{\\circ}}{2}=\\frac{\\sin 45^{\\circ}}{1+\\cos 45^{\\circ}}<\/span><\/span><\/p>\n=\\frac{\\frac{1}{\\sqrt{2}}}{1+\\frac{1}{\\sqrt{2}}} <\/span>\n
=\\frac{1}{1+\\sqrt{2}}<\/span>
<\/span><\/span><\/p>\n
=\\frac{\\sqrt{2}+1}{4} s^{2}<\/span><\/span><\/p>\nArea of an Irregular octagon<\/b><\/h2>\n
Solved examples<\/b><\/h2>\n
=2(\\sqrt{2}+1) 15^{2}<\/span><\/span><\/p>\n
=1086.3961 \\mathrm{~cm}^{2}<\/span><\/span><\/p>\n
\\Rightarrow 82=2(\\sqrt{2}+1) s^{2} <\/span><\/span><\/p>\n
\\Rightarrow s^{2}=\\frac{82}{2(\\sqrt{2}+1)}=16.98 <\/span><\/span><\/p>\n
\\Rightarrow s=\\sqrt{16.98}=4.12 \\mathrm{~cm}<\/span><\/span><\/p>\n
\n” _builder_version=”4.9.11″ _module_preset=”default” header_font=”|700|||||||” header_font_size=”25px” text_orientation=”center” custom_margin=”0px||0px||true|false” custom_padding=”8px|15px|0px|15px|false|true” locked=”off” global_colors_info=”{}”]<\/p>\nExplore Other Topics<\/h1>\n
\n” _builder_version=”4.9.11″ _module_preset=”default” header_font=”|700|||||||” header_font_size=”25px” text_orientation=”center” custom_margin=”0px||0px||true|false” custom_padding=”8px|15px|0px|15px|false|true” locked=”off” global_colors_info=”{}”]<\/p>\nRelated Concepts<\/h1>\n