Geometry · Textbook solutions

Mensuration

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 72 questions

Surface area and volume of solids

3 q

Solved Examples

Worked · 3
  1. Surface area and volume of solids SolvedEx.1
    The radius and height of a cylindrical water reservoir is 2.8 m and 3.5 m respectively. How much maximum water can the tank hold ? A person needs 70 litre of water per day. For how many persons is the water sufficient for a day ? (π=227)\left(\pi = \dfrac{22}{7}\right)
  2. Surface area and volume of solids SolvedEx.2
    How many solid cylinders of radius 10 cm and height 6 cm can be made by melting a solid sphere of radius 30 cm ?
  3. Surface area and volume of solids SolvedEx.3
    A tent of a circus is such that its lower part is cylindrical and upper part is conical. The diameter of the base of the tent is 48 m and the height of the cylindrical part is 15 m. Total height of the tent is 33 m. Find area of canvas required to make the tent. Also find volume of air in the tent.

Practice set 7.1

12 q
  1. Ex 7.1 Q.1
    Find the volume of a cone if the radius of its base is 1.5 cm and its perpendicular height is 5 cm.
  2. Ex 7.1 Q.2
    Find the volume of a sphere of diameter 6 cm.
  3. Ex 7.1 Q.3
    Find the total surface area of a cylinder if the radius of its base is 5 cm and height is 40 cm.
  4. Ex 7.1 Q.4
    Find the surface area of a sphere of radius 7 cm.
  5. Ex 7.1 Q.5
    The dimensions of a cuboid are 44 cm, 21 cm, 12 cm. It is melted and a cone of height 24 cm is made. Find the radius of its base.
  6. Ex 7.1 Q.6
    Observe the measures of pots in figure 7.8 and 7.9. How many jugs of water can the cylindrical pot hold ? (From fig. 7.8, the conical water jug has base radius 3.5 cm and depth 10 cm. From fig. 7.9, the cylindrical water pot has base radius 7 cm and height 10 cm.)
  7. Ex 7.1 Q.7
    A cylinder and a cone have equal bases. The height of the cylinder is 3 cm and the area of its base is 100 cm2^2. The cone is placed upon the cylinder. Volume of the solid figure so formed is 500 cm3^3. Find the total height of the figure.
  8. Ex 7.1 Q.8
    In figure 7.11, a toy made from a hemisphere, a cylinder and a cone is shown. Find the total area of the toy. (From fig. 7.11: the hemisphere, the cylinder and the cone all have radius 3 cm; the cylindrical part is 40 cm long and the height of the cone is 4 cm.)
  9. Ex 7.1 Q.9
    In the figure 7.12, a cylindrical wrapper of flat tablets is shown. The radius of a tablet is 7 mm and its thickness is 5 mm. How many such tablets are wrapped in the wrapper ? (From fig. 7.12: the wrapper is 10 cm long and 14 mm in diameter.)
  10. Ex 7.1 Q.10
    Figure 7.13 shows a toy. Its lower part is a hemisphere and the upper part is a cone. Find the volume and the surface area of the toy from the measures shown in the figure. (π=3.14)(\pi = 3.14) (From fig. 7.13: the height of the cone is 4 cm and the common radius of the cone and the hemisphere is 3 cm.)
  11. Ex 7.1 Q.11
    Find the surface area and the volume of a beach ball shown in the figure. (From fig. 7.14: the diameter of the ball is 42 cm.)
  12. Ex 7.1 Q.12
    As showm in the figure, a cylindrical glass contains water. A metal sphere of diameter 2 cm is immersed in it. Find the volume of the water. (From fig. 7.15: the glass is 30 cm high and 14 cm in diameter, and the water fills it completely.)

Frustum of a cone

2 q

Solved Examples

Worked · 2
  1. Frustum of a cone SolvedEx.1
    A bucket is frustum shaped. Its height is 28 cm. Radii of circular faces are 12 cm and 15 cm. Find the capacity of the bucket. (π=227)\left(\pi = \dfrac{22}{7}\right)
  2. Frustum of a cone SolvedEx.2
    Radii of the top and the base of a frustum are 14 cm, 8 cm respectively. Its height is 8 cm. Find its i) curved surface area ii) total surface area iii) volume.

Practice set 7.2

5 q
  1. Ex 7.2 Q.1
    The radii of two circular ends of frustum shape bucket are 14 cm and 7 cm. Height of the bucket is 30 cm. How many liters of water it can hold ? (1 litre = 1000 cm3^3)
  2. The radii of ends of a frustum are 14 cm and 6 cm respectively and its height is 6 cm. Find its (π=3.14)(\pi = 3.14)
    Ex 7.2 Q.2(i)
    i) curved surface area
  3. Ex 7.2 Q.2(ii)
    ii) total surface area.
  4. Ex 7.2 Q.2(iii)
    iii ) volume
  5. Ex 7.2 Q.3
    The circumferences of circular faces of a frustum are 132 cm and 88 cm and its height is 24 cm. To find the curved surface area of the frustum complete the following activity. (π=227)\left(\pi = \dfrac{22}{7}\right). circumference1=2πr1=132_1 = 2\pi r_1 = 132 r1=1322π=r_1 = \dfrac{132}{2\pi} = \boxed{\quad} circumference2=2πr2=88_2 = 2\pi r_2 = 88 r2=882π=r_2 = \dfrac{88}{2\pi} = \boxed{\quad} slant height of frustum, l=h2+(r1−r2)2l = \sqrt{h^2 + (r_1 - r_2)^2} = 2+ 2= \sqrt{\boxed{\quad}^{\,2} + \boxed{\quad}^{\,2}} == \boxed{\quad} cm curved surface area of the frustum =π(r1+r2) l= \pi(r_1 + r_2)\,l =π××= \pi \times \boxed{\quad} \times \boxed{\quad} == \boxed{\quad} sq.cm.

A relation between length of an arc and area of the sector

2 q

Solved Examples

Worked · 2
  1. A relation between length of an arc and area of the sector SolvedEx.1
    The measure of a central angle of a circle is 150∘^\circ and radius of the circle is 21 cm. Find the length of the arc and area of the sector associated with the central angle.
  2. A relation between length of an arc and area of the sector SolvedEx.2
    In figure 7.29, P is the centre of the circle of radius 6 cm. Seg QR is a tangent at Q. If PR = 12, find the area of the shaded region. (3=1.73)(\sqrt{3} = 1.73)

Practice set 7.3

21 q
  1. Ex 7.3 Q.1
    Radius of a circle is 10 cm. Measure of an arc of the crcleis 54∘^\circ. Find the area of the sector associated with the arc. (π=3.14)(\pi = 3.14)
  2. Ex 7.3 Q.2
    Measure of an arc of a circle is 80 cm and its radius is 18 cm. Find the length of the arc. (π=3.14)(\pi = 3.14)
  3. Ex 7.3 Q.3
    Radius of a sector of a circle is 3.5 cm and length of its arc is 2.2 cm. Find the area of the sector.
  4. Ex 7.3 Q.4
    Radius of a circle is 10 cm. Area of a sector of the sector is 100 cm2^2. Find the area of its corresponding major sector. (π=3.14)(\pi = 3.14)
  5. Ex 7.3 Q.5
    Area of a sector of a circle of radius 15 cm is 30 cm2^2. Find the length of the arc of the sector.
  6. In the figure 7.31, radius of the circle is 7 cm and mm(arc MBN) = 60°, find
    Ex 7.3 Q.6(1)
    (1) Area of the circle .
  7. Ex 7.3 Q.6(2)
    (2) A(O - MBN) .
  8. Ex 7.3 Q.6(3)
    (3) A(O - MCN) .
  9. Ex 7.3 Q.7
    In figure 7.32, radius of circle is 3.4 cm and perimeter of sector P-ABC is 12.8 cm. Find A(P-ABC).
  10. Ex 7.3 Q.8
    In figure 7.33 O is the centre of the sector. ∠\angle ROQ = ∠\angle MON = 60∘^\circ. OR = 7 cm, and OM = 21 cm. Find the lengths of arc RXQ and arc MYN. (π=227)\left(\pi = \dfrac{22}{7}\right)
  11. In figure 7.34, if A(P-ABC) = 154 cm2^2 radius of the circle is 14 cm, find
    Ex 7.3 Q.9(1)
    (1) ∠\angle APC.
  12. Ex 7.3 Q.9(2)
    (2) ll(arc ABC) .
  13. Radius of a sector of a circle is 7 cm. If measure of arc of the sector is – find the area of the sector in each case.
    Ex 7.3 Q.10(1)
    (1) 30∘^\circ
  14. Ex 7.3 Q.10(2)
    (2) 210∘^\circ
  15. Ex 7.3 Q.10(3)
    (3 ) three right angles;
  16. Ex 7.3 Q.11
    The area of a minor sector of a circle is 3.85 cm2^2 and the measure of its central angle is 36∘^\circ. Find the radius of the circle.
  17. Ex 7.3 Q.12
    In figure 7.35, □\square PQRS is a rectangle. If PQ = 14 cm, QR = 21 cm, find the areas of the parts xx, yy and zz. (From fig. 7.35: xx is the quarter circle drawn with centre Q and radius QP, yy is the quarter circle drawn with centre R whose radius is the remaining part of QR, and zz is the rest of the rectangle.)
  18. △\triangle LMN is an equilateral triangle. LM = 14 cm. As shown in figure, three sectors are drawn with vertices as centres and radius 7 cm. Find,
    Ex 7.3 Q.13(1)
    (1) A (△\triangle LMN)
  19. Ex 7.3 Q.13(2)
    (2) Area of any one of the sectors.
  20. Ex 7.3 Q.13(3)
    (3) Total area of all the three sectors.
  21. Ex 7.3 Q.13(4)
    (4) Area of the shaded region.

Area of a Segment

3 q

Solved Examples

Worked · 3
  1. Area of a Segment SolvedEx.1
    In the figure 7.40, ∠\angle AOB = 30∘^\circ, OA = 12 cm . Find the area of the segment. (π=3.14)(\pi = 3.14)
  2. Area of a Segment SolvedEx.2
    The radius of a circle with centre P is 10 cm. If chord AB of the circle substends a right angle at P, find areas of the minor segment and the major segment. (π=3.14)(\pi = 3.14)
  3. Area of a Segment SolvedEx.3
    A regular hexagon is inscribed in a circle of radius 14 cm. Find the area of the region between the circle and the hexagon. (π=227, 3=1.732)\left(\pi = \dfrac{22}{7},\ \sqrt{3} = 1.732\right)

Practice set 7.4

5 q
  1. Ex 7.4 Q.1
    In figure 7.43, A is the centre of the circle. ∠\angle ABC = 45∘^\circ and AC = 727\sqrt{2} cm. Find the area of segment BXC.
  2. Ex 7.4 Q.2
    In the figure 7.44, O is the centre of the circle. mm(arc PQR) = 60∘^\circ OP = 10 cm. Find the area of the shaded region. (π=3.14, 3=1.73)(\pi = 3.14,\ \sqrt{3} = 1.73)
  3. Ex 7.4 Q.3
    In the figure 7.45, if A is the centre of the circle. ∠\angle PAR = 30∘^\circ, AP = 7.5, find the area of the segment PQR (π=3.14)(\pi = 3.14)
  4. Ex 7.4 Q.4
    In the figure 7.46, if O is the centre of the circle, PQ is a chord. ∠\angle POQ = 90∘^\circ, area of shaded region is 114 cm2^2, find the radius of the circle. (π=3.14)(\pi = 3.14)
  5. Ex 7.4 Q.5
    A chord PQ of a circle with radius 15 cm subtends an angle of 60∘^\circ with the centre of the circle. Find the area of the minor as well as the major segment. (π=3.14, 3=1.73)(\pi = 3.14,\ \sqrt{3} = 1.73)

Problem set 7

19 q
  1. Choose the correct alternative answer for each of the following questions.
    PS7 Q.1(1)
    The ratio of circumference and area of a circle is 2:7. Find its circumference.
    1. A.
      14π14\pi
    2. B.
      7π\dfrac{7}{\pi}
    3. C.
      7π7\pi
    4. D.
      14π\dfrac{14}{\pi}
  2. PS7 Q.1(2)
    If measure of an arc of a circle is 160∘^\circ and its length is 44 cm, find the circumference of the circle.
    1. A.
      66 cm
    2. B.
      44 cm
    3. C.
      160 cm
    4. D.
      99 cm
  3. PS7 Q.1(3)
    Find the perimeter of a sector of a circle if its measure is 90 ∘^\circ and radius is 7 cm.
    1. A.
      44 cm
    2. B.
      25 cm
    3. C.
      36 cm
    4. D.
      56 cm
  4. PS7 Q.1(4)
    Find the curved surface area of a cone of radius 7 cm and height 24 cm.
    1. A.
      440 cm2^2
    2. B.
      550 cm2^2
    3. C.
      330 cm2^2
    4. D.
      110 cm2^2
  5. PS7 Q.1(5)
    The curved surface area of a cylinder is 440 cm2^2 and its radius is 5 cm. Find its height.
    1. A.
      44π\dfrac{44}{\pi} cm
    2. B.
      22π22\pi cm
    3. C.
      44π44\pi cm
    4. D.
      22π\dfrac{22}{\pi} cm
  6. PS7 Q.1(6)
    A cone was melted and cast into a cylinder of the same radius as that of the base of the cone. If the height of the cylinder is 5 cm, find the height of the cone.
    1. A.
      15 cm
    2. B.
      10 cm
    3. C.
      18 cm
    4. D.
      5 cm
  7. PS7 Q.1(7)
    Find the volume of a cube of side 0.01 cm.
    1. A.
      1 cm3^3
    2. B.
      0.001 cm3^3
    3. C.
      0.0001 cm3^3
    4. D.
      0.000001 cm3^3
  8. PS7 Q.1(8)
    Find the side of a cube of volume 1 m3^3.
    1. A.
      1 cm
    2. B.
      10 cm
    3. C.
      100 cm
    4. D.
      1000 cm
  9. PS7 Q.2
    A washing tub in the shape of a frustum of a cone has height 21 cm. The radii of the circular top and bottom are 20 cm and 15 cm respectively. What is the capacity of the tub ? (π=227)\left(\pi = \dfrac{22}{7}\right)
  10. PS7 Q.3
    Some plastic balls of radius 1 cm were melted and cast into a tube. The thickness, length and outer radius of the tube were 2 cm, 90 cm and 30 cm respectively. How many balls were melted to make the tube ?
  11. PS7 Q.4
    A metal parallelopiped of measures 16 cm ×\times 11cm ×\times 10 cm was melted to make coins. How many coins were made if the thickness and diameter of each coin was 2 mm and 2 cm respectively ?
  12. PS7 Q.5
    The diameter and length of a roller is 120 cm and 84 cm respectively. To level the ground, 200 rotations of the roller are required. Find the expenditure to level the ground at the rate of Rs. 10 per sq.m.
  13. PS7 Q.6
    The diameter and thickness of a hollow metals sphere are 12 cm and 0.01 m respectively. The density of the metal is 8.88 gm per cm3^3. Find the outer surface area and mass of the sphere.
  14. PS7 Q.7
    A cylindrical bucket of diameter 28 cm and height 20 cm was full of sand. When the sand in the bucket was poured on the ground, the sand got converted into a shape of a cone. If the height of the cone was 14 cm, what was the base area of the cone ?
  15. PS7 Q.8
    The radius of a metallic sphere is 9 cm. It was melted to make a wire of diameter 4 mm. Find the length of the wire.
  16. PS7 Q.9
    The area of a sector of a circle of 6 cm radius is 15π15\pi sq.cm. Find the measure of the arc and length of the arc corresponding to the sector.
  17. PS7 Q.10
    In the figure 7.47, seg AB is a chord of a circle with centre P. If PA = 8 cm and distance of chord AB from the centre P is 4 cm, find the area of the shaded portion. (π=3.14, 3=1.73)(\pi = 3.14,\ \sqrt{3} = 1.73)
  18. PS7 Q.11
    In the figure 7.48, square ABCD is inscribed in the sector A-PCQ. The radius of sector C-BXD is 20 cm. Complete the following activity to find the area of shaded region. Side of square ABCD = radius of sector C-BXD = \boxed{\quad} cm Area of square = (side)2= 2=(\text{side})^2 = \boxed{\quad}^{\,2} = \boxed{\quad} ..... (I) Area of shaded region inside the square = Area of square ABCD −- Area of sector C-BXD =−θ360×πr2= \boxed{\quad} - \dfrac{\theta}{360} \times \pi r^2 =−90360×3.141×4001= \boxed{\quad} - \dfrac{90}{360} \times \dfrac{3.14}{1} \times \dfrac{400}{1} =−314= \boxed{\quad} - 314 == \boxed{\quad} Radius of bigger sector = Length of diagonal of square ABCD =202= 20\sqrt{2} Area of the shaded regions outside the square = Area of sector A-PCQ −- Area of square ABCD = A(A-PCQ) −- A(□\squareABCD) =(θ360×π×r2)− 2= \left(\dfrac{\theta}{360} \times \pi \times r^2\right) - \boxed{\quad}^{\,2} =90360×3.14 (202)2−(20)2= \dfrac{90}{360} \times 3.14\,(20\sqrt{2})^2 - (20)^2 =−= \boxed{\quad} - \boxed{\quad} == \boxed{\quad} ∴\therefore total area of the shaded region = 86 + 228 = 314 sq.cm.
  19. PS7 Q.12
    In the figure 7.49, two circles with centres O and P are touching internally at point A. If BQ = 9, DE = 5, complete the following activity to find the radii of the circles. Let the radius of the bigger circle be R and that of smaller circle be r. OA, OB, OC and OD are the radii of the bigger circle ∴\therefore OA = OB = OC = OD = R PQ = PA = r OQ = OB −- BQ = \boxed{\quad} OE = OD −- DE = \boxed{\quad} As the chords QA and EF of the circle with centre P intersect in the interior of the circle, so by the property of internal division of two chords of a circle, OQ ×\times OA = OE ×\times OF ×R=×\boxed{\quad} \times \text{R} = \boxed{\quad} \times \boxed{\quad} .......... (∵(\because OE = OF) R2−9R=R2−10R+25\text{R}^2 - 9\text{R} = \text{R}^2 - 10\text{R} + 25 R = \boxed{\quad} AQ = 2r = AB −- BQ 2r = 50 −- 9 = 41 r = \boxed{\quad} = \boxed{\quad}