{"id":3254,"date":"2021-10-13T08:33:11","date_gmt":"2021-10-13T08:33:11","guid":{"rendered":"http:\/\/stgwebsite.mindspark.in\/wordpress\/?page_id=3254"},"modified":"2021-12-02T11:18:23","modified_gmt":"2021-12-02T11:18:23","slug":"area-of-parallelogram-using-diagonals","status":"publish","type":"page","link":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/math-concepts\/area-of-parallelogram-using-diagonals\/","title":{"rendered":"AREA OF PARALLELOGRAM USING DIAGONALS"},"content":{"rendered":"

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AREA OF PARALLELOGRAM USING DIAGONALS<\/strong><\/h2>\n

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All of you must have seen Kaju katli in your life. The shape of this delicious sweet is a parallelogram. There are many other examples in our surroundings that have the shape of a parallelogram like tiles, staircases, roofs etc. This makes the role of a quadrilateral more important and conceptual knowledge becomes necessary to understand these shapes in our real life. In this article, we are going to learn about the meaning, properties, area of a parallelogram using its diagonals, derivation of formulae and some illustrations.<\/p>\n

PARALLELOGRAM<\/strong><\/h2>\n

<\/strong><\/h3>\n

MEANING AND PROPERTIES OF PARALLELOGRAM<\/strong><\/h3>\n

A parallelogram is a 2-dimensional quadrilateral whose opposite sides and opposite angles are equal.\u00a0 A rectangle is a special type of parallelogram whose interior angles are 90\u00b0 and the diagonals are also equal and bisect each other. In the case of a general parallelogram, the diagonals are not equal but they bisect each other.<\/p>\n

Some of the properties of the parallelogram are –<\/strong><\/p>\n

    \n
  1. The opposite angles of a parallelogram are equal to each other.<\/li>\n
  2. The opposite sides of a parallelogram are parallel and equal.<\/li>\n
  3. The diagonals of a parallelogram bisect each other.<\/li>\n
  4. The sum of all the angles of a \u2225gram is 360\u00b0.<\/li>\n
  5. The sum of the adjacent angles is 180\u00b0.<\/li>\n
  6. Either of the diagonals of a \u2225gram divides it into two triangles of equal area.<\/li>\n<\/ol>\n

     <\/p>\n

    AREA OF PARALLELOGRAM USING DIAGONALS<\/strong><\/h2>\n

    We already know that the diagonals of a parallelogram bisect each other.<\/p>\n

    The area of a parallelogram can be determined with the help of diagonals.<\/p>\n

    Formula<\/strong><\/h3>\n

    Area = \\frac{1}{2}\\times d_1\\times d_2\\times \\sin\\theta<\/span><\/p>\n

    Where d_1<\/span> = length of one diagonal, d_2<\/span> = length of other diagonal and \\theta<\/span>= \u2220BOC between both the diagonals. The two angles that are formed are \u2220BOC and \u2220BOA when the diagonals intersect, we can use either of the angles because both will lead to the same result. \u2220DOA and \u2220DOC are equal to \u2220BOC and \u2220BOA respectively because they are vertically opposite angles.<\/p>\n

     <\/p>\n

    \"\"<\/p>\n

     <\/p>\n

     <\/p>\n

    DERIVATION OF AREA OF PARALLELOGRAM USING DIAGONALS<\/strong><\/h2>\n

     <\/p>\n

    \"\"<\/p>\n

     <\/p>\n

    We know that either of the two diagonals of a parallelogram divides it into two congruent triangles of equal area.<\/p>\n

    So, area of ABCD = 2 \u00d7 area of \u25b3BCD\u00a0\u00a0 – (1) (We are taking diagonal BD here)<\/p>\n

    Now, draw perpendicular CE on BD to find the area of \u25b3BCD.<\/p>\n

    Area of \u25b3BCD = \u00bd \u00d7 CE \u00d7 BD\u00a0\u00a0 – (2)<\/p>\n

    We can see that the \u25b3CEO is a right-angled triangle.<\/p>\n

    So, Sin\ud835\udeb9 = CE\/CO<\/p>\n

    \u21d2 CE = CO \u00d7 Sin\ud835\udeb9<\/p>\n

    \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 = Sin\ud835\udeb9 \u00d7 AC\/2\u00a0 ( AC = 2CO. Since, \u201cdiagonals of parallelogram bisect each other\u201d )<\/p>\n

    Putting the value of CE in (2)<\/p>\n

    Area of \u25b3BCD = \u00bd \u00d7 \u00bd \u00d7 Sin\ud835\udeb9 \u00d7 AC \u00d7 BD<\/p>\n

    \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 = \u00bc\u00a0 Sin\ud835\udeb9 \u00d7 AC \u00d7 BD – (3)<\/p>\n

    Putting the value of (3) in (1)<\/p>\n

    area of ABCD = 2 \u00d7 area of \u25b3BCD<\/p>\n

    \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0= 2 \u00d7 \u00bc\u00a0 Sin\ud835\udeb9 \u00d7 AC \u00d7 BD<\/p>\n

    \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0= \u00bd \u00d7 Sin\ud835\udeb9 \u00d7 AC \u00d7 BD<\/p>\n

    \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0=1\/2\\times d_1\\times d_2\\times \\sin\\theta \\text{ } {\\text{unit}}^{2}<\/span><\/p>\n

     <\/p>\n

    ILLUSTRATION<\/strong><\/h2>\n

    Q1. Prove that either of the diagonals of a parallelogram divides it into two triangles of equal area.<\/strong><\/p>\n

    \"\"<\/strong><\/p>\n

    <\/strong><\/p>\n

    Sol.-<\/strong><\/p>\n

    In parallelogram ABCD, considering diagonal AC, we get – \u25b3ADC and \u25b3ABC.<\/p>\n

    In \u25b3ADC and \u25b3CBA,<\/p>\n

    AD = BC\u00a0\u00a0 (opposite sides are equal to each other)<\/p>\n

    DC = AB\u00a0\u00a0 (opposite sides are equal to each other)<\/p>\n

    AC = AC\u00a0\u00a0 (Common side)<\/p>\n

    By Side Side Side (SSS) congruence rule,<\/p>\n

    \u25b3ADC \u2245 \u25b3CBA<\/p>\n

    Since they are congruent, their areas are the same.<\/p>\n

     <\/p>\n

    Q2. In a parallelogram ABCD, the lengths of the diagonals are 8 cm and 12 cm. The angle forming at the intersection of both the diagonals is 30\u00b0. Find the area of the parallelogram.<\/strong><\/p>\n

    Sol.-<\/strong><\/p>\n

    Given – d_1<\/span> = 8 cm, d_2<\/span> = 12 cm, \ud835\udeb9 = 30\u00b0<\/p>\n

    Area of parallelogram = 1\/2\\times d_1\\times d_2\\times \\sin\\theta<\/span><\/p>\n

    \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 = \u00bd \u00d7 8 \u00d7 12 \u00d7 sin30\u00b0<\/p>\n

    \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 =\u00a0 \u00bd \u00d7 8 \u00d7 12 \u00d7 \u00bd<\/p>\n

    \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 = 24 {\\text{cm}}^2<\/span><\/p>\n

     <\/p>\n

    Q3. Prove that the diagonals of a parallelogram divide each other into two portions of equal length.<\/strong><\/p>\n

    <\/strong><\/p>\n

    \"\"<\/strong><\/p>\n

    <\/strong><\/p>\n

    Sol.-<\/strong><\/p>\n

    For proving that the diagonals bisect each other, we need to show AO = OC and DO = OB.<\/p>\n

    In \u25b3AOD and \u25b3COB,<\/p>\n

    \u2220OAD = \u2220OCB\u00a0\u00a0\u00a0\u00a0 ( Alternate interior angles of a \u2225gram )<\/p>\n

    DA = CB\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0\u00a0\u00a0( Opposite sides are equal in a \u2225gram )<\/p>\n

    \u2220ADO = \u2220CBO\u00a0\u00a0\u00a0\u00a0 ( Alternate interior angles of a \u2225gram )<\/p>\n

    By Angle side angle(ASA) congruence rule,<\/p>\n

    \u25b3AOD \u2245 \u25b3COB<\/p>\n

    Thus, AO = CO & OD = OB\u00a0 (Since they are corresponding parts of congruent triangles)<\/p>\n

    Hence, it is proved that the diagonals of a parallelogram divide each other into two portions<\/p>\n

    of equal length.<\/p>\n

     <\/p>\n

    Q4. The height and base of a parallelogram is 16 cm and 10 cm respectively. Find the area of the parallelogram.<\/strong><\/p>\n

    Sol.-<\/strong><\/p>\n

    Area of parallelogram = Base (b) \u00d7 Height (h)<\/p>\n

    \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 = 16 \u00d7 10<\/p>\n

    \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 = 160 {\\text{cm}}^2<\/span><\/p>\n

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    Frequently Asked Questions<\/h1>\n

    Q1. What is the area of a parallelogram?<\/span><\/strong><\/p>\n

    Ans.<\/strong> When we talk about the area of a parallelogram, it is the region covered by the parallelogram within its perimeter in a 2-D plane.<\/span><\/p>\n

    Formula- Area of parallelogram = Base(b) \u00d7 Height(h) {\\text {unit}}^2<\/span><\/span>\u00a0<\/span><\/p>\n

    Q2. Are the diagonals of a parallelogram equal to each other?<\/span><\/strong><\/p>\n

    Ans. No, the diagonals of a parallelogram are not equal to each other but they bisect each other.<\/span><\/p>\n

    [\/et_pb_text][\/et_pb_column][\/et_pb_row][\/et_pb_section]<\/p>\n","protected":false},"excerpt":{"rendered":"

    AREA OF PARALLELOGRAM USING DIAGONALSAll of you must have seen Kaju katli in your life. The shape of this delicious sweet is a parallelogram. There are many other examples in our surroundings that have the shape of a parallelogram like tiles, staircases, roofs etc. This makes the role of a quadrilateral more important and conceptual […]<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":714,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_et_pb_use_builder":"on","_et_pb_old_content":"","_et_gb_content_width":"","footnotes":""},"yoast_head":"\nAREA OF PARALLELOGRAM USING DIAGONALS - mydomain<\/title>\n<meta name=\"description\" content=\"The area of a parallelogram using diagonals is \u00bd d1\u00d7d2sin\ud835\udeb9, where d1 = length of one diagonal, d2 = length of another diagonal & \ud835\udeb9 = angle between both the diagonals.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/stgwebsite.mindspark.in\/studymaterial\/math-concepts\/area-of-parallelogram-using-diagonals\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"AREA OF PARALLELOGRAM USING DIAGONALS - 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