Surface Area, Volume, and Capacity of Solids
 Most solids are a combination of two or more plane shapes. Examples:
 A Cuboid (Rectangular Solid or Rectangular Prism) is a combination of rectangles.
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 A Cuboid (Rectangular Solid or Rectangular Prism) is a combination of rectangles.

 A Triangular Prism is a combination of rectangles and triangles.
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 A Triangular Prism is a combination of rectangles and triangles.

 A Cube is a combination of squares.
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 A Cube is a combination of squares.

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

 A Cone is a combination of a triangle and a circle.
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 A Cone is a combination of a triangle and a circle.
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 A SquareBased Pyramid is a combination of a square and triangles.
 A SquareBased Pyramid is a combination of a square and triangles.
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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 openended 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}/_{3}πr³
 cone = ^{1}/_{3}πr²h
 pyramid = ^{1}/_{3}Bh
 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
 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
 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}/_{3}πr²h
 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.
 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
 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 = 2^{1}/_{2} m
 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
 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
 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
 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 :
 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
 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
 its curved surface area.
 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.