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#### 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. Photo Credit: Math-Salamanders Photo Credit: TutorialsPoint.Dev

• A Triangular Prism is a combination of rectangles and triangles. Photo Credit: Math-Salamanders Photo Credit: GigaCalculator

• A Cube is a combination of squares. Photo Credit: Math-Salamanders • A Cylinder is a combination of circles and a rectangle. Photo Credit: Math-Salamanders Photo Credit: NCalculators

• A Sphere is a combination of circles. Photo Credit: Varsity Tutors

• 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.