Where:<\/span><\/p>\n A Chord divides the circle into two regions, where one is larger than the other forming two types of segments, that are:<\/span><\/p>\n Segment of a Circle<\/i><\/b><\/p>\n <\/i><\/b><\/p>\n Area of a Segment<\/i><\/b><\/p>\n <\/span><\/p>\n In the circle above A is the centre of the circle, CD is a chord that forms a minor segment which is shown as the shaded region. Let \u2018r\u2019 be the radius of the circle and the angle subtended at the centre by the arc CD be <\/span>\u03b8<\/span>.<\/span><\/p>\n Now to calculate the area of the segment, first, we need to find the area of the <\/span>\u2206<\/span>ACD and area of the sector ACD. The area of the segment is the difference between the area of the sector ACD and the area of the <\/span>\u2206<\/span>ACD.\u00a0<\/span><\/p>\n From trigonometry, we know,<\/span><\/p>\n The area of <\/span>\u2206<\/span>ACD<\/span> =\\frac{1}{2} r^{2} \\sin \\theta<\/span><\/span><\/p>\n Now, we also know the formula for the area of a sector is,<\/p>\n Area of sector =\\left(\\frac{\\theta}{360^{\\circ}}\\right) \\times \\pi r^{2}<\/span>, if <\/span>\u03b8 is in degrees.<\/span><\/p>\n And, Area of sector =\\frac{1}{2} \\times r^{2} \\theta<\/span> <\/span>if \u03b8 is in radians.<\/span><\/p>\n Thus, Area of the segment ACD (when <\/span>\u03b8<\/span> is in degrees)<\/span><\/p>\n \u00a0= Area of sector ACD – Area of \u2206ACD<\/span><\/p>\n = \\left(\\left(\\frac{\\theta}{360^{\\circ}}\\right) \\times \\pi r^{2}\\right)-\\left(\\frac{1}{2} r^{2} \\sin \\theta\\right)<\/span><\/span><\/p>\n Area of the segment ACD (when <\/span>\u03b8<\/span> is in radians)<\/span><\/p>\n = Area of sector ACD – Area of \u2206ACD<\/span><\/p>\n = \\left(\\frac{1}{2} \\times r^{2} \\theta\\right)-\\left(\\frac{1}{2} r^{2} \\sin \\theta\\right)<\/span><\/span><\/p>\n This formula is used to calculate the area of a minor segment when the value of the angle subtended at the centre and the radius of the circle is provided.<\/span><\/p>\n Unless specified segment is usually referred to as a minor segment only. But if asked to determine the area of the major segment, then the difference between the area of the circle and the area of the minor segment is calculated.<\/span><\/p>\n <\/span><\/p>\n <\/b><\/p>\n The perimeter of the segment = length of the chord + length of the arc.<\/span><\/p>\n Now, we know the formula for the length of an arc is, which is:<\/span><\/p>\n Length of Arc <\/span>= r\u03b8, if \u03b8 is in radians,<\/span><\/b><\/p>\n \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0= \\pi r \\frac{\\theta}{180^{\\circ}}<\/span> if \u03b8 is in degrees.<\/span><\/b><\/p>\n Length of the Chord =2 r \\sin \\left(\\frac{8}{2}\\right)<\/span><\/span><\/b><\/p>\n Therefore, the perimeter of a segment can be easily calculated.<\/span><\/p>\n <\/span><\/p>\n There are two important theorems based on the segment of a circle, which are:<\/span><\/p>\n 1. Angles in the same segment theorem<\/b><\/p>\n Statement:<\/strong> The angles subtended in the segment of a circle are always equal.<\/span><\/p>\n <\/span><\/p>\n There are two angles subtended in the segment AB. According to the theorem:<\/span><\/p>\n \u2220AOB = \u2220AQB<\/span><\/p>\n <\/span><\/p>\n <\/b><\/p>\n 2. Alternate segment theorem<\/b><\/p>\n <\/b>Statement:<\/strong> The angle formed by the tangent and the chord at the point of contact is equal to the angle formed in the alternate segment on the circumference of the circle through the endpoints of the chord.<\/span><\/p>\n <\/span><\/p>\n According to the theorem \u2220PRS = \u2220PQR and also \u2220RPQ = \u2220QRT.<\/span><\/p>\n <\/strong><\/p>\n 1. Find the area and perimeter of the segment of a circle that extends an angle of measurement and the radius of the circle is 5 units. Take \u03c0 = 3.14.<\/strong><\/p>\n Solution:<\/strong><\/p>\n The radius of the given circle is 5 units and <\/span>\u03b8<\/span> = 60<\/span>\u00b0.<\/span><\/p>\n Area of the segment OCD (when <\/span>\u03b8<\/span> is in degrees)<\/span><\/p>\n = Area of sector OCD – Area of \u2206OCD<\/span><\/p>\n = \\left(\\left(\\frac{\\theta}{360^{\\circ}}\\right) \\times \\pi r^{2}\\right)-\\left(\\frac{1}{2} r^{2} \\sin \\theta\\right)<\/span><\/span><\/p>\n = \\left(\\left(\\frac{60^{\\circ}}{360^{\\circ}}\\right) \\times 3.14 \\times 5^{2}\\right)-\\left(\\frac{1}{2} \\times 5^{2} \\times \\sin 60^{\\circ}\\right) \\text { sq units }<\/span><\/span><\/p>\n = \\left(\\frac{1}{6} \\times 3.14 \\times 25\\right)-\\left(\\frac{1}{2} \\times 25 \\times \\frac{\\sqrt{3}}{2}\\right) \\text { sq units }<\/span>,\\sin 60^{\\circ}=\\frac{\\sqrt{3}}{2}<\/span><\/span><\/p>\n = 13.08 – 10.83 sq units = 2.25 sq units<\/span><\/p>\n \u00a0<\/span>\u2234<\/span> Area of the segment = 2.25 sq units.<\/span><\/p>\n The perimeter of segment OCD<\/span><\/p>\n = length of chord CD + length of arc CD<\/span><\/p>\n =2 r \\sin \\left(\\frac{\\theta}{2}\\right)+\\pi r \\frac{\\theta}{180^{\\circ}}<\/span><\/span><\/p>\n =\\left(2 \\times 5 \\times \\sin \\left(\\frac{60^{\\circ}}{2}\\right)\\right)+\\left(3.14 \\times 5 \\times \\frac{60^{\\circ}}{180^{\\circ}}\\right) \\text { units }<\/span><\/span><\/span><\/p>\n =\\left(2 \\times 5 \\times \\sin 30^{\\circ}\\right)+\\left(3.14 \\times 5 \\times \\frac{1}{3}\\right) \\text { units }<\/span><\/span><\/p>\n =\\left(2 \\times 5 \\times \\frac{1}{2}\\right)+(5.23) \\text { units }<\/span><\/span><\/p>\n =5+5.23 \\text { units }=10.23 \\text { units }<\/span><\/span><\/p>\n \u2234 <\/span>The perimeter of the segment of the circle is 10.23 units.<\/span><\/p>\n <\/b><\/p>\n <\/span><\/p>\n <\/span><\/p>\n <\/b><\/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 [\/et_pb_text][\/et_pb_column][\/et_pb_row][et_pb_row admin_label=”Row for space” _builder_version=”4.10.6″ _module_preset=”default” locked=”off” 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global_colors_info=”{}”][et_pb_image src=”https:\/\/stgwebsite.mindspark.in\/wordpress\/wp-content\/uploads\/2021\/08\/calloutImage.png” title_text=”calloutImage” show_bottom_space=”off” admin_label=”Image” module_class=”img1″ _builder_version=”4.10.8″ _module_preset=”default” width=”25px” height=”60px” custom_padding=”2px||2px||true|false” global_colors_info=”{}”][\/et_pb_image][et_pb_text module_class=”ftstyle” _builder_version=”4.9.10″ _module_preset=”default” text_orientation=”center” global_colors_info=”{}”]<\/p>\n\n
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![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/pie-300x137.png\")
<\/span><\/p>\nArea of a Segment of a Circle<\/b><\/h2>\n
![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/pie-2-300x235.png\")
<\/b><\/p>\nPerimeter of a Segment of a Circle<\/b><\/h2>\n
Theorems<\/b><\/h2>\n
![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/segment2-300x258.png\")
<\/span><\/p>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/pie-4-300x228.png\")
<\/span><\/p>\n<\/b><\/h2>\n
Example<\/b><\/h2>\n
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\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