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Surface Area, Volume, and Capacity of Solids

  • Most solids are a combination of two or more plane shapes. Examples:

 

    • A Cylinder is a combination of circles and a rectangle.

      Photo Credit: Math-Salamanders

 

Photo Credit: NCalculators

    • A Cone is a combination of a triangle and a circle.

      Photo Credit: Math-Salamanders

 

Photo Credit: MathWorld.Wolfram

    • A Square-Based Pyramid is a combination of a square and triangles.

Photo Credit: Math-Salamanders

Photo Credit: Mammoth Memory

SURFACE AREA OF SOLIDS
  • The surface area of a solid is a measure of the distance around the solid.
  • Surface area is measured in units such as cm² or m² which are derived from the SI units of length.
  • The surface area (SA) of a:
    • cuboid (or a rectangular solid, or a rectangular prism) = 2lw + 2lh + 2wh = 2(lw + lh + wh)
    • triangular prism = 2B + Ph
    • cube = 6S²
    • cylinder = 2πr² + 2πrh (for a closed cylinder), and πr² + 2πrh (for an open-ended cylinder)
    • sphere = 4πr²
    • cone = πr² + πrs
    • pyramid = B + ½Ps
    • NOTE: l – length, w – breadth or width, h – height, B – base area, P – base perimeter, S – side length, r – radius, s – slant height.
  • When solving surface area questions, make sure the units are the same throughout the equation.
VOLUME OF SOLIDS
  • The volume of a solid is a measure of the space it takes up.
  • Volume is measured in units such as cm³ or m³ which are derived from the SI units of length.
  • The volume of a:
    • cuboid (or a rectangular solid, or a rectangular prism) = lwh
    • triangular prism = Bh
    • cube = S³
    • cylinder = πr²h
    • sphere = 4/3πr³
    • cone = 1/3πr²h
    • pyramid = 1/3Bh
    • NOTE: l – length, w – breadth or width, h – height, B – base area, P – base perimeter, S – side length, r – radius.
  • When solving volume questions, make sure the units are the same throughout the equation.
CAPACITY OF SOLIDS
  • The capacity of a container is a measure of the space inside it. It is the amount of substance, solid, liquid or gaseous, a volume can hold.
  • The basic unit of capacity is the liter.
  • Liquids and gases are usually measured in liters and ml (milliliters)
  • 1 ml = 1 cm³
Examples
  1. Find the surface area of a cube, if the area of one of its faces is 40 cm².
    • The surface area of a cube = 6S²
    • Remember, a cube is a combination of six squares.
    • The area of a square = l² = s² = 40 cm²
    • ∴ The surface area of the cube = 6 × 40 = 240 cm²
  2. A cone and a cylinder have the same volume of 180 cm³, and the same height, h = 15 cm. Which of these two solids has the larger surface area? (Take π = 22/7)
    • The volume of a cylinder = πr²h
    • The volume of the cylinder = 180 cm³
    • The height of the cylinder = 15 cm
    • ∴ The radius of the cylinder = √(180/22/7 × 15)
    • ∴ The radius of the cylinder = 1.95 cm
    • The surface of a cylinder = 2πr² + 2πrh
    • ∴ The surface of a cylinder = 2 × 22/7 × 1.95² + 2 × 22/7 × 1.95 × 15
    • ∴ The surface of a cylinder = 207.76 cm²
    • The volume of a cone = 1/3πr²h
    • The volume of the cone = 180 cm³
    • The height of the cone = 15 cm
    • ∴ The radius of the cone = √(3 × 180/22/7 × 15)
    • ∴ The radius of the cone = 3.38 cm
    • The surface area of a cone = πr² + πrs
    • ∴The surface area of the cone = 22/7 × 3.38² + 22/7 × 3.38 × s
    • To get the slant height, s, use the pythagoras’ theorem:
    • s = √(3.38² + 15²)
    • ∴ s = 15.38 cm
    • ∴ The surface area of the cone = 22/7 × 3.38² + 22/7 × 3.38 × 15.38
    • ∴ The surface area of the cone = 199.28 cm²
    • Therefore, the cylinder has the larger surface area.
  3. Calculate the volume of a rectangular box which measures 30 cm × 15 cm × 10 cm.
    • Volume of a Box = (30 × 15 × 10) cm³
    •                                 = 4500 cm³
  4. A rectangular room 4 m long by 3m wide contains 30 m³ of air. Calculate the height of the room.
    • Volume = Area × height
    • Volume of the room = 30 m³
    • Area of the floor (base) = 4 m × 3 m = 12 m
    • ∴ Height = Volume/Area
    •                  = 30/12
    • ∴ The height of the room  = 21/2 m
  5. A concrete beam is 20 m long. Its end face is a rectangle 60 cm by 40 cm. Calculate the volume of the beam. Find the mass of the beam if 1 m³ of concrete has a mass of 2.5 tonnes.
    • Volume of the beam = Area of the Rectangle × height
    • Area of the rectangle = Length × breadth
    •                                              = 0.6 × 0.4
    • ∴ The area of the rectangle = 0.24 m²
    • ∴ The volume of the beam = 0.24 × 20
    • ∴ The volume of the beam = 4.8 m³
    • 1 m³ of beam = 2.5 tonnes
    • ∴ 4.8 m³ of beam = 4.8 × 2.5
    • ∴ 4.8 m³ of beam = 12 tonnes
  6. How many liters of water does a 5 m × 4 m × 3 m tank hold?
    • Volume of a tank = (5 × 4 × 3) m³
    • ∴ Volume of a tank = 60 m³
    • But 1 m³ = 1000 liters
    • ∴ The capacity of the tank = 60 × 1000
    • ∴ The capacity of the tank = 60000 liters
  7. An edge of a cube is 2 cm. How many such cubes can fill a cuboid 3 m long, 2 m wide, and 1 m high?
    • For the cube: L = W = H = 2 cm
    • ∴ The volume of the cube = 2 × 2 × 2 = 8 cm³
    • For the cuboid: Length = 3 m = 300 cm,
    •                            Width = 2 m = 200 cm,
    •                            Height = 1 m = 100 cm
    • ∴ The volume of the cuboid = 300 × 200 × 100
    •                                                  = 6000000 cm³
    • The number of cubes that can fill the cuboid = volume of the cuboid ÷ volume of the cube
    •                                                                               = 6000000/8
    •                                                                               = 750000 cubes
  8. A cylindrical cup has a base of radius 7 cm and a height of 10 cm. Taking the value of π to be 22/7, calculate :
    1. its curved surface area.
      • the area of the curved surface of a cylinder of radius r and height h is 2πrh.
      • The curved surface area of the cup = 2 × 22/7 × 7 × 10
      • ∴ The curved surface area of the cup = 440 cm²
    2. the area of its circular base.
      • the area of the circular base of the cup = 22/7 × 7 × 7
      • ∴ The area of its circular base = 154 cm²
  9. How many liters can be held by a cylindrical can 14 cm in diameter and 20 cm high?
    • Volume of can = πr²h
    •                            = 22/7 × 7 × 7 × 20
    •                            = 3080 cm³
    • 1 liter = 1000 cm³
    • ∴ Capacity of the can = 3.08 liters.
    • The can holds about 3 liters.
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