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Downloaded from https:// www.studiestoday.com, , , , , , , , , e Area and Perimeter of Plane e Volume and Surface Area of, Figures Cuboids, , , , 1. Find the area of a square, given its perimeter is 15.2 cm., 9. Find the perimeter of a 9.8 cm long rectangle, given its area is 53.9 cm?., 3. Find the total surface area of a cube, given its volume is 216 cm’., , aunit of measure for area equal to 100 square metres. The word, and the unit of measure, seems to have, : the French and derived from the Latin word ‘area’ with its current meaning. The are is seldom used today,, erivative form, the hectare is still a common unit of land measure in some countries., , er The word perimeter comes from the Greek word ‘peri’ (around) + ‘metron’ (measure)., Val a e is derived from the Latin word ‘volvere’, which means to turn or roll. The idea comes from the practice, g m scrolls that were then rolled into a cylinder., , , , , , , , , , , , , , , , , , , , , , , We'll have a 240 cm = 150 cm rectangular, | soft board hung on the wall, on which we, : can pin small charts. If the charts are of, size 80 cm x 50 cm, then how many would we, require to cover the entire board?, , Thanks for your help Rahul! Well, the area, of the rectangular board is 36000 cm*, and the area of one chart is 4000 cm’., That means, we can have nine such, , charts displayed on the board., , By Jove! That was quick, Priya, , , , , , , , , , a Why don't you answer, _| that yourself?, , , , Downloaded from hitps:// www.studiestoday.com @

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Downloaded from https:// www.studiestoday.com, , AREA AND PERIMETER OF, , PLANE FIGURES, , , , e Area, e Perimeter, , Let us first recall the units of area and their, conversions., , , , 1 km? = | km X 1 km = 1000 m x 1000 m = 1000000 m?, 1 hectare = 1 hm X 1 hm = 100 m x 100 m = 10000 m?, 1 are = 1 dem X 1 dem = 10m x 10 m= 100 m?, 1 m?=1mx1m=100cm x 100 cm = 10000 cm?, 1 m?=1mx1m=1000 mm x 1000 mm, , = 1000000 mm?, 1 cm? = 1 cm X 1 cm= 10 mm X 10 mm = 100 mm?, , , , , , , , Area and Perimeter of, A Squares and Rectangles, , , , , , Formulae, > P=4, , (P = Perimeter of square, £ = length), > P=2£+ 4), , (P = Perimeter of rectangle, £ = length, 6 = breadth), > A= (A= Area of square, £ = length), > A=fxb, , (A = Area of rectangle, £ = length, 5 = breadth), > Diagonal of a square = ,/2 length”, = J2 length oy, , >» Diagonal of a rectangle = length? + breadth™ 3 eee, , (by Pythagoras’ the a, , , , , , , , Example 1: The area of a plot of land whose length, is 7x and breadth is 5x is given as 2835 m2. Find the, cost of erecting a fence along its boundary at the, rate of Rs 12.50 per metre., , @, , , , Area = length x breadth = 2835 m?, => = 7xx 5x= 2835 m?, , 2835, => aa m? = 8] m?, , — x= V81 m=9m, , Thus, the length of the plot = 7 x 9 = 63 m and, breadth of the plot =5 x9=45m, Perimeter = 2(length + breadth), , = 2(63 + 45)m=2 x 108 = 216m, Cost of erecting fence along the boundary of the, plot = 216 m X Rs 12.50 = Rs 2700.00, , Example 2: The grass from a square lawn whose, diagonal measures 6/2 m is transplanted to a, rectangular lawn that is 4m wide. What is the length, of the rectangular lawn?, , Diagonal of a square = length /2 = 6/2 m, , 6/2, => length = mo =6m, => Area of the square lawn = 6 x 6 = 36 m?, Area of rectangular lawn is also 36 m?, Width or breadth of the rectangular lawn = 4 m, As length x breadth = Area of rectangular lawn,, , => length x 4 m = 36 m?, 36, , = _ length of the rectangular lawn = em, , =9m, Example 3: The area of a square field is 9 hectares., A 3 m wide road is constructed along its boundary, inside the field., , Downloaded from https:// www.studiestoday.com

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Downloaded from https:// www.studiestoday.com, , (i) How much area remains free for cultivation?, (ii) At Rs 18.50 per sq. m, how much did it cost, to construct the road?, , , , , , , , , , , , 300 m, , (i) Area of the field, = 9 hectares = 90000 m?, => length of the field = 90000 m = 300 m, Length of cultivable land, = 300m-3m —3m=294m, Area free for cultivation = 294 m x 294 m, = 86436 m? or 8.6436 hectares, (ii) Area of road constructed, = Area of the field — Area free for cultivation, = 90000 m2 — 86436 m? = 3564 m?, At Rs 18.50 per sq. m, cost of constructing, road = 3564 m? x Rs 18.50 = Rs 65934.00, , , , , , , , , , , , , , , , , , , , , , , , , , , , ina ed te dogo, 4 a a a, BEE ESEREee rh, A Area of a Triangle, , , , , , In equilateral AABC with all sides measuring, a units,, , Downloaded from https:// www.studiestoday.com, , AB? = AO? + OB? = AO? = AB? - OB?, (by Pythagoras’ theorem), , 2 2, , , , Pe ORLA ANe ac 92 7 A, = AO a (5) a i, 4a’ -a? 3a’ . c, x SORT Or Fy, , , , i ac ut, = AO =,{/— =—-v3 B, je ae, , , , , , a, , > Aude ofan equilateral ang = 7v3, a = enath of a side), , sa As area of equilateral AABC = + base x altitude, , € 1 oe, , Pau mus =, , > Area of equilateral triangle = — 53, Jj Ge = length of a side), > The Hero’s formula, discovered by a Greek, , mathematician, gives the area of a triangle with sides, 28 bs and semi-perimeter s as, , _ Area of a triangle = Js(s - a) (s—b) (s—c) unit?, , * atbte, | where s = 9, , , , , , , , , , Example 4: In rectangular farm ABCE, a tractor, has ploughed plot BCD, the dimensions of which, , are given in Figure 32.1. Find the area that remains, to be ploughed., , A B, , , , <, S, , , , , , , , ESS CO, , 10mD 240 m, Fig. 32.1, In right-angled ABCD,, , hypotenuse BD = 260 m and base DC = 240 m, , zo RPS Bp? Dc*, , = 260? — 2407, , = 67600 m? — 57600 m2 = 10000 m?, > BC = 100m, , @

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Downloaded from https:// www.studiestoday.com, , l, — base X altitude, , Thus, area ploughed = 9, , ], me x 240 x 100 =, , = 1.2 hectares, Now length of the farm = 240 + 10 = 250m, Breadth of the farm = 100 m, (from altitude of triangle calculated above), = Area of farm = 250 m x 100 m = 25000 m?, , = 2.5 hectares, Thus, area to be ploughed = 2.5 — 1.2, = 1.3 hectares, , Example 5: AD is the altitude of AABC, drawn, from A to side BC. Find the measure of AD, given, AB = 8 cm, BC = 10 cm, and AC = 6 cm., , Area of triangle = ,/s(s — a)(s — b)(s —c) unit”, , (Hero’s formula), => Area of AABC, , = ,/12(12—10)(12—8)(12—6) cm?, , , , 10+8+6 24, as 5 = —————- = — = 12cm, memes, , = Jl2x2x4x6 = J/576 cm?, , = Area of AABC = 24 cm?, But area of AABC, , l, = 9 xX base X altitude, , 1, = 24 cm? = ; x 10cm x AD, , 24x2 48, —— AND) =—- = 4.8 cm, , 10 10, , , , 12000 m?, , , , Example 6: Find the area and altitude of an, equilateral triangle, correct upto 3 decimal places,, in which each side measures 12 cm., , 2, Area of an equilateral triangle = e 3 unit?, , When a = 12 cm, area of the equilateral triangle, , 2, = <3 = 3603 cm?, , V3 = 1.732 (from the chapter on powers asnd roots), = Area of the equilateral triangle = 36 x 1.732, = 62.352 cm?, , a, 5¥3 units, 12 cm, altitude of the equilateral triangle, , V3, , = “7 x12 =6v3 cm, , => Altitude of the equilateral triangle = 6 x 1.732, = 10.392 cm, , Altitude of an equilateral triangle =, When a=, , cee eee i, Find the area of a Colca a, triangle given its base ‘measures, cm and its altitude is 6 cm., , , , , , Exercise 32.1, , 1. Find the area of a square whose perimeter is, 27.6 cm., , 2. Find the area of a 7.45 cm long rectangle whose, perimeter is 24.5 cm., , 3. The area of a rectangle whose sides are in the, ratio 5 : 2 is given as 3.6 m’. Find the perimeter, of the rectangle., , @, , 4. The area of a park whose length is 5x and., breadth is 4x is given as 1280 m?. Find the cost, of erecting a fence along its boundary at the ©, rate of Rs 11.35 per metre., , 5. Find the area of a square, given that its diagonal, measures 5/2 cm., , Downloaded from https:// www.studiestoday.com

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10., , whe, , iz:, , 13,, , 14., , bes, , 16., , $f., , Downloaded from https:// www.studiestoday.com, , . Find the measure of the diagonal of a square,, , whose area is given as 100 cm’, correct up to 2, decimal places., , . Find the area of a 14.4 cm long rectangle, given, , that its diagonal measures 15 cm., , . Find the measure of the diagonal of a 30 cm, , long rectangle whose area is given as 480 cm’., The floor tiles from a square room whose, diagonal measures 12,/2 m are removed and, fitted in a rectangular room that is 8 m wide., What is the length of the rectangular room?, The area of a square park is 25 ares. A gravel, path, 1.5 m wide, is laid along its boundary, inside the park. At Rs 6.50 per sq. m, how much, did it cost to lay the gravel path?, , Three 2 m wide paths criss-cross each other, as, shown in Figure 32.2, in a rectangular field, whose area is 6 hectares. If the field is 500 m, long, find the area that remains free for, cultivation., , , , Fig. 32.2, , At Rs 8.50 per sq. m, it costs Rs 2754 to lay, grass in a square lawn. At Rs 11.50 per metre,, find how much it would cost to erect a fence, along its boundary., , Find the area of a right-angled triangle, given, its base measures 14 cm and its altitude is 0.6, times its base., , Find the area of a right-isosceles triangle, given, that one of its equal sides measures 11 cm., Find the area of an equilateral triangle, given, that one of its equal sides measures 6 cm., (Take /3 = 1.73), , Find the altitude of an equilateral triangle, given, that one of its equal sides measures 5/3 cm., Find the area of a scalene triangle, given the, , measures of its three sides as 9 cm, 12 cm, 15 cm., , Downloaded from hitps:// www.studiestoday.com, , 18., , 19., , 20., , eb., , 22., , 2., , The three sides of a scalene triangle measure, 3 cm, 4 cm, and 5 cm. Find the altitude of the, triangle when the shortest side is its base., In APQR (Figure 32.3), Q, PQ = 8 cm and, , PR = 10 cm. If altitude, , RY on PQ measures 9 cm,, , find the measure of Xx, altitude QX on PR. Fig. 32.3, In AABC (Figure 32.4), Cc, AB = 6 cm and AC = 9 cm. If, , altitude BX on AC measures 5.5, , cm, find the measure of altitude x, , CY on side AB. A Y B, , The base of a triangle is given _— Fig, 32.4, , as 7x while its altitude is given, , as 5x. If the area of the, , triangle is given as 2117.5 cm”, find its base and, altitude., , Find the area of trapezium ABCE (Figure 32.5),, given ABCD is a square and ADE is a rightangled triangle., , A, , , , D, , <—_><+_—, , 38cm 6 cm, , o 6cm WD, , , , , , E, , , , Fig. 32.5, , Find the area of the shaded portion in, Figure 32.6., , 5 cm 15cm 5cm, , <>< —$$_____> <>, , , , , , , , , , | ey |S, 5 cm 15 cm 5 cm, Fig. 32.6