Volume and Capacity Chapter Notes | Mathematics Class 8 ICSE PDF Download

Introduction

This chapter introduces the concepts of volume, capacity, and surface area for three-dimensional shapes like cuboids, cubes, and cylinders. Volume refers to the space occupied by a solid, capacity is the internal volume of a container, and surface area is the total area of all faces of a solid. The following notes provide a clear, step-by-step explanation of each concept, accompanied by examples and practical applications that illustrate the calculations.

Key Points

• Volume is the measure of how much space an object occupies. It is usually measured in cubic units, such as cubic meters (m³), cubic centimeters (cm³), or liters (L). The formula to calculate the volume of a cuboid is:
  Volume of a Cuboid = Length × Breadth × Height
• Capacity refers to the maximum amount of space that a container can hold. It is often measured in litres or gallons. For example, a water tank with a capacity of 1000 litres can hold 1000 litres of water. The capacity of a cuboid can be calculated using the same formula as volume, since it refers to the space inside the container.
• Surface area is the total area of all the surfaces of a three-dimensional object. It is measured in square units, such as square meters (m²) or square centimetres (cm²). The formula to calculate the surface area of a cuboid is:
  Surface Area of a Cuboid = 2 × (Length × Breadth + Length × Height + Breadth × Height)

Units of measurement:

• Length: metre (m), centimetre (cm), millimetre (mm).
• Volume: cubic metre (m³), cubic centimetre (cm³), cubic millimetre (mm³).
• Surface area: square metre (m²), square centimetre (cm²), square millimetre (mm²).

Conversion between units:

• 1 m³ = 1,000,000 cm³ (100 × 100 × 100 cm³).
• 1 cm³ = 1/1,000,000 m³.
• 1 cm³ = 1,000 mm³ (10 × 10 × 10 mm³).
• 1 mm³ = 1/1,000 cm³.

Volume of liquids or gases is measured in litres:

• 1 m³ = 1,000 litres.
• 1 litre = 1,000 cm³ (also called millilitres or c.c.).

Example: Understand that if a container has a capacity of 1 litre, its internal volume is 1,000 cm³.

Cuboid (A Rectangular Solid)

A cuboid is a solid shape with six rectangular faces.

Key measurements

• Volume: The volume of a cuboid is calculated by multiplying its length, breadth, and height. The formula is: Volume = Length × Breadth × Height (l × b × h).
• Total Surface Area: The total surface area of a cuboid is found by using the formula: Total Surface Area = 2(l × b + b × h + h × l).

Area of Faces

• Area of Top and Bottom Faces: The area of the top and bottom faces is calculated as 2(l × b).
• Area of Front and Back Faces: The area of the front and back faces is calculated as 2(b × h).
• Area of Left and Right Faces: The area of the left and right faces is calculated as 2(h × l).

Steps to calculate Volume

• To calculate the volume of a cuboid, measure the length, breadth, and height in the same unit.
• Multiply these three dimensions together: Volume = Length × Breadth × Height.
• Express the result in cubic units, such as cm³ or m³.

Steps to Calculate the Total Surface Area

• Measure the length, breadth, and height of the cuboid in the same unit.
• Calculate the following products: Length × Breadth, Breadth × Height, and Height × Length.
• Add these three products together and multiply the sum by 2. 
  Total Surface Area = 2 × (Length × Breadth + Breadth × Height + Height × Length).
• Express the result in square units, such as cm² or m².

Example: A cuboid has length, breadth, and height in the ratio 6:5:4, with a volume of 15,000 cm³.

Steps to find dimensions:

• Let length = 6x cm, breadth = 5x cm, height = 4x cm.
• Volume = 6x × 5x × 4x = 15,000 cm³.
• Simplify: 120x³ = 15,000 → x³ = 15,000/120 = 125.
• Find x: x = ∛125 = 5.
• Dimensions: length = 6 × 5 = 30 cm, breadth = 5 × 5 = 25 cm, height = 4 × 5 = 20 cm.

Steps to find surface area:

• Use formula: 2(l × b + b × h + h × l).
• Substitute: 2(30 × 25 + 25 × 20 + 20 × 30) = 2(750 + 500 + 600) = 2 × 1850 = 3,700 cm².

Cube

A cube is a special kind of cuboid where all the faces are squares, meaning the length, breadth, and height are all the same. Let's say the edge of the cube is a.

• Volume = a × a × a = a³ ( edge³ ).
• Total surface area = 6 × (edge)² = 6a² (since each face is a square).

Steps to calculate Volume

• Measure the edge length ( a ).
• Calculate the volume by finding a³.
• Express the result in cubic units.

Steps to calculate the Total Surface Area

• Measure the edge length (a).
• Calculate a² for one face.
• Multiply by 6 (since a cube has 6 faces).
• Express the result in square units.

Example:

• A cube has a total surface area of 294 cm². Find its volume.
• Steps:
  1. Total surface area = 6a² = 294 cm².
  2. Find a²: a² = 294/6 = 49.
  3. Find a: a = √49 = 7 cm.
  4. Volume = a³ = 7³ = 343 cm³.

Application

Application 1: For a Room

• A room has four walls: two along its length and two along its width.
• Area calculations:
  1. Area of each wall along length = length × height (l × h).
  2. Area of each wall along width = breadth × height (b × h).
  3. Total area of four walls = 2(l × h) + 2(b × h) = 2(l + b) × h.
  4. This includes the area of doors and windows.
  5. Area of roof = area of floor = length × breadth (l × b).
• Steps to calculate the area to be whitewashed:
  1. Calculate the area of four walls: 2(l + b) × h.
  2. Calculate the area of the roof: l × b.
  3. Subtract the area of doors and windows if they are not to be whitewashed.
     
  4. Add the remaining areas to get the total area to be whitewashed.

Example: A room has internal dimensions 6 m (length), 5.2 m (breadth), 4.5 m (height), with two doors (1.2 m × 2 m) and three windows (1 m × 0.8 m). Find the area to be whitewashed and cost at ₹6 per m².
Steps:

• Area of four walls = 2(6 + 5.2) × 4.5 = 2 × 11.2 × 4.5 = 100.8 m².
• Area of roof = 6 × 5.2 = 31.2 m².
• Area of two doors = 2 × (1.2 × 2) = 4.8 m².
• Area of three windows = 3 × (1 × 0.8) = 2.4 m².
• Total area to whitewash = (100.8 + 31.2) - (4.8 + 2.4) = 124.8 m².
• Cost = 124.8 × 6 = ₹748.80.

Application 2: For a Box

• Space occupied = external volume.
• Capacity = internal volume.
• Volume of material = external volume - internal volume.
• For a closed box with external dimensions l, b, h and wall thickness t:
  1. Internal length = l - 2t.
  2. Internal breadth = b - 2t.
  3. Internal height = h - 2t.
• If internal dimensions are l, b, h, and thickness is t:
  1. External length = l + 2t.
  2. External breadth = b + 2t.
  3. External height = h + 2t.
• Steps to calculate capacity:
  1. Find internal dimensions by subtracting twice the thickness from external dimensions.
  2. Calculate the internal volume: length × breadth × height
• Steps to calculate the volume of material:
  1. Calculate external volume: l × b × h.
  2. Calculate internal volume: (l - 2t) × (b - 2t) × (h - 2t).
  3. Subtract internal volume from external volume.

Example: A closed wooden box has external dimensions 30 cm, 18 cm, 20 cm, with walls 1.5 cm thick. Find capacity, volume of wood, and weight if 1 cm³ of wood weighs 0.8 g.
Steps:

• External volume = 30 × 18 × 20 = 10,800 cm³.
• Internal dimensions: length = 30 - 2 × 1.5 = 27 cm, breadth = 18 - 2 × 1.5 = 15 cm, height = 20 - 2 × 1.5 = 17 cm.
• Capacity (internal volume) = 27 × 15 × 17 = 6,885 cm³.
• Volume of wood = 10,800 - 6,885 = 3,915 cm³.
• Weight = 3,915 × 0.8 = 3,132 g = 3.132 kg.

Cylinder

A cylinder is a solid with a uniform circular cross-section. Let the radius of the base = r cm, height = h cm.

Key measurements

• Area of cross-section = πr².
• Perimeter of cross-section = 2πr.
• Curved surface area = perimeter × height = 2πrh.
• Total surface area = curved surface area + 2 × area of cross-section = 2πrh + 2πr² = 2πr(h + r).
• Volume = area of cross-section × height = πr²h.
• Use π = 22/7 unless specified otherwise.

Steps to calculate Curved Surface Area

• Measure radius (r) and height (h).
• Calculate 2πrh.
• Express in square units.

Steps to calculate the Total Surface Area

• Calculate curved surface area: 2πrh.
• Calculate the area of two bases: 2 × πr².
• Add both: 2πr(h + r).
• Express in square units.

Steps to calculate Volume

• Measure radius (r) and height (h).
• Calculate πr²h.
• Express in cubic units.

Example:

• A cylinder has a curved surface area of 17,600 cm² and a base circumference of 220 cm. Find height and volume.
• Steps:
  • Curved surface area = 2πrh = 17,600 cm², circumference = 2πr = 220 cm.
  • Find height: h = 17,600 / 220 = 80 cm.
  • Find radius: 2πr = 220 → 2 × 22/7 × r = 220 → r = (220 × 7) / (2 × 22) = 35 cm.
  • Volume = πr²h = 22/7 × 35 × 35 × 80 = 308,000 cm³.