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Class 8 · Tamil Nadu State Board

Measurements Exercise Solutions | KalviGavani

Samacheer Kalvi 8th Maths Measurements exercise solutions with MCQs, fill in blanks, true or false, match the following, short and detailed answers, common mistakes, and exam checklist.

Updated 12 June 202631 min read

How to Use This Page

Solve every exercise question on your own first, then compare your answers below. Focus on understanding why each step works — that builds exam confidence. Show all working, include units at every step, and verify your final answer by estimation.

Key Concepts (Quick Recap)

  • Circle: Area = pi r^2; Circumference = 2 pi * r; Diameter d = 2r
  • Semi-circle: Area = (1/2) pi r^2; Perimeter = pi * r + 2r
  • Trapezium: Area = (1/2) (a + b) h, where a and b are parallel sides, h is the perpendicular distance between them
  • Rhombus: Area = (1/2) d1 d2, where d1 and d2 are diagonals; also Area = base * height
  • Quadrilateral (with diagonals): Area = (1/2) d (h1 + h2), where d is one diagonal and h1, h2 are perpendiculars from opposite vertices
  • Cylinder:
  • CSA = 2 pi r * h
  • TSA = 2 pi r * (r + h)
  • Volume = pi r^2 h
  • Cone:
  • CSA = pi r l (where l is slant height)
  • TSA = pi r (l + r)
  • Volume = (1/3) pi r^2 * h
  • l = sqrt(r^2 + h^2)
  • Sphere:
  • Surface area = 4 pi r^2
  • Volume = (4/3) pi r^3
  • Hemisphere:
  • CSA = 2 pi r^2
  • TSA = 3 pi r^2 (CSA + base circle)
  • Volume = (2/3) pi r^3
  • Unit conversions: 1 sq m = 10,000 sq cm; 1 cu m = 10,00,000 cu cm; 1 litre = 1000 cu cm; 1 kl = 1000 litres = 1 cu m

Exercise 2.1 — Area of Circle and Quadrilaterals

I. Choose the Best Answer (MCQs)

Q1. If the radius of a circle is 3.5 cm, then its area is (a) 38.5 sq cm (b) 77 sq cm (c) 22 sq cm (d) 11 sq cm

Answer(a) 38.5 sq cm
Area = pi * r^2 = (22/7) * 3.5 * 3.5 = (22/7) * 12.25 = 22 * 1.75 = 38.5 sq cm

Q2. The circumference of a circle of diameter 14 cm is (a) 44 cm (b) 88 cm (c) 22 cm (d) 154 cm

Answer(a) 44 cm
r = d/2 = 14/2 = 7 cm Circumference = 2 * pi * r = 2 * (22/7) * 7 = 44 cm

Q3. The area of a trapezium whose parallel sides are 10 cm and 16 cm and the distance between them is 8 cm is (a) 104 sq cm (b) 208 sq cm (c) 52 sq cm (d) 128 sq cm

Answer(a) 104 sq cm
Area = (1/2) * (a + b) * h = (1/2) * (10 + 16) * 8 = (1/2) * 26 * 8 = 13 * 8 = 104 sq cm

Q4. The area of a rhombus whose diagonals are 10 cm and 8 cm is (a) 80 sq cm (b) 40 sq cm (c) 20 sq cm (d) 160 sq cm

Answer(b) 40 sq cm
Area = (1/2) * d1 * d2 = (1/2) * 10 * 8 = 40 sq cm

Q5. If the area of a circle is 154 sq cm, then its radius is (a) 7 cm (b) 14 cm (c) 3.5 cm (d) 21 cm

Answer(a) 7 cm
pi * r^2 = 154 (22/7) * r^2 = 154 r^2 = 154 * 7 / 22 = 49 r = 7 cm

Q6. The perimeter of a semi-circle of radius 7 cm is (a) 22 cm (b) 36 cm (c) 44 cm (d) 29 cm

Answer(b) 36 cm
Perimeter of semi-circle = pi * r + 2r = (22/7) * 7 + 2 * 7 = 22 + 14 = 36 cm

Q7. The area of a quadrant of a circle of radius 7 cm is (a) 77 sq cm (b) 38.5 sq cm (c) 19.25 sq cm (d) 154 sq cm

Answer(b) 38.5 sq cm
Area of circle = pi * r^2 = (22/7) * 49 = 154 sq cm Area of quadrant = (1/4) * 154 = 38.5 sq cm

Q8. If the diagonals of a rhombus are 12 cm and 16 cm, then its side is (a) 10 cm (b) 8 cm (c) 14 cm (d) 20 cm

Answer(a) 10 cm
Diagonals of a rhombus bisect each other at right angles. Half of d1 = 6 cm, Half of d2 = 8 cm Side = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 cm

II. Fill in the Blanks

1. The area of a circle of radius 7 cm is 154 sq cm.

Area = (22/7) * 7 * 7 = 154 sq cm

2. The circumference of a circle of radius 5.6 cm is 35.2 cm.

Circumference = 2 * (22/7) * 5.6 = 2 * 22 * 0.8 = 35.2 cm

3. The area of a trapezium is 180 sq cm. If the distance between parallel sides is 10 cm and one of the parallel sides is 20 cm, then the other parallel side is 16 cm.

180 = (1/2) * (20 + b) * 10 180 = 5 * (20 + b) 36 = 20 + b b = 16 cm

4. The area of a rhombus is 96 sq cm. If one diagonal is 12 cm, then the other diagonal is 16 cm.

96 = (1/2) * 12 * d2 96 = 6 * d2 d2 = 16 cm

5. If the circumference of a circle is 88 cm, then its diameter is 28 cm.

2 * pi * r = 88 2 * (22/7) * r = 88 r = 88 * 7 / 44 = 14 cm d = 2r = 28 cm

III. True or False (with corrections for false statements)

Q1. The area of a circle is equal to the product of pi and the square of its radius.
AnswerTrue
Q2. The circumference of a circle is pi times the diameter.
AnswerTrue — Circumference = 2 pi r = pi * d (since d = 2r).
Q3. The area of a trapezium is the product of the sum of parallel sides and the distance between them.
AnswerFalse — Correction: The area of a trapezium is half the product of the sum of parallel sides and the distance between them. Area = (1/2)(a + b) * h.
Q4. The diagonals of a rhombus are equal in length.
AnswerFalse — Correction: The diagonals of a rhombus are generally not equal. They bisect each other at right angles, but d1 and d2 can be different lengths.
Q5. The area of a semi-circle is half the area of the corresponding full circle.
AnswerTrue
Q6. If the radius of a circle is doubled, its area becomes four times.
AnswerTrue — Area = pi r^2. If r becomes 2r, new area = pi (2r)^2 = 4 pi r^2 = 4 times original.

IV. Match the Following

Column AColumn BAnswer
1. Area of circle(a) (1/2) d1 d2(d) pi * r^2
2. Circumference of circle(b) (1/4) pi r^2(e) 2 pi r
3. Area of trapezium(c) (1/2) pi r^2(f) (1/2)(a + b) * h
4. Area of rhombus(d) pi * r^2(a) (1/2) d1 d2
5. Area of semi-circle(e) 2 pi r(c) (1/2) pi r^2
6. Area of quadrant(f) (1/2)(a + b) * h(b) (1/4) pi r^2

V. Short Answer Questions

Q1. Find the area of a circle whose circumference is 44 cm.

Circumference = 2 * pi * r = 44 2 * (22/7) * r = 44 r = 44 * 7 / 44 = 7 cm Area = pi * r^2 = (22/7) * 7 * 7 = 154 sq cm

Q2. Find the area of a trapezium with parallel sides 6 cm and 10 cm and height 5 cm.

Area = (1/2) * (a + b) * h = (1/2) * (6 + 10) * 5 = (1/2) * 16 * 5 = 8 * 5 = 40 sq cm

Q3. The diagonals of a rhombus are 8 cm and 6 cm. Find its area.

Area = (1/2) * d1 * d2 = (1/2) * 8 * 6 = 24 sq cm

Q4. A wire is bent in the form of a circle of radius 7 cm. It is rebent to form a square. Find the side of the square.

Circumference of circle = Perimeter of square 2 * pi * r = 4 * side 2 * (22/7) * 7 = 4 * side 44 = 4 * side side = 11 cm

Q5. Find the area of a path 2 m wide surrounding a circular park of radius 7 m.

Outer radius = 7 + 2 = 9 m Area of path = pi * R^2 - pi * r^2 = pi * (R^2 - r^2) = (22/7) * (81 - 49) = (22/7) * 32 = 704/7 = 100.57 sq m (approx)

Q6. Find the area of the quadrilateral ABCD where diagonal AC = 12 cm and the perpendiculars from B and D to AC are 5 cm and 7 cm respectively.

Area = (1/2) * d * (h1 + h2) = (1/2) * 12 * (5 + 7) = (1/2) * 12 * 12 = 72 sq cm

VI. Detailed Answer Questions

Q1. A circular garden of radius 17.5 m has a path of width 3.5 m running around it on the outside. Find the area of the path.

Inner radius r = 17.5 m Outer radius R = 17.5 + 3.5 = 21 m Area of path = pi * R^2 - pi * r^2 = pi * (R^2 - r^2) = (22/7) * (21^2 - 17.5^2) = (22/7) * (441 - 306.25) = (22/7) * 134.75 = 2964.5 / 7 = 423.5 sq m

Q2. The parallel sides of a trapezium are 20 cm and 30 cm and its area is 375 sq cm. Find the distance between the parallel sides.

Area = (1/2) * (a + b) * h 375 = (1/2) * (20 + 30) * h 375 = (1/2) * 50 * h 375 = 25 * h h = 375 / 25 = 15 cm

Q3. The diagonals of a rhombus are 24 cm and 18 cm. Find (i) its area, (ii) its side, (iii) its perimeter.

(i) Area = (1/2) * d1 * d2 = (1/2) * 24 * 18 = 216 sq cm (ii) Diagonals bisect each other at right angles. Half of d1 = 12 cm, Half of d2 = 9 cm Side = sqrt(12^2 + 9^2) = sqrt(144 + 81) = sqrt(225) = 15 cm (iii) Perimeter = 4 * side = 4 * 15 = 60 cm

Q4. The inner circumference of a circular track is 220 m and the outer circumference is 308 m. Find the width of the track.

Inner circumference: 2 * pi * r = 220 r = 220 * 7 / 44 = 35 m Outer circumference: 2 * pi * R = 308 R = 308 * 7 / 44 = 49 m Width of track = R - r = 49 - 35 = 14 m

Exercise 2.2 — Surface Area of Cylinder and Cone

I. Choose the Best Answer (MCQs)

Q1. The curved surface area of a right circular cylinder of radius 7 cm and height 10 cm is (a) 440 sq cm (b) 220 sq cm (c) 154 sq cm (d) 110 sq cm

Answer(a) 440 sq cm
CSA = 2 * pi * r * h = 2 * (22/7) * 7 * 10 = 2 * 22 * 10 = 440 sq cm

Q2. The total surface area of a cylinder of radius 7 cm and height 10 cm is (a) 440 sq cm (b) 594 sq cm (c) 748 sq cm (d) 297 sq cm

Answer(c) 748 sq cm
TSA = CSA + 2 * base area = 440 + 2 * (22/7) * 49 = 440 + 308 = 748 sq cm

Q3. The curved surface area of a cone with radius 6 cm and slant height 8 cm is (a) 48 pi sq cm (b) 96 pi sq cm (c) 24 pi sq cm (d) 30 pi sq cm

Answer(a) 48 pi sq cm
CSA = pi * r * l = pi * 6 * 8 = 48 pi sq cm

Q4. The total surface area of a cone whose radius is 7 cm and slant height is 10 cm is (a) 220 sq cm (b) 374 sq cm (c) 154 sq cm (d) 440 sq cm

Answer(b) 374 sq cm
TSA = pi * r * (l + r) = (22/7) * 7 * (10 + 7) = 22 * 17 = 374 sq cm

Q5. If the height and radius of a cone are 8 cm and 6 cm respectively, then its slant height is (a) 14 cm (b) 12 cm (c) 10 cm (d) 9 cm

Answer(c) 10 cm
l = sqrt(r^2 + h^2) = sqrt(6^2 + 8^2) = sqrt(36 + 64) = sqrt(100) = 10 cm

Q6. A cylinder has the same radius as a cone and the same height as the cone. The ratio of the volume of the cylinder to the volume of the cone is (a) 1:3 (b) 3:1 (c) 1:2 (d) 2:1

Answer(b) 3:1
Volume of cylinder = pi * r^2 * h Volume of cone = (1/3) * pi * r^2 * h Ratio = pi * r^2 * h : (1/3) * pi * r^2 * h = 1 : (1/3) = 3 : 1

II. Fill in the Blanks

1. The CSA of a cylinder of radius 3.5 cm and height 14 cm is 308 sq cm.

CSA = 2 * (22/7) * 3.5 * 14 = 2 * 22 * 0.5 * 14 = 308 sq cm

2. The total surface area of a cone of radius 3 cm and slant height 5 cm is 75.43 sq cm (approx).

TSA = pi * r * (l + r) = (22/7) * 3 * (5 + 3) = (22/7) * 24 = 528/7 = 75.43 sq cm

3. If the CSA of a cylinder is 440 sq cm and its radius is 7 cm, then its height is 10 cm.

2 * pi * r * h = 440 2 * (22/7) * 7 * h = 440 44 * h = 440 h = 10 cm

4. The CSA of a cone whose radius is 14 cm and slant height is 21 cm is 924 sq cm.

CSA = pi * r * l = (22/7) * 14 * 21 = 22 * 2 * 21 = 924 sq cm

5. If the radius of a cylinder is doubled and height is halved, the CSA remains the same (unchanged).

Original CSA = 2 * pi * r * h New CSA = 2 * pi * (2r) * (h/2) = 2 * pi * r * h The two changes cancel out, so CSA stays the same.

Correction: The CSA remains the same (unchanged).

6. The slant height of a cone of radius 8 cm and height 6 cm is 10 cm.

l = sqrt(8^2 + 6^2) = sqrt(64 + 36) = sqrt(100) = 10 cm

III. True or False

Q1. The CSA of a cylinder equals the area of a rectangle with sides 2pir and h.
AnswerTrue — When unrolled, the curved surface of a cylinder is a rectangle of length 2pir and breadth h.
Q2. The slant height of a cone is always greater than its vertical height.
AnswerTrue — l = sqrt(r^2 + h^2), so l > h always (since r > 0).
Q3. If the radius of a cylinder is doubled, its TSA also doubles.
AnswerFalse — Correction: TSA = 2 pi r * (r + h). Doubling r does not simply double TSA; it changes it by a more complex factor.
Q4. The CSA of a cone is pi r l where l is the slant height.
AnswerTrue
Q5. A cone can have the same radius and height as a cylinder.
AnswerTrue

IV. Short Answer Questions

Q1. Find the CSA and TSA of a cylinder of radius 7 cm and height 10 cm.

CSA = 2 * pi * r * h = 2 * (22/7) * 7 * 10 = 44 * 10 = 440 sq cm TSA = 2 * pi * r * (r + h) = 2 * (22/7) * 7 * (7 + 10) = 44 * 17 = 748 sq cm

Q2. The CSA of a cylinder is 264 sq cm. If its radius is 7 cm, find its height.

2 * pi * r * h = 264 2 * (22/7) * 7 * h = 264 44 * h = 264 h = 264 / 44 = 6 cm

Q3. Find the slant height of a cone whose radius is 5 cm and height is 12 cm.

l = sqrt(r^2 + h^2) = sqrt(25 + 144) = sqrt(169) = 13 cm

Q4. Find the CSA of a cone whose radius is 14 cm and slant height is 25 cm.

CSA = pi * r * l = (22/7) * 14 * 25 = 22 * 2 * 25 = 1100 sq cm

Q5. A cylindrical vessel has radius 7 cm and height 15 cm. Find the area of the metal sheet required to make it (open at the top).

Area needed = CSA + Base area CSA = 2 * pi * r * h = 2 * (22/7) * 7 * 15 = 44 * 15 = 660 sq cm Base area = pi * r^2 = (22/7) * 49 = 154 sq cm Total area = 660 + 154 = 814 sq cm

V. Detailed Answer Questions

Q1. A solid cylinder has radius 7 cm and height 10 cm. Find (i) CSA, (ii) TSA, (iii) volume.

Given: r = 7 cm, h = 10 cm (i) CSA = 2 * pi * r * h = 2 * (22/7) * 7 * 10 = 440 sq cm (ii) TSA = 2 * pi * r * (r + h) = 2 * (22/7) * 7 * 17 = 44 * 17 = 748 sq cm (iii) Volume = pi * r^2 * h = (22/7) * 49 * 10 = 1540 cu cm

Q2. A cone has radius 7 cm and height 24 cm. Find (i) slant height, (ii) CSA, (iii) TSA.

Given: r = 7 cm, h = 24 cm (i) l = sqrt(r^2 + h^2) = sqrt(49 + 576) = sqrt(625) = 25 cm (ii) CSA = pi * r * l = (22/7) * 7 * 25 = 22 * 25 = 550 sq cm (iii) TSA = pi * r * (l + r) = (22/7) * 7 * (25 + 7) = 22 * 32 = 704 sq cm

Q3. The ratio of the CSA to the base area of a right circular cylinder is 2:1. Find the height of the cylinder if its radius is 7 cm.

CSA / Base area = 2 : 1 (2 * pi * r * h) / (pi * r^2) = 2 / 1 2h / r = 2 2h / 7 = 2 h = 7 cm

Q4. A rectangular sheet of paper 44 cm long and 20 cm wide is rolled along its length to form a cylinder. Find the volume of the cylinder.

When rolled along length, the length becomes the circumference and width becomes height. Circumference = 2 * pi * r = 44 2 * (22/7) * r = 44 r = 44 * 7 / 44 = 7 cm Height h = 20 cm Volume = pi * r^2 * h = (22/7) * 49 * 20 = 22 * 7 * 20 = 3080 cu cm

Exercise 2.3 — Volume of Cylinder, Cone, Sphere

I. Choose the Best Answer (MCQs)

Q1. The volume of a cylinder of radius 7 cm and height 10 cm is (a) 1540 cu cm (b) 770 cu cm (c) 440 cu cm (d) 220 cu cm

Answer(a) 1540 cu cm
Volume = pi * r^2 * h = (22/7) * 7 * 7 * 10 = 22 * 7 * 10 = 1540 cu cm

Q2. The volume of a cone of radius 7 cm and height 9 cm is (a) 462 cu cm (b) 154 cu cm (c) 924 cu cm (d) 308 cu cm

Answer(a) 462 cu cm
Volume = (1/3) * pi * r^2 * h = (1/3) * (22/7) * 49 * 9 = (1/3) * 22 * 7 * 9 = (1/3) * 1386 = 462 cu cm

Q3. The volume of a sphere of radius 3.5 cm is (a) 154.83 cu cm (b) 179.67 cu cm (c) 309.66 cu cm (d) 577.27 cu cm

Answer(b) 179.67 cu cm (approx)
Volume = (4/3) * pi * r^3 = (4/3) * (22/7) * 3.5 * 3.5 * 3.5 = (4/3) * (22/7) * 42.875 = (4/3) * 134.75 = 539/3 = 179.67 cu cm (approx)

Q4. The volume of a hemisphere of radius 7 cm is (a) 718.67 cu cm (b) 1437.33 cu cm (c) 359.33 cu cm (d) 615.83 cu cm

Answer(a) 718.67 cu cm (approx)
Volume = (2/3) * pi * r^3 = (2/3) * (22/7) * 7 * 7 * 7 = (2/3) * 22 * 49 = (2/3) * 1078 = 2156/3 = 718.67 cu cm (approx)

Q5. How many times is the volume of a sphere compared to a hemisphere of the same radius? (a) 2 (b) 3 (c) 4/3 (d) 1/2

Answer(a) 2
Volume of sphere = (4/3) * pi * r^3 Volume of hemisphere = (2/3) * pi * r^3 Ratio = (4/3) / (2/3) = 2

Q6. If the radius of a sphere is doubled, its volume becomes (a) 2 times (b) 4 times (c) 8 times (d) 16 times

Answer(c) 8 times
Original volume = (4/3) * pi * r^3 New volume = (4/3) * pi * (2r)^3 = (4/3) * pi * 8r^3 = 8 times original

II. Fill in the Blanks

1. The volume of a cylinder of radius 3.5 cm and height 20 cm is 770 cu cm.

V = pi * r^2 * h = (22/7) * 3.5 * 3.5 * 20 = (22/7) * 245 = 770 cu cm

2. If the volume of a cone is 308 cu cm and its base radius is 7 cm, then its height is 6 cm.

(1/3) * pi * r^2 * h = 308 (1/3) * (22/7) * 49 * h = 308 (1/3) * 154 * h = 308 (154/3) * h = 308 h = 308 * 3 / 154 = 6 cm

3. The surface area of a sphere of radius 7 cm is 616 sq cm.

SA = 4 * pi * r^2 = 4 * (22/7) * 49 = 4 * 22 * 7 = 616 sq cm

4. The total surface area of a hemisphere of radius 7 cm is 462 sq cm.

TSA = 3 * pi * r^2 = 3 * (22/7) * 49 = 3 * 154 = 462 sq cm

5. The volume of a cone is 1/3 the volume of a cylinder of the same base and height.

III. Short Answer Questions

Q1. Find the volume of a cylinder of radius 5 cm and height 14 cm.

Volume = pi * r^2 * h = (22/7) * 25 * 14 = (22/7) * 350 = 1100 cu cm

Q2. Find the volume of a cone of radius 3.5 cm and height 12 cm.

Volume = (1/3) * pi * r^2 * h = (1/3) * (22/7) * 3.5 * 3.5 * 12 = (1/3) * (22/7) * 147 = (1/3) * 462 = 154 cu cm

Q3. Find the volume and surface area of a sphere of radius 10.5 cm.

Volume = (4/3) * pi * r^3 = (4/3) * (22/7) * 10.5 * 10.5 * 10.5 = (4/3) * (22/7) * 1157.625 = (4/3) * 3638.25 = 4851 cu cm Surface area = 4 * pi * r^2 = 4 * (22/7) * 110.25 = 4 * 346.5 = 1386 sq cm

Q4. Find the volume of a hemisphere of radius 3.5 cm.

Volume = (2/3) * pi * r^3 = (2/3) * (22/7) * 3.5 * 3.5 * 3.5 = (2/3) * (22/7) * 42.875 = (2/3) * 134.75 = 269.5 / 3 = 89.83 cu cm (approx)

Q5. A metallic sphere of radius 7 cm is melted and recast into a cylinder of radius 7 cm. Find the height of the cylinder.

Volume of sphere = Volume of cylinder (4/3) * pi * r^3 = pi * R^2 * H (4/3) * pi * 343 = pi * 49 * H (4/3) * 343 = 49 * H H = (4 * 343) / (3 * 49) = 1372 / 147 = 28/3 = 9.33 cm (approx)

IV. Detailed Answer Questions

Q1. A cylindrical tank has radius 1.5 m and height 7 m. Find how many litres of water it can hold.

Volume = pi * r^2 * h = (22/7) * 1.5 * 1.5 * 7 = 49.5 cu m Capacity = 49.5 * 1000 = 49,500 litres (1 cu m = 1000 litres)

Q2. A cone of radius 6 cm and height 8 cm is reshaped into a sphere. Find the radius of the sphere.

Volume of cone = Volume of sphere (1/3) * pi * 36 * 8 = (4/3) * pi * R^3 96 = (4/3) * R^3 R^3 = 72 => R = 4.16 cm (approx)

Q3. The volume of a sphere is 38808 cu cm. Find its radius and surface area.

(4/3) * (22/7) * r^3 = 38808 r^3 = 38808 * 21 / 88 = 9261 => r = 21 cm Surface area = 4 * (22/7) * 441 = 5544 sq cm

Q4. A hemisphere is cut out from a solid cylinder of radius 7 cm and height 14 cm such that the hemisphere's base coincides with one base of the cylinder. Find the volume of the remaining solid.

Volume of cylinder = (22/7) * 49 * 14 = 2156 cu cm Volume of hemisphere = (2/3) * (22/7) * 343 = 718.67 cu cm Remaining volume = 2156 - 718.67 = 1437.33 cu cm (approx)

Exercise 2.4 — Word Problems and Applications

Q1. A well of radius 3.5 m is dug 14 m deep. The earth taken out is spread over a rectangular field of dimensions 50 m x 44 m. Find the height of the platform formed.

Volume of earth dug out = pi * r^2 * h = (22/7) * 3.5 * 3.5 * 14 = (22/7) * 171.5 = 539 cu m This is spread over area = 50 * 44 = 2200 sq m Height of platform = Volume / Area = 539 / 2200 = 0.245 m = 24.5 cm

Q2. A road roller of radius 35 cm and length 1 m takes 500 revolutions to level a road. Find the area of the road levelled.

Area covered in 1 revolution = CSA of roller = 2 * pi * r * h = 2 * (22/7) * 0.35 * 1 = 2 * (22/7) * 0.35 = 2 * 1.1 = 2.2 sq m Total area = 2.2 * 500 = 1100 sq m

Q3. The radius of a sphere is increased by 10%. Find the percentage increase in its surface area.

Original SA = 4 * pi * r^2 New radius = 1.1r New SA = 4 * pi * (1.1r)^2 = 4 * pi * 1.21 * r^2 = 1.21 * (4 * pi * r^2) Percentage increase = (1.21 - 1) * 100 = 21%

Q4. A cylindrical bucket 32 cm high and with radius 18 cm is filled with sand. The sand from the bucket is poured on the ground to form a conical heap whose height is 24 cm. Find the radius of the conical heap.

Volume of cylinder = Volume of cone pi * r1^2 * h1 = (1/3) * pi * r2^2 * h2 18^2 * 32 = (1/3) * r2^2 * 24 324 * 32 = 8 * r2^2 10368 = 8 * r2^2 r2^2 = 1296 r2 = 36 cm

Q5. How many spherical balls of radius 1 cm can be made from a solid sphere of radius 6 cm?

Volume of big sphere = (4/3) * pi * 6^3 = (4/3) * pi * 216 Volume of small sphere = (4/3) * pi * 1^3 = (4/3) * pi * 1 Number of balls = Volume of big / Volume of small = 216 / 1 = 216 balls

Q6. A hollow cylinder has outer radius 14 cm and inner radius 10 cm and height 21 cm. Find the volume of the hollow cylinder.

Volume = pi * (R^2 - r^2) * h = (22/7) * (196 - 100) * 21 = (22/7) * 96 * 21 = 22 * 96 * 3 = 6336 cu cm

Q7. A tent is in the shape of a cylinder surmounted by a cone. The radius of the base is 7 m. The cylindrical part has height 10 m and the conical part has height 4 m. Find the total surface area of the tent (excluding the base).

Slant height of cone: l = sqrt(7^2 + 4^2) = sqrt(65) = 8.06 m (approx) CSA of cylinder = 2 * pi * r * h = 2 * (22/7) * 7 * 10 = 440 sq m CSA of cone = pi * r * l = (22/7) * 7 * 8.06 = 177.32 sq m (approx) Total SA (excluding base) = 440 + 177.32 = 617.32 sq m (approx)

Q8. A juice seller has a cylindrical vessel of radius 21 cm and height 35 cm full of juice. He sells the juice in cone-shaped cups of radius 3.5 cm and height 7 cm. How many cups can be filled?

Volume of cylinder = pi * R^2 * H = (22/7) * 441 * 35 = 48510 cu cm Volume of cone = (1/3) * pi * r^2 * h = (1/3) * (22/7) * 12.25 * 7 = 89.83 cu cm Number of cups = 48510 / 89.83 = 540 cups (approx)

Q9. A hemispherical bowl of radius 10.5 cm is filled with water. The water is poured into a cylindrical bottle of radius 3.5 cm. Find the height of water in the bottle.

Volume of hemisphere = (2/3) * pi * r^3 = (2/3) * (22/7) * 10.5^3 = 2425.5 cu cm Volume of cylinder = pi * R^2 * H = (22/7) * 12.25 * H = 38.5 * H 38.5 * H = 2425.5 => H = 63 cm

Q10. A solid is composed of a cylinder with hemispherical ends. The radius is 7 cm and total height is 20 cm. Find the volume.

Height of cylindrical part = 20 - 14 = 6 cm Volume of cylinder = (22/7) * 49 * 6 = 924 cu cm Volume of two hemispheres = (4/3) * (22/7) * 343 = 1437.33 cu cm Total volume = 924 + 1437.33 = 2361.33 cu cm (approx)

Common Mistakes to Avoid

  • Using diameter instead of radius: The formulas for area (pi r^2) and circumference (2 pi r) use the radius*. If the diameter is given, always convert: r = d/2 before substituting.
  • Confusing CSA and TSA: CSA includes only the curved surface. TSA includes CSA plus the area of base(s). For a cylinder: TSA = CSA + 2 pi r^2; for a cone: TSA = CSA + pi * r^2.
  • Forgetting the slant height in cones: The height h of a cone is not the same as the slant height l. Use l = sqrt(r^2 + h^2) before computing CSA or TSA of a cone.
  • Mixing up volume formulas: Cylinder volume = pi r^2 h; Cone volume = (1/3) pi r^2 * h. The cone is one-third of the cylinder with the same base and height.
  • Forgetting to convert units: If dimensions are in cm, area is in sq cm and volume is in cu cm. When mixing m and cm, convert everything to the same unit first. 1 sq m = 10,000 sq cm; 1 cu m = 10,00,000 cu cm.
  • Wrong hemisphere TSA: The total surface area of a hemisphere includes the curved surface and the flat circular base. TSA = 3 pi r^2 (not 2 pi r^2, which is only CSA).
  • Missing base areas in open vessels: An open cylinder or cone does not have a top base. For open-top problems, TSA = CSA + one base area only.
  • Not reading "recast" problems correctly: When a solid is melted and recast, volumes are equal. Set Volume (original) = Volume (recast shape) and solve for the unknown.

Exam Checklist

  • [ ] Can you find the area and circumference of a circle given radius or diameter?
  • [ ] Can you find the area of a trapezium, rhombus, and quadrilateral with diagonals?
  • [ ] Can you compute CSA, TSA, and volume of a cylinder given radius and height?
  • [ ] Can you compute slant height, CSA, TSA, and volume of a cone?
  • [ ] Can you compute surface area and volume of a sphere and hemisphere?
  • [ ] Can you handle "recast" problems (sphere to cylinder, cone to sphere, etc.) where volumes are equal?
  • [ ] Can you solve composite shape problems (cylinder + cone, cylinder + hemispheres)?
  • [ ] Do you always convert diameter to radius before using formulas?
  • [ ] Can you distinguish between CSA and TSA for cylinders, cones, and hemispheres?
  • [ ] Do you include correct units (sq cm, sq m, cu cm, cu m, litres) in your answers?
  • [ ] Can you solve word problems involving circular paths, road rollers, and water tanks?
  • [ ] Can you find the percentage change in area/volume when radius is changed by a percentage?

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Frequently Asked Questions

Area = pi * r^2, where r is the radius. If the diameter d is given, use r = d/2. Example: For radius 7 cm, Area = (22/7) * 7^2 = 154 sq cm.
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