TRY THESE - NCERT

1 MENSURATION IntroductionWe have learnt that for a closed plane figure, the perimeter is the distance around itsboundary and its area is the region covered by it. We found the area and perimeter ofvarious plane figures such as triangles, rectangles, circles etc. We have also learnt to findthe area of pathways or borders in rectangular this chapter, we will try to solve problems related to perimeter and area of otherplane closed figures like will also learn about surface area and volume of solids such as cube, cuboid Let us RecallLet us take an example to review our previous is a figure of a rectangular park (Fig ) whose length is 30 m and width is 20 m.(i) What is the total length of the fence surrounding it? To find the length of the fence weneed to find the perimeter of this park, which is 100 m.(Check it)(ii)How much land is occupied by the park? To find theland occupied by this park we need to find the area ofthis park which is 600 square meters (m2) (How?)

2 (iii)There is a path of one metre width running inside alongthe perimeter of the park that has to be 1 bag of cement is required to cement 4 m2 area, howmany bags of cement would be required to construct thecemented path?We can say that the number of cement bags used = area of the patharea cemented by 1 of cemented path = Area of park Area of park not is 1 m wide, so the rectangular area not cemented is (30 2) (20 2) is 28 18 number of cement bags used =------------------(iv)There are two rectangular flower beds of size m 2 m each in the park asshown in the diagram (Fig ) and the rest has grass on it. Find the area coveredby MATHEMATICSTRY THESEArea of rectangular beds = ------------------Area of park left after cementing the path = ------------------Area covered by the grass = ------------------We can find areas of geometrical shapes other than rectangles also if certainmeasurements are given to us.

3 Try to recall and match the following:DiagramShapeArearectanglea asquareb htriangle b2parallelogram12bh circlea bCan you write an expression for the perimeter of each of the above shapes?49 cm277 cm298 cm2(a) Match the following figures with their respective areas in the box.(b) Write the perimeter of each 171 EXERCISE square and a rectangular field withmeasurements as given in the figure have the sameperimeter. Which field has a larger area? Kaushik has a square plot with themeasurement as shown in the figure. She wants toconstruct a house in the middle of the plot . A garden is developedaround the house. Find the total cost of developing a garden aroundthe house at the rate of ` 55 per shape of a garden is rectangular in the middle and semi circularat the ends as shown in the diagram. Find the area and the perimeterof this garden [Length of rectangle is20 ( + ) metres].

4 Flooring tile has the shape of a parallelogram whose base is 24 cm and thecorresponding height is 10 cm. How many such tiles are required to cover a floor ofarea 1080 m2? (If required you can split the tiles in whatever way you want to fill upthe corners). ant is moving around a few food pieces of different shapes scattered on the which food -piece would the ant have to take a longer round? Remember,circumference of a circle can be obtained by using the expression c = 2 r, where ris the radius of the circle.(a)(b)(c)(b)(a) Area of TrapeziumNazma owns a plot near a main road(Fig ). Unlike some other rectangularplots in her neighbourhood, the plot hasonly one pair of parallel opposite , it is nearly a trapezium in shape. Canyou find out its area?Let us name the vertices of this plot asshown in Fig drawing EC || AB, we can divide itinto two parts, one of rectangular shapeand the other of triangular shape, (whichis right angled at C), as shown in Fig (b = c + a = 30 m)Fig MATHEMATICSDO THISTRY THESEArea of ECD = 12h c = 112 102 = 60 of rectangle ABCE = h a = 12 20 = 240 of trapezium ABDE = area of ECD + Area of rectangle ABCE = 60 + 240 = 300 can write the area by combining the two areas and write the area of trapezium asarea of ABDE =12h c + h a = h 2ca + =h 222cacaah+++ = =()2bah+ = (sum of parallel sides)height2By substituting the values of h, b and a in this expression, we find ()2bah+ = 300 Nazma s sister also has a trapezium shaped plot .

5 Divide it into three parts as shown(Fig ). Show that the area of trapezium WXYZ ()2abh+=. Fig h = 10 cm, c = 6 cm, b = 12 cm,d = 4 cm, find the values of each ofits parts separetely and add to findthe area WXYZ. Verify it by puttingthe values of h, a and b in theexpression ()2ha b+.Fig Draw any trapezium WXYZ on a pieceof graph paper as shown in the figureand cut it out (Fig ).2. Find the mid point of XY by foldingthe side and name it A (Fig ).Fig 173DO THISTRY THESE3. Cut trapezium WXYZ into two pieces by cutting along ZA. Place ZYA as shownin Fig , where AY is placed on is the length of the base of the largertriangle? Write an expression for the area ofthis triangle (Fig ).4. The area of this triangle and the area of the trapezium WXYZ are same (How?).Get the expression for the area of trapezium by using the expression for the areaof to find the area of a trapezium we need to know the length of the parallel sides and theperpendicular distance between THESE two parallel sides.

6 Half the product of the sum ofthe lengths of parallel sides and the perpendicular distance between them gives the area the area of the following trapeziums (Fig ).(i)(ii)Fig Class VII we learnt to draw parallelograms of equal areas with different it be done for trapezium? Check if the following trapeziums are of equal areas buthave different perimeters (Fig ).Fig MATHEMATICSTRY THESEWe know that all congruent figures are equal in area. Can we say figures equal in areaneed to be congruent too? Are THESE figures congruent?Draw at least three trapeziums which have different areas but equal perimeters on asquared Area of a General QuadrilateralA general quadrilateral can be split into two triangles by drawing one of its diagonals. This triangulation helps us to find a formula for any general quadrilateral. Study the Fig of quadrilateral ABCD= (area of ABC) + (area of ADC)=(12AC h1) + (12AC h2)=12AC ( h1 + h2)=12 d ( h1 + h2) where d denotes the length of diagonal 1: Find the area of quadrilateral PQRS shown in Fig : In this case, d = cm, h1 = , h2 = cm,Area =12 d ( h1 + h2)=12 ( + ) cm2=12 4 cm2 = 11 cm2We know that parallelogram is also a quadrilateral.

7 Let usalso split such a quadrilateral into two triangles, find theirareas and hence that of the parallelogram. Does this agreewith the formula that you know already? (Fig ) Area of special quadrilateralsWe can use the same method of splitting into triangles (which we called triangulation ) tofind a formula for the area of a rhombus. In Fig ABCD is a rhombus. Therefore, itsdiagonals are perpendicular bisectors of each of rhombus ABCD = (area of ACD) + (area of ABC)Fig 175 TRY THESEFig , DISCUSS AND WRITE=(12 AC OD) + (12 AC OB) = 12AC (OD + OB)=12AC BD = 12d1 d2 where AC = d1 and BD = d2In other words, area of a rhombus is half the product of its 2: Find the area of a rhombus whose diagonals are of lengths 10 cm and :Area of the rhombus =12d1 d2 where d1, d2 are lengths of 10 cm2 = 41 parallelogram is divided into two congruent triangles by drawing a diagonal acrossit.

8 Can we divide a trapezium into two congruent triangles?Find the areaof thesequadrilaterals(Fig ). Area of a PolygonWe split a quadrilateral into triangles and find its area. Similar methods can be used to findthe area of a polygon. Observe the following for a pentagon: (Fig , )By constructing one diagonal AD and two perpendiculars BFand CG on it, pentagon ABCDE is divided into four parts. So,area of ABCDE = area of right angled AFB + area oftrapezium BFGC + area of right angled CGD + area of AED. (Identify the parallel sides of trapezium BFGC.)By constructing two diagonals AC and AD thepentagon ABCDE is divided into three , area ABCDE = area of ABC + area of ACD + area of (ii)(iii)Fig (i)176 MATHEMATICSTRY THESEFig (i) Divide the following polygons (Fig ) into parts (triangles and trapezium) tofind out its is a diagonal of polygon EFGHINQ is a diagonal of polygon MNOPQR(ii)Polygon ABCDE is divided into parts as shown below (Fig ).

9 Find its area ifAD = 8 cm, AH = 6 cm, AG = 4 cm, AF = 3 cm and perpendiculars BF = 2 cm,CH = 3 cm, EG = of Polygon ABCDE = area of AFB + ..Area of AFB = 12 AF BF = 12 3 2 = ..Area of trapezium FBCH = FH (BF CH)2+=3 (2 3)2+ [FH = AH AF]Area of CHD = 12 HD CH = ..; Area of ADE = 12 AD GE = ..So, the area of polygon ABCDE =..(iii)Find the area of polygon MNOPQR (Fig ) ifMP = 9 cm, MD = 7 cm, MC = 6 cm, MB = 4 cm,MA = 2 cmNA, OC, QD and RB are perpendiculars todiagonal 1: The area of a trapezium shaped field is 480 m2, the distance between twoparallel sides is 15 m and one of the parallel side is 20 m. Find the other parallel : One of the parallel sides of the trapezium is a = 20 m, let another parallelside be b, height h = 15 given area of trapezium =480 of a trapezium =12h (a + b)So 480 =12 15 (20 + b) or 480215 = 20 + bor 64 = 20 + b or b = 44 mHence the other parallel side of the trapezium is 44 177 Example 2: The area of a rhombus is 240 cm2 and one of the diagonals is 16 cm.

10 Findthe other : Let length of one diagonal d1 = 16 cmandlength of the other diagonal =d2 Area of the rhombus =12 d1 . d2 = 240So,21162d = 240 Therefore,d2 = 30 cmHence the length of the second diagonal is 30 3: There is a hexagon MNOPQR of side 5 cm (Fig ). Aman andRidhima divided it in two different ways (Fig ).Find the area of this hexagon using both : Aman s method:Since it is a hexagon so NQ divides the hexagon into two congruent trapeziums. You canverify it by paper folding (Fig ).Now area of trapezium MNQR = (115)42+ = 2 16 = 32 the area of hexagon MNOPQR = 2 32 = 64 s method: MNO and RPQ are congruent triangles with altitude3 cm (Fig ).You can verify this by cutting off THESE two triangles andplacing them on one of MNO = 12 8 3 = 12 cm2 = Area of RPQArea of rectangle MOPR = 8 5 = 40 , area of hexagon MNOPQR = 40 + 12 + 12 = 64 shape of the top surface of a table is a trapezium.