NCERT Textbook: Surface Area and Volumes | Mathematics (Maths) Class 9 PDF Download (2024)

Please wait while the PDF view is loading

 Page 1SURFACE AREAS AND VOLUMES 137CHAPTER 11SURFACE AREAS AND VOLUMES11.1 Surface Area of a Right Circular ConeWe have already studied the surface areas of cube, cuboid and cylinder. We will nowstudy the surface area of cone.So far, we have been generating solids by stacking up congruent figures. Incidentally,such figures are called prisms. Now let us look at another kind of solid which is not aprism (These kinds of solids are called pyramids.). Let us see how we can generatethem.Activity : Cut out a right-angled triangle ABC right angled at B. Paste a long thickstring along one of the perpendicular sides say AB of the triangle [see Fig. 11.1(a)].Hold the string with your hands on either sides of the triangle and rotate the triangleabout the string a number of times. What happens? Do you recognize the shape thatthe triangle is forming as it rotates around the string [see Fig. 11.1(b)]? Does it remindyou of the time you had eaten an ice-cream heaped into a container of that shape [seeFig. 11.1 (c) and (d)]?Fig. 11.1Rationalised 2023-24Page 2SURFACE AREAS AND VOLUMES 137CHAPTER 11SURFACE AREAS AND VOLUMES11.1 Surface Area of a Right Circular ConeWe have already studied the surface areas of cube, cuboid and cylinder. We will nowstudy the surface area of cone.So far, we have been generating solids by stacking up congruent figures. Incidentally,such figures are called prisms. Now let us look at another kind of solid which is not aprism (These kinds of solids are called pyramids.). Let us see how we can generatethem.Activity : Cut out a right-angled triangle ABC right angled at B. Paste a long thickstring along one of the perpendicular sides say AB of the triangle [see Fig. 11.1(a)].Hold the string with your hands on either sides of the triangle and rotate the triangleabout the string a number of times. What happens? Do you recognize the shape thatthe triangle is forming as it rotates around the string [see Fig. 11.1(b)]? Does it remindyou of the time you had eaten an ice-cream heaped into a container of that shape [seeFig. 11.1 (c) and (d)]?Fig. 11.1Rationalised 2023-24138 MATHEMA TICSThis is called a right circular cone. In Fig. 11.1(c)of the right circular cone, the point A is called thevertex, AB is called the height, BC is called the radiusand AC is called the slant height of the cone. Here Bwill be the centre of circular base of the cone. Theheight, radius and slant height of the cone are usuallydenoted by h, r and l respectively. Once again, let ussee what kind of cone we can not call a right circularcone. Here, you are (see Fig. 11.2)! What you see inthese figures are not right circular cones; because in(a), the line joining its vertex to the centre of its baseis not at right angle to the base, and in (b) the base isnot circular.As in the case of cylinder, since we will be studying only about right circular cones,remember that by ‘cone’ in this chapter, we shall mean a ‘right circular cone.’Activity : (i) Cut out a neatly made paper cone that does not have any overlappedpaper, straight along its side, and opening it out, to see the shape of paper that formsthe surface of the cone. (The line along which you cut the cone is the slant height ofthe cone which is represented by l). It looks like a part of a round cake.(ii) If you now bring the sides marked A and B at the tips together, you can see thatthe curved portion of Fig. 11.3 (c) will form the circular base of the cone.Fig. 11.3(iii) If the paper like the one in Fig. 11.3 (c) is now cut into hundreds of little pieces,along the lines drawn from the point O, each cut portion is almost a small triangle,whose height is the slant height l of the cone.(iv) Now the area of each triangle = 12 × base of each triangle × l.So, area of the entire piece of paperFig. 11.2Rationalised 2023-24Page 3SURFACE AREAS AND VOLUMES 137CHAPTER 11SURFACE AREAS AND VOLUMES11.1 Surface Area of a Right Circular ConeWe have already studied the surface areas of cube, cuboid and cylinder. We will nowstudy the surface area of cone.So far, we have been generating solids by stacking up congruent figures. Incidentally,such figures are called prisms. Now let us look at another kind of solid which is not aprism (These kinds of solids are called pyramids.). Let us see how we can generatethem.Activity : Cut out a right-angled triangle ABC right angled at B. Paste a long thickstring along one of the perpendicular sides say AB of the triangle [see Fig. 11.1(a)].Hold the string with your hands on either sides of the triangle and rotate the triangleabout the string a number of times. What happens? Do you recognize the shape thatthe triangle is forming as it rotates around the string [see Fig. 11.1(b)]? Does it remindyou of the time you had eaten an ice-cream heaped into a container of that shape [seeFig. 11.1 (c) and (d)]?Fig. 11.1Rationalised 2023-24138 MATHEMA TICSThis is called a right circular cone. In Fig. 11.1(c)of the right circular cone, the point A is called thevertex, AB is called the height, BC is called the radiusand AC is called the slant height of the cone. Here Bwill be the centre of circular base of the cone. Theheight, radius and slant height of the cone are usuallydenoted by h, r and l respectively. Once again, let ussee what kind of cone we can not call a right circularcone. Here, you are (see Fig. 11.2)! What you see inthese figures are not right circular cones; because in(a), the line joining its vertex to the centre of its baseis not at right angle to the base, and in (b) the base isnot circular.As in the case of cylinder, since we will be studying only about right circular cones,remember that by ‘cone’ in this chapter, we shall mean a ‘right circular cone.’Activity : (i) Cut out a neatly made paper cone that does not have any overlappedpaper, straight along its side, and opening it out, to see the shape of paper that formsthe surface of the cone. (The line along which you cut the cone is the slant height ofthe cone which is represented by l). It looks like a part of a round cake.(ii) If you now bring the sides marked A and B at the tips together, you can see thatthe curved portion of Fig. 11.3 (c) will form the circular base of the cone.Fig. 11.3(iii) If the paper like the one in Fig. 11.3 (c) is now cut into hundreds of little pieces,along the lines drawn from the point O, each cut portion is almost a small triangle,whose height is the slant height l of the cone.(iv) Now the area of each triangle = 12 × base of each triangle × l.So, area of the entire piece of paperFig. 11.2Rationalised 2023-24SURFACE AREAS AND VOLUMES 139= sum of the areas of all the triangles=1 2 31 1 12 2 2b l b l b l + + +? = ( )1 2 312l b b b + + +?=12 × l × length of entire curved boundary of Fig. 11.3(c)(as b1 + b2 + b3 + . . . makes up the curved portion of the figure)But the curved portion of the figure makes up the perimeter of the base of the coneand the circumference of the base of the cone = 2pr, where r is the base radius of thecone.So, Curved Surface Area of a Cone = 12 × l × 2p p p p pr = p p p p prlwhere r is its base radius and l its slant height.Note that l2 = r2 + h2 (as can be seen from Fig. 11.4), byapplying Pythagoras Theorem. Here h is the height of thecone.Therefore, l = 2 2r h +Now if the base of the cone is to be closed, then a circular piece of paper of radius ris also required whose area is pr2.So, Total Surface Area of a Cone = p p p p prl + p p p p pr2 = p p p p pr(l + r)Example 1 : Find the curved surface area of a right circular cone whose slant heightis 10 cm and base radius is 7 cm.Solution : Curved surface area = prl=227 × 7 × 10 cm2= 220 cm2Example 2 : The height of a cone is 16 cm and its base radius is 12 cm. Find thecurved surface area and the total surface area of the cone (Use p = 3.14).Solution : Here, h = 16 cm and r = 12 cm.So, from l2 = h2 + r2, we havel =2 216 12 + cm = 20 cmFig. 11.4Rationalised 2023-24Page 4SURFACE AREAS AND VOLUMES 137CHAPTER 11SURFACE AREAS AND VOLUMES11.1 Surface Area of a Right Circular ConeWe have already studied the surface areas of cube, cuboid and cylinder. We will nowstudy the surface area of cone.So far, we have been generating solids by stacking up congruent figures. Incidentally,such figures are called prisms. Now let us look at another kind of solid which is not aprism (These kinds of solids are called pyramids.). Let us see how we can generatethem.Activity : Cut out a right-angled triangle ABC right angled at B. Paste a long thickstring along one of the perpendicular sides say AB of the triangle [see Fig. 11.1(a)].Hold the string with your hands on either sides of the triangle and rotate the triangleabout the string a number of times. What happens? Do you recognize the shape thatthe triangle is forming as it rotates around the string [see Fig. 11.1(b)]? Does it remindyou of the time you had eaten an ice-cream heaped into a container of that shape [seeFig. 11.1 (c) and (d)]?Fig. 11.1Rationalised 2023-24138 MATHEMA TICSThis is called a right circular cone. In Fig. 11.1(c)of the right circular cone, the point A is called thevertex, AB is called the height, BC is called the radiusand AC is called the slant height of the cone. Here Bwill be the centre of circular base of the cone. Theheight, radius and slant height of the cone are usuallydenoted by h, r and l respectively. Once again, let ussee what kind of cone we can not call a right circularcone. Here, you are (see Fig. 11.2)! What you see inthese figures are not right circular cones; because in(a), the line joining its vertex to the centre of its baseis not at right angle to the base, and in (b) the base isnot circular.As in the case of cylinder, since we will be studying only about right circular cones,remember that by ‘cone’ in this chapter, we shall mean a ‘right circular cone.’Activity : (i) Cut out a neatly made paper cone that does not have any overlappedpaper, straight along its side, and opening it out, to see the shape of paper that formsthe surface of the cone. (The line along which you cut the cone is the slant height ofthe cone which is represented by l). It looks like a part of a round cake.(ii) If you now bring the sides marked A and B at the tips together, you can see thatthe curved portion of Fig. 11.3 (c) will form the circular base of the cone.Fig. 11.3(iii) If the paper like the one in Fig. 11.3 (c) is now cut into hundreds of little pieces,along the lines drawn from the point O, each cut portion is almost a small triangle,whose height is the slant height l of the cone.(iv) Now the area of each triangle = 12 × base of each triangle × l.So, area of the entire piece of paperFig. 11.2Rationalised 2023-24SURFACE AREAS AND VOLUMES 139= sum of the areas of all the triangles=1 2 31 1 12 2 2b l b l b l + + +? = ( )1 2 312l b b b + + +?=12 × l × length of entire curved boundary of Fig. 11.3(c)(as b1 + b2 + b3 + . . . makes up the curved portion of the figure)But the curved portion of the figure makes up the perimeter of the base of the coneand the circumference of the base of the cone = 2pr, where r is the base radius of thecone.So, Curved Surface Area of a Cone = 12 × l × 2p p p p pr = p p p p prlwhere r is its base radius and l its slant height.Note that l2 = r2 + h2 (as can be seen from Fig. 11.4), byapplying Pythagoras Theorem. Here h is the height of thecone.Therefore, l = 2 2r h +Now if the base of the cone is to be closed, then a circular piece of paper of radius ris also required whose area is pr2.So, Total Surface Area of a Cone = p p p p prl + p p p p pr2 = p p p p pr(l + r)Example 1 : Find the curved surface area of a right circular cone whose slant heightis 10 cm and base radius is 7 cm.Solution : Curved surface area = prl=227 × 7 × 10 cm2= 220 cm2Example 2 : The height of a cone is 16 cm and its base radius is 12 cm. Find thecurved surface area and the total surface area of the cone (Use p = 3.14).Solution : Here, h = 16 cm and r = 12 cm.So, from l2 = h2 + r2, we havel =2 216 12 + cm = 20 cmFig. 11.4Rationalised 2023-24140 MATHEMA TICSSo, curved surface area = prl= 3.14 × 12 × 20 cm2= 753.6 cm2Further, total surface area = prl + pr2= (753.6 + 3.14 × 12 × 12) cm2= (753.6 + 452.16) cm2= 1205.76 cm2Example 3 : A corn cob (see Fig. 11.5), shaped somewhatlike a cone, has the radius of its broadest end as 2.1 cm andlength (height) as 20 cm. If each 1 cm2 of the surface of thecob carries an average of four grains, find how many grainsyou would find on the entire cob.Solution : Since the grains of corn are found only on the curved surface of the corncob, we would need to know the curved surface area of the corn cob to find the totalnumber of grains on it. In this question, we are given the height of the cone, so weneed to find its slant height.Here, l =2 2r h + = 2 2(2.1) 20 + cm=404.41 cm = 20.11 cmTherefore, the curved surface area of the corn cob = prl=227 × 2.1 × 20.11 cm2 = 132.726 cm2 = 132.73 cm2 (approx.)Number of grains of corn on 1 cm2 of the surface of the corn cob = 4Therefore, number of grains on the entire curved surface of the cob= 132.73 × 4 = 530.92 = 531 (approx.)So, there would be approximately 531 grains of corn on the cob.EXERCISE 11.1Assume p = 227, unless stated otherwise.1. Diameter of the base of a cone is 10.5 cm and its slant height is 10 cm. Find its curvedsurface area.2. Find the total surface area of a cone, if its slant height is 21 m and diameter of its baseis 24 m.Fig. 11.5Rationalised 2023-24Page 5SURFACE AREAS AND VOLUMES 137CHAPTER 11SURFACE AREAS AND VOLUMES11.1 Surface Area of a Right Circular ConeWe have already studied the surface areas of cube, cuboid and cylinder. We will nowstudy the surface area of cone.So far, we have been generating solids by stacking up congruent figures. Incidentally,such figures are called prisms. Now let us look at another kind of solid which is not aprism (These kinds of solids are called pyramids.). Let us see how we can generatethem.Activity : Cut out a right-angled triangle ABC right angled at B. Paste a long thickstring along one of the perpendicular sides say AB of the triangle [see Fig. 11.1(a)].Hold the string with your hands on either sides of the triangle and rotate the triangleabout the string a number of times. What happens? Do you recognize the shape thatthe triangle is forming as it rotates around the string [see Fig. 11.1(b)]? Does it remindyou of the time you had eaten an ice-cream heaped into a container of that shape [seeFig. 11.1 (c) and (d)]?Fig. 11.1Rationalised 2023-24138 MATHEMA TICSThis is called a right circular cone. In Fig. 11.1(c)of the right circular cone, the point A is called thevertex, AB is called the height, BC is called the radiusand AC is called the slant height of the cone. Here Bwill be the centre of circular base of the cone. Theheight, radius and slant height of the cone are usuallydenoted by h, r and l respectively. Once again, let ussee what kind of cone we can not call a right circularcone. Here, you are (see Fig. 11.2)! What you see inthese figures are not right circular cones; because in(a), the line joining its vertex to the centre of its baseis not at right angle to the base, and in (b) the base isnot circular.As in the case of cylinder, since we will be studying only about right circular cones,remember that by ‘cone’ in this chapter, we shall mean a ‘right circular cone.’Activity : (i) Cut out a neatly made paper cone that does not have any overlappedpaper, straight along its side, and opening it out, to see the shape of paper that formsthe surface of the cone. (The line along which you cut the cone is the slant height ofthe cone which is represented by l). It looks like a part of a round cake.(ii) If you now bring the sides marked A and B at the tips together, you can see thatthe curved portion of Fig. 11.3 (c) will form the circular base of the cone.Fig. 11.3(iii) If the paper like the one in Fig. 11.3 (c) is now cut into hundreds of little pieces,along the lines drawn from the point O, each cut portion is almost a small triangle,whose height is the slant height l of the cone.(iv) Now the area of each triangle = 12 × base of each triangle × l.So, area of the entire piece of paperFig. 11.2Rationalised 2023-24SURFACE AREAS AND VOLUMES 139= sum of the areas of all the triangles=1 2 31 1 12 2 2b l b l b l + + +? = ( )1 2 312l b b b + + +?=12 × l × length of entire curved boundary of Fig. 11.3(c)(as b1 + b2 + b3 + . . . makes up the curved portion of the figure)But the curved portion of the figure makes up the perimeter of the base of the coneand the circumference of the base of the cone = 2pr, where r is the base radius of thecone.So, Curved Surface Area of a Cone = 12 × l × 2p p p p pr = p p p p prlwhere r is its base radius and l its slant height.Note that l2 = r2 + h2 (as can be seen from Fig. 11.4), byapplying Pythagoras Theorem. Here h is the height of thecone.Therefore, l = 2 2r h +Now if the base of the cone is to be closed, then a circular piece of paper of radius ris also required whose area is pr2.So, Total Surface Area of a Cone = p p p p prl + p p p p pr2 = p p p p pr(l + r)Example 1 : Find the curved surface area of a right circular cone whose slant heightis 10 cm and base radius is 7 cm.Solution : Curved surface area = prl=227 × 7 × 10 cm2= 220 cm2Example 2 : The height of a cone is 16 cm and its base radius is 12 cm. Find thecurved surface area and the total surface area of the cone (Use p = 3.14).Solution : Here, h = 16 cm and r = 12 cm.So, from l2 = h2 + r2, we havel =2 216 12 + cm = 20 cmFig. 11.4Rationalised 2023-24140 MATHEMA TICSSo, curved surface area = prl= 3.14 × 12 × 20 cm2= 753.6 cm2Further, total surface area = prl + pr2= (753.6 + 3.14 × 12 × 12) cm2= (753.6 + 452.16) cm2= 1205.76 cm2Example 3 : A corn cob (see Fig. 11.5), shaped somewhatlike a cone, has the radius of its broadest end as 2.1 cm andlength (height) as 20 cm. If each 1 cm2 of the surface of thecob carries an average of four grains, find how many grainsyou would find on the entire cob.Solution : Since the grains of corn are found only on the curved surface of the corncob, we would need to know the curved surface area of the corn cob to find the totalnumber of grains on it. In this question, we are given the height of the cone, so weneed to find its slant height.Here, l =2 2r h + = 2 2(2.1) 20 + cm=404.41 cm = 20.11 cmTherefore, the curved surface area of the corn cob = prl=227 × 2.1 × 20.11 cm2 = 132.726 cm2 = 132.73 cm2 (approx.)Number of grains of corn on 1 cm2 of the surface of the corn cob = 4Therefore, number of grains on the entire curved surface of the cob= 132.73 × 4 = 530.92 = 531 (approx.)So, there would be approximately 531 grains of corn on the cob.EXERCISE 11.1Assume p = 227, unless stated otherwise.1. Diameter of the base of a cone is 10.5 cm and its slant height is 10 cm. Find its curvedsurface area.2. Find the total surface area of a cone, if its slant height is 21 m and diameter of its baseis 24 m.Fig. 11.5Rationalised 2023-24SURFACE AREAS AND VOLUMES 1413. Curved surface area of a cone is 308 cm2 and its slant height is 14 cm. Find(i) radius of the base and (ii) total surface area of the cone.4. A conical tent is 10 m high and the radius of its base is 24 m. Find(i) slant height of the tent.(ii) cost of the canvas required to make the tent, if the cost of 1 m2 canvas is ` 70.5. What length of tarpaulin 3 m wide will be required to make conical tent of height 8 mand base radius 6 m? Assume that the extra length of material that will be required forstitching margins and wastage in cutting is approximately 20 cm (Use p = 3.14).6. The slant height and base diameter of a conical tomb are 25 m and 14 m respectively.Find the cost of white-washing its curved surface at the rate of ` 210 per 100 m2.7. A joker’s cap is in the form of a right circular cone of base radius 7 cm and height24 cm. Find the area of the sheet required to make 10 such caps.8. A bus stop is barricaded from the remaining part of the road, by using 50 hollowcones made of recycled cardboard. Each cone has a base diameter of 40 cm and height1 m. If the outer side of each of the cones is to be painted and the cost of painting is` 12 per m2, what will be the cost of painting all these cones? (Use p = 3.14 and take1.04 = 1.02)11.2 Surface Area of a SphereWhat is a sphere? Is it the same as a circle? Can you draw a circle on a paper? Yes,you can, because a circle is a plane closed figure whose every point lies at a constantdistance (called radius) from a fixed point, which is called the centre of the circle.Now if you paste a string along a diameter of a circular disc and rotate it as you hadrotated the triangle in the previous section, you see a new solid (see Fig 11.6). Whatdoes it resemble? A ball? Yes. It is called a sphere.Fig. 11.6Rationalised 2023-24

Read More

Join Course

Download as PDF

Explore Courses