{"id":57399,"date":"2025-03-23T12:55:40","date_gmt":"2025-03-23T07:25:40","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=57399"},"modified":"2026-02-25T04:17:34","modified_gmt":"2026-02-24T22:47:34","slug":"determine-area-of-an-irregular-plane-figure-using-geogebra-classic-5","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=57399","title":{"rendered":"Determine Area of an Irregular Plane Figure using Geogebra Classic 5"},"content":{"rendered":"\n<h5 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-89bfda0d449bddcb20c4924e685a3000\">Aim:<\/h5>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.\\ \\text{To mark the required number of points}\\ \\hspace{19cm}\\\\ \\text{on the boundary of the figure.}\\ \\hspace{10cm}\\]<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ \\text{To draw the boundary of the figure}\\ \\hspace{19cm}\\\\ \\text{by joining the points}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.\\ \\text{To divide the figure into trapeziums using}\\ \\hspace{19cm}\\\\ \\text{the points on the boundary.}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4.\\ \\text{To calculate the approximate are of the figure.}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<h5 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-caaadd60b535c837691a2eb0c3ab170e\">Procedure:<\/h5>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 1:}\\ \\text{Open Geogebra classic 5 (by double clicking on the icon)}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 2:}\\ \\text{Import the given image of the given irregular}\\ \\hspace{18cm}\\\\ \\text{plane figure by using the Image tool.}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 3:}\\  \\hspace{25cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[i.\\ \\hspace{1cm}\\ \\text{Two points A and B are provided to the set}\\ \\hspace{15cm}\\\\ \\text{the coordinates of the image.}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ii.\\ \\hspace{1cm}\\ \\text{Without changing the dimension of the image, drag the}\\ \\hspace{15cm}\\\\ \\text{image and fix the points A and B on the graphics view.}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[iii.\\ \\hspace{1cm}\\ \\text{Right click on the image, go to Object Properties}\\ \\hspace{15cm}\\\\ \\text{and set a Background image.}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 4:}\\ \\hspace{1cm}\\ \\text{To mark the vertices of the irregular}\\ \\hspace{18cm}\\\\ \\text{polygon using point tool.}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 5:}\\  \\hspace{25cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[i.\\ \\hspace{1cm}\\ \\text{For each point representing a vertex, mark another point on the}\\ \\hspace{15cm}\\\\ \\text{boundary of the polygon just vertically above.}\\ \\hspace{13cm}\\\\ \\text{or below the point if possible}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ii.\\ \\hspace{1cm}\\ \\text{After marking the point, adjust the x &#8211; coordinate of the newly}\\ \\hspace{15cm}\\\\ \\text{marked point in the input bar to match with the}\\ \\hspace{13cm}\\\\ \\text{x-coordinate of the vertex point}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 6:}\\ \\text{Let the vertices are marked with the points C,D,E,F,G and}\\ \\hspace{18cm}\\\\ \\text{corresponding vertical opposite points on the boundary of the}\\ \\hspace{14cm}\\\\ \\text{polygon corresponding to D,F and G are marked with}\\ \\hspace{14cm}\\\\ \\text{the points H, I and J respectively}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Note:}\\ \\text{Record x and y coordinates of the points C,D,E,F,G, H, I}\\ \\hspace{18cm}\\\\ \\text{and J by looking them in the input bar}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 7:}\\ \\text{Now the entire irregular polygon can be divided into triagnles}\\ \\hspace{18cm}\\\\ \\text{and trapeziums using the vertices and their vertically}\\ \\hspace{14cm}\\\\ \\text{opposite points on the boundary of the polygon}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Note:}\\ \\text{In this problem the polygon is divided into triangle CCJ, trapezium jGHD,}\\ \\hspace{15cm}\\\\ \\text{trapezium HFID and triangle FEI}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 8:}\\ \\text{Calculate the area of each subdivision of the polygon as following,}\\ \\hspace{15cm}\\\\ \\text{for illustration, we take the trapezium GHDJ}\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 9:}\\ \\text{Join the points J and D by a straight line}\\ \\hspace{18cm}\\\\ \\text{using the command l1:Line(J,D)}\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 10:}\\ \\text{Join the points G and H by a straight line}\\ \\hspace{18cm}\\\\ \\text{using the command l2:Line(G,H)}\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 11:}\\ \\text{Calculate the Area of the trapezium CADJ }\\ \\hspace{18cm}\\\\ \\text{using the command A2=Integral between (l1,l2,2.74,4.38)}\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 12:}\\ \\text{Determine the area of other subdivisions in the similar number and let }\\ \\hspace{15cm}\\\\ \\text{A1=Area of CGJ, A3= Area of HFID and A4=Area of FEI}\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 13:}\\ \\text{Now calculate the area of the whole polygon}\\ \\hspace{17cm}\\\\ \\text{using the command Total = A1+A2+A3+A4}\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{Step\\ 14:}\\ \\text{Verify the measurement of the area}\\ \\hspace{18cm}\\\\ \\text{using the input command Area(C,J,D,I,E,F,H,G,C)} \\hspace{10cm}\\]<\/div>\n\n\n\n<h5 class=\"wp-block-heading has-vivid-purple-color has-text-color has-link-color wp-elements-883bedab9864fd79dff46cd9aa765019\">Output:<\/h5>\n\n\n\n<p class=\"has-text-align-center\">Output for modelling the boundary of the polygon<\/p>\n\n\n\n<table id=\"tablepress-1\" class=\"tablepress tablepress-id-1 tbody-has-connected-cells\">\n<thead>\n<tr class=\"row-1\">\n\t<th class=\"column-1\">Vertex<\/th><th class=\"column-2\">x-coordinate<\/th><th class=\"column-3\">y-coordinate<\/th><th class=\"column-4\">Vertically opposite vertex<\/th><th class=\"column-5\">x-coordinate<\/th><th class=\"column-6\">y-coordinate<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-striping row-hover\">\n<tr class=\"row-2\">\n\t<td class=\"column-1\">C<br \/>\n<\/td><td class=\"column-2\">2.33<\/td><td class=\"column-3\">0.86<\/td><td colspan=\"3\" class=\"column-4\">No vertically opposite Vertex \/ Leave Blank<\/td>\n<\/tr>\n<tr class=\"row-3\">\n\t<td class=\"column-1\">D<\/td><td class=\"column-2\">4.38<\/td><td class=\"column-3\">2.71<\/td><td class=\"column-4\">H<\/td><td class=\"column-5\">4.38<\/td><td class=\"column-6\">0.43<\/td>\n<\/tr>\n<tr class=\"row-4\">\n\t<td class=\"column-1\">E<\/td><td class=\"column-2\">5.58<\/td><td class=\"column-3\">1.57<\/td><td colspan=\"3\" class=\"column-4\">No vertically opposite Vertex \/ Leave Blank<\/td>\n<\/tr>\n<tr class=\"row-5\">\n\t<td class=\"column-1\">F<\/td><td class=\"column-2\">4.96<\/td><td class=\"column-3\">0.42<\/td><td class=\"column-4\">I<\/td><td class=\"column-5\">4.96<\/td><td class=\"column-6\">2.16<\/td>\n<\/tr>\n<tr class=\"row-6\">\n\t<td class=\"column-1\">G<\/td><td class=\"column-2\">2.74<\/td><td class=\"column-3\">0.4<\/td><td class=\"column-4\">J<\/td><td class=\"column-5\">2.74<\/td><td class=\"column-6\">1.24<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-1 from cache -->\n\n\n\n<p class=\"has-text-align-center\">Output for Area of the Irregular Polygon<\/p>\n\n\n\n<table id=\"tablepress-2\" class=\"tablepress tablepress-id-2 tbody-has-connected-cells\">\n<thead>\n<tr class=\"row-1\">\n\t<th class=\"column-1\">Sub-division polygon<\/th><th class=\"column-2\">Equation of upper envelope<\/th><th class=\"column-3\">Equation of lower envelope<\/th><th class=\"column-4\">Left limit<\/th><th class=\"column-5\">Right limit<\/th><th class=\"column-6\">Area<\/th>\n<\/tr>\n<\/thead>\n<tbody class=\"row-striping row-hover\">\n<tr class=\"row-2\">\n\t<td class=\"column-1\">Triangle<br \/>\nCGJ<\/td><td class=\"column-2\">j=0.61<\/td><td class=\"column-3\">g=0.55<\/td><td class=\"column-4\">2.74<\/td><td class=\"column-5\">2.33<\/td><td class=\"column-6\">A1= 0.17<\/td>\n<\/tr>\n<tr class=\"row-3\">\n\t<td class=\"column-1\">Trapezium<br \/>\nGHDJ<\/td><td class=\"column-2\">g1=1.64<\/td><td class=\"column-3\">d=2.21<\/td><td class=\"column-4\">4.38<\/td><td class=\"column-5\">2.74<\/td><td class=\"column-6\">A2=2.56<\/td>\n<\/tr>\n<tr class=\"row-4\">\n\t<td class=\"column-1\">Trapezium<br \/>\nHFID<\/td><td class=\"column-2\">f=0.58<\/td><td class=\"column-3\">d1=0.81<\/td><td class=\"column-4\">4.96<\/td><td class=\"column-5\">4.38<\/td><td class=\"column-6\">A3=1.17<\/td>\n<\/tr>\n<tr class=\"row-5\">\n\t<td class=\"column-1\">Triangle FEI<\/td><td class=\"column-2\">i1=1.31<\/td><td class=\"column-3\">f1=0.85<\/td><td class=\"column-4\">5.58<\/td><td class=\"column-5\">4.96<\/td><td class=\"column-6\">A4=0.54<\/td>\n<\/tr>\n<tr class=\"row-6\">\n\t<td colspan=\"5\" class=\"column-1\">Area of the Given Polygon (A1+A2+A3+A4)<\/td><td class=\"column-6\">4.44<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<!-- #tablepress-2 from cache -->\n\n\n<p><iframe width=\"976\" height=\"549\" src=\"https:\/\/www.youtube.com\/embed\/Hlm1PJoagbY?list=PLQIom4Rz29vyp5ZiCGuc0-zCEqs8Pvslf\" title=\"Irregular polygon - Area - Application of Integration - Geogebra Classic 5 - Explained in Telugu\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Aim: Procedure: Output: Output for modelling the boundary of the polygon Output for Area of the Irregular 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