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.
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- A Triangular Prism is a combination of rectangles and triangles.
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- A Triangular Prism is a combination of rectangles and triangles.
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- A Cube is a combination of squares.
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- A Cube is a combination of squares.
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- 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.
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- 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 Square-Based Pyramid is a combination of a square and triangles.
- A Square-Based 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 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
- 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 = 1/3πr²h
- 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.
- 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 = 21/2 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.