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State Board · Class 9 · Mathematics Part 2

Lesson 9: Surface Area and Volume

4 concepts · 32 practice questions with answers · Free to read

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Revision: cuboid, cube and cylinder

From daily life

A milk-powder tin (cylinder), a biscuit packet (cuboid) and a dice (cube). To paint them we need the surface area; to know how much they hold we need the volume.

Simple definition

Surface area is the total area of all the faces (square units). Volume is the space inside (cubic units). The curved surface area is only the curved part, while the total surface area includes the top and bottom too.

Formulas of the cuboid, cube and cylinderRevision: cuboid, cube and cylinderlhbCuboid: lateral area 2(l + b) × htotal 2(lb + bh + lh); volume l × b × hCube (side a): total 6a²; volume a³Cylinder: curved 2πrh; total 2πr(r + h)volume πr²hNow the cone and the sphere: two new solids

In the figure: The formulas of three solids together.

Live figure

Revision: cuboid, cube and cylinder

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Rule 1: Cuboid and cube — Cuboid: total 2(lb + bh + lh), volume l × b × h. Cube: total 6 × side², volume side³.

  • Easy: Cube, side 6: total 6 × 36 = 216, volume 216.
  • Medium: Cuboid 8 × 5 × 4: total 2(40 + 20 + 32) = 184, volume 160.
  • Exam type: A tank is 5 m long, 4 m wide and 3 m deep. Painting four inner walls and the floor: 2(5 + 4) × 3 + 5 × 4 = 54 + 20 = 74 m².

Rule 2: Cylinder — Curved area 2πrh, total 2πr(r + h), volume πr²h. Here π = 22/7.

  • Easy: r = 7, h = 10: curved 2 × 22/7 × 7 × 10 = 440.
  • Medium: For the same cylinder the total is 440 + 2 × (22/7 × 49) = 440 + 308 = 748 and the volume is 22/7 × 49 × 10 = 1540.
  • Exam type: Tin r = 14 cm, h = 20 cm: volume 22/7 × 196 × 20 = 12320 cm³ = 12.32 litres.

Rule 3: Units and conversion — Length in cm, area in cm², volume in cm³. 1 litre = 1000 cm³ and 1 m³ = 1000 litres.

  • Easy: 2000 cm³ = 2 litres.
  • Medium: 3 m³ = 3000 litres.
  • Exam type: A tank 2 m × 1 m × 1.5 m: volume 3 m³ = 3000 litres. At 12 litres a day it lasts 250 days.
Remember

Surface area is area (cm²), volume is space (cm³). Curved surface area leaves out the top and bottom.

Common mistake

Mixing up radius and diameter. The formulas use r; halve the diameter if it is given.

Shortcut

Questions on filling water are about volume; questions on painting are about surface area. Memory clue: Paint = surface area (square units), water = volume (cubic units).

Spot the difference

ShapeTotal surface areaVolume
Cube (side a)6a²a³
Cuboid2(lb + bh + lh)l × b × h
Cylinder2πr(r + h)πr²h

Practice questions

Easy
  1. 1If the side of a cube is 5, find its volume.
  2. 2Find the volume of the cuboid 6 × 4 × 3.
  3. 3How many litres is 5000 cm³?
Medium
  1. 4Find the curved surface area of a cylinder with r = 7, h = 5.
  2. 5Find the volume of the same cylinder.
  3. 6Find the total surface area of the cuboid 10 × 6 × 2.
Challenging
  1. 7A cylindrical tank has diameter 14 m and height 5 m. Find its volume in litres.
  2. 8The total surface area of a cube is 150. Find its side and volume.
Show answers (try them yourself first)
  1. 1125. 2) 72. 3) 5 litres.
  2. 42 × 22/7 × 7 × 5 = 220. 5) 22/7 × 49 × 5 = 770. 6) 2(60 + 12 + 20) = 184.
  3. 7r = 7: 22/7 × 49 × 5 = 770 m³ = 770000 litres. 8) 6a² = 150 ⇒ a² = 25 ⇒ a = 5. Volume 125.

Surface area of a cone

From daily life

An ice-cream cone, a joker's cap and a conical lamp. If the cap is opened out, it becomes a part of a circle (a sector). Its area is the curved surface area of the cone.

Simple definition

A cone has a base (a circle of radius r), a height h and a slant height l. l² = r² + h². Curved surface area = πrl and total surface area = πr(r + l) (including the base).

Parts of a cone and the opened-out shape of its curved surfaceCone: r, h and the slant height lhrll² = r² + h²r = 7, h = 24 ⇒ l = 25larc length = 2πrcurved surface = πrltotal = πr(r + l)

In the figure: r = 7, h = 24 ⇒ l = 25. The arc of the opened-out part is 2πr.

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Surface area of a cone

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Rule 1: First find the slant height l — l = √(r² + h²). The right triangle r, h, l follows Pythagoras.

  • Easy: r = 3, h = 4 ⇒ l = 5.
  • Medium: r = 7, h = 24 ⇒ l = √(49 + 576) = 25.
  • Exam type: r = 9, h = 12 ⇒ l = √(81 + 144) = 15. r = 5, h = 12 ⇒ l = 13.

Rule 2: Curved surface area = πrl — This is the paper needed for a cap. It tells how much paper a joker's cap needs.

  • Easy: r = 7, l = 25 ⇒ 22/7 × 7 × 25 = 550.
  • Medium: r = 3, l = 5 ⇒ 22/7 × 3 × 5 = 330/7 ≈ 47.14.
  • Exam type: Diameter 14 cm and height 24 cm, a cap: r = 7, l = 25, curved area 550 cm². 20 caps ⇒ 11000 cm².

Rule 3: Total surface area = πr(r + l) — Curved area + base area πr². For an open cone (a cap) take only the curved area.

  • Easy: r = 7, l = 25: base 22/7 × 49 = 154, total 550 + 154 = 704.
  • Medium: Check by formula: 22/7 × 7 × (7 + 25) = 22 × 32 = 704.
  • Exam type: r = 5, h = 12 ⇒ l = 13: total 22/7 × 5 × 18 = 1980/7 ≈ 282.86.
Remember

l² = r² + h². Curved area πrl, total πr(r + l). A cap = only the curved area.

Common mistake

Using h in place of l. Surface area needs l, while volume needs h.

Shortcut

Keep 7-24-25, 5-12-13, 3-4-5 in mind; then no square root needs working out. Memory clue: Cone: slant height l = √(r² + h²); "the slant height is the hypotenuse".

Spot the difference

ConeFormular = 7, h = 24
Slant height l√(r² + h²)25
Curved areaπrl550
Total areaπr(r + l)704

Practice questions

Easy
  1. 1If r = 6 and h = 8, find l.
  2. 2If r = 7 and l = 10, find the curved surface area.
  3. 3Does the curved surface area use h or l?
Medium
  1. 4If r = 5 and h = 12, find the curved surface area.
  2. 5If r = 14 and l = 50, find the curved surface area.
  3. 6If r = 7 and l = 25, find the total surface area.
Challenging
  1. 7A joker's cap has base diameter 14 cm and height 24 cm. How much paper for one cap and for 10 caps?
  2. 8The base area of a cone is 154 and l = 25. Find the radius and the curved surface area.
Show answers (try them yourself first)
  1. 1l = √(36 + 64) = 10. 2) 22/7 × 7 × 10 = 220. 3) l.
  2. 4l = 13 ⇒ 22/7 × 5 × 13 = 1430/7 ≈ 204.29. 5) 22/7 × 14 × 50 = 2200. 6) 22/7 × 7 × (7 + 25) = 704.
  3. 7r = 7, l = 25 ⇒ 550 cm² for one; 5500 cm² for 10. 8) 22/7 × r² = 154 ⇒ r² = 49 ⇒ r = 7. Curved area 22/7 × 7 × 25 = 550.

Volume of a cone

From daily life

An ice-cream cone and a cup with the same base and height. The ice cream of three cones fills the cup. So the volume of a cone is one third of the cup.

Simple definition

Volume of a cone = ⅓ × πr²h. Here h is the height (not the slant height). A cylinder with the same base and height has volume πr²h and the cone is one third of that.

Volumes of a cone and a cylinder with the same base and heightVolume of a cone = one third of the cylindersame r and same hCone: V = ⅓ πr²hCylinder: V = πr²hWater of 3 cones = 1 cylinderr = 7, h = 24 ⇒ V = ⅓ × 22/7 × 49 × 24= 1232

In the figure: r = 7, h = 24 ⇒ V = ⅓ × 22/7 × 49 × 24 = 1232.

Rule 1: Use the formula: ⅓ πr²h — First r², then π, then h, and divide by 3 at the end.

  • Easy: r = 3, h = 7 ⇒ ⅓ × 22/7 × 9 × 7 = 66.
  • Medium: r = 7, h = 24 ⇒ ⅓ × 22/7 × 49 × 24 = 1232.
  • Exam type: r = 14, h = 15 ⇒ ⅓ × 22/7 × 196 × 15 = 3080.

Rule 2: If h is not given, find it from l — h = √(l² − r²). First h, then the volume.

  • Easy: l = 5, r = 3 ⇒ h = 4.
  • Medium: r = 7, l = 25 ⇒ h = 24 ⇒ V = 1232.
  • Exam type: r = 6, l = 10 ⇒ h = 8 ⇒ V = ⅓ × 22/7 × 36 × 8 = 2112/7 ≈ 301.71.

Rule 3: Comparing cone and cylinder — Same r and h ⇒ cone : cylinder = 1 : 3. Cutting the largest cone from a cylinder keeps one third; two thirds is wasted.

  • Easy: Cylinder volume 3000 ⇒ cone of the same base and height 1000.
  • Medium: Cone volume 154 ⇒ cylinder 462.
  • Exam type: r = 7, h = 24: cylinder 22/7 × 49 × 24 = 3696 and cone 1232 ⇒ 3696 ÷ 1232 = 3.
Remember

Cone volume = cylinder volume ÷ 3. Volume uses h (straight height), surface area uses l (slant).

Common mistake

Forgetting the ⅓ or using l in place of h.

Shortcut

Check: the cone volume must be exactly one third of the cylinder volume. Memory clue: A cone is a third of the cylinder; 3 cones = 1 cylinder.

Spot the difference

CylinderCone
Basecircle rcircle r
Volumeπr²h⅓ πr²h
r = 7, h = 2436961232

Practice questions

Easy
  1. 1Find the volume of a cone with r = 3, h = 7.
  2. 2If the volume of a cylinder is 90, what is that of a cone of the same base and height?
  3. 3Does the volume of a cone use h or l?
Medium
  1. 4Find the volume of a cone with r = 7, h = 6.
  2. 5If r = 6 and l = 10, find h and the volume.
  3. 6Find the volume of a cone with r = 14, h = 3.
Challenging
  1. 7A conical heap of sand has r = 7 m and h = 12 m. Find its volume and how many loads of 20 m³ it needs.
  2. 8A cone and a cylinder have equal r and h and the cylinder volume is 2310. Find the cone volume and, if r = 7, find h.
Show answers (try them yourself first)
  1. 166. 2) 30. 3) h.
  2. 4⅓ × 22/7 × 49 × 6 = 308. 5) h = 8 ⇒ V = 2112/7 ≈ 301.71. 6) ⅓ × 22/7 × 196 × 3 = 616.
  3. 7V = ⅓ × 22/7 × 49 × 12 = 616 m³; 616 ÷ 20 = 30.8 ⇒ 31 loads. 8) Cone 2310 ÷ 3 = 770. 22/7 × 49 × h = 2310 ⇒ 154h = 2310 ⇒ h = 15.

Sphere and hemisphere

From daily life

A cricket ball and a football are spheres. Cut a sphere through the middle and you get two equal hemispheres, like half a watermelon. A bowl, a dome and a basin are hemispheres.

Simple definition

Every point on the surface of a sphere is equally far from its centre (radius r). Sphere: surface area 4πr², volume (4/3)πr³. Hemisphere: curved area 2πr², total 3πr² (including the flat circle), volume (2/3)πr³.

Formulas of the sphere and the hemisphereSphere and hemisphererSphere: S = 4πr²V = (4/3)πr³rCurved 2πr²; total 3πr²V = (2/3)πr³Hemisphere = half of a sphere

In the figure: Formulas of the sphere and hemisphere; r is the same in both.

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Sphere and hemisphere

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Rule 1: Surface area and volume of a sphere — r = 7: surface area 4 × 22/7 × 49 = 616. r = 21: volume (4/3) × 22/7 × 9261 = 38808.

  • Easy: r = 7 ⇒ surface area 616.
  • Medium: r = 21 ⇒ volume (4/3) × 22/7 × 21 × 21 × 21 = 38808.
  • Exam type: The surface area of a ball with diameter 14 cm: r = 7 ⇒ 616 cm². If the diameter is given, find r first.

Rule 2: Curved and total surface of a hemisphere — Curved: 2πr² (half the sphere), total: 3πr² (curved + the circle πr²). To paint a bowl inside, use the curved area.

  • Easy: r = 7 ⇒ curved 2 × 22/7 × 49 = 308.
  • Medium: r = 7 ⇒ total 3 × 22/7 × 49 = 462 (308 + 154).
  • Exam type: r = 14 ⇒ curved 2 × 22/7 × 196 = 1232 and total 1848.

Rule 3: Volume of a hemisphere = (2/3)πr³ — Half the volume of the sphere. Use it to find how much milk a bowl holds.

  • Easy: r = 21 ⇒ (2/3) × 22/7 × 9261 = 19404.
  • Medium: The same sphere is 38808 ⇒ half is 19404 (it matches).
  • Exam type: Bowl r = 7 cm: (2/3) × 22/7 × 343 = 718.67 cm³ ≈ 0.72 litres.
Remember

Sphere: 4πr² and (4/3)πr³. Hemisphere: curved 2πr², total 3πr², volume (2/3)πr³.

Common mistake

Taking the total surface of a hemisphere as 2πr². Adding the circle πr² gives 3πr².

Shortcut

If a diameter is given, halve it first; all formulas use r. Memory clue: Sphere 4πr², hemisphere (curved) 2πr²; half a sphere, but count the "base" separately.

Spot the difference

ShapeSurface areaVolume
Sphere4πr²(4/3)πr³
Hemisphere (curved)2πr²(2/3)πr³
Hemisphere (total)3πr²(2/3)πr³

Practice questions

Easy
  1. 1What is the surface area of a sphere with r = 7?
  2. 2What is the curved surface area of a hemisphere with r = 7?
  3. 3If the volume of a sphere is 100, what is that of the hemisphere?
Medium
  1. 4Find the surface area of a ball with diameter 28 cm.
  2. 5Find the total surface area of a hemisphere with r = 14.
  3. 6Find the volume of a hemisphere with r = 21.
Challenging
  1. 7The inner diameter of a hemispherical bowl is 14 cm. Find its volume and the area to be painted inside.
  2. 8If the surface area of a sphere is 616, find r and then the volume.
Show answers (try them yourself first)
  1. 1616. 2) 308. 3) 50.
  2. 4r = 14 ⇒ 4 × 22/7 × 196 = 2464. 5) 3 × 22/7 × 196 = 1848. 6) (2/3) × 22/7 × 9261 = 19404.
  3. 7r = 7 ⇒ volume ≈ 718.67 cm³; inner area 2πr² = 308 cm². 8) 4πr² = 616 ⇒ r² = 49 ⇒ r = 7. Volume (4/3) × 22/7 × 343 = 1437.33.