Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Calculate the area of compound shapes.
• Find the surface area of cubes and rectangular prisms.
• Calculate the volume of cubes and rectangular prisms.
• Solve practical problems involving these concepts.
• Convert between different units of area and volume.

Lesson summary

This week, we delve into the fascinating world of measurement, specifically focusing on area, surface area, and volume. Understanding these concepts is crucial not only for academic success but also for everyday life in South Africa. Imagine planning a garden for your home in Khayelitsha, estimating the amount of paint needed for a classroom in Limpopo, or calculating the capacity of a water tank in a drought-stricken area of the Northern Cape. These are just a few examples where a solid grasp of area, surface area, and volume proves invaluable. We will build upon your previous knowledge of shapes and units to tackle more complex calculations and applications.

Lesson notes

2.1 Area of Compound Shapes: A compound shape is a shape made up of two or more basic shapes (squares, rectangles, triangles, circles). To find the area of a compound shape, we need to: Decompose: Break down the compound shape into its basic component shapes.

Calculate: Find the area of each of the basic shapes separately.

Combine: Add or subtract the areas of the basic shapes, depending on how they are arranged, to find the total area.

Example 1: A floor plan consists of a rectangle with a semi-circle attached to one of its sides. The rectangle measures 8m by 6m, and the diameter of the semi-circle is 6m. Find the total area of the floor plan.

Solution: Area of Rectangle = length x width = 8m x 6m = 48 m² Radius of Semi-circle = diameter / 2 = 6m / 2 = 3m Area of Semi-circle = (1/2) x π x radius² = (1/2) x 3.14 x (3m)² = (1/2) x 3.14 x 9 m² ≈ 14.13 m² Total Area = Area of Rectangle + Area of Semi-circle = 48 m² + 14.13 m² = 62.13 m² Therefore, the total area of the floor plan is approximately 62.13 m².

Example 2: A shape is formed by a square (side = 5cm) with a triangle on top. The base of the triangle is the same as the side of the square (5cm), and the height of the triangle is 4cm. Find the total area.

Solution: Area of Square = side x side = 5cm x 5cm = 25 cm² Area of Triangle = (1/2) x base x height = (1/2) x 5cm x 4cm = 10 cm² Total Area = Area of Square + Area of Triangle = 25 cm² + 10 cm² = 35 cm² Therefore, the total area of the shape is 35 cm². 2.2 Surface Area of Cubes and Rectangular Prisms: The surface area of a 3D object is the total area of all its faces. To find the surface area, we can use two methods: Using Nets: A net is a 2D representation of a 3D shape that can be folded to form the shape. Draw the net of the cube or rectangular prism, find the area of each face, and then add all the areas together.

Using Formulas: Cube:* A cube has 6 identical square faces. If the side length of the cube is 's', then the surface area is 6s².

Rectangular Prism:* A rectangular prism has 6 rectangular faces. If the length, width, and height are 'l', 'w', and 'h' respectively, then the surface area is 2(lw + lh + wh).

Example 3: Find the surface area of a cube with a side length of 4 cm.

Solution: Using Formula: Surface Area = 6s² = 6 x (4cm)² = 6 x 16 cm² = 96 cm² Therefore, the surface area of the cube is 96 cm².

Example 4: Find the surface area of a rectangular prism with length 5cm, width 3cm, and height 2cm.

Solution: Using Formula: Surface Area = 2(lw + lh + wh) = 2((5cm x 3cm) + (5cm x 2cm) + (3cm x 2cm)) = 2(15 cm² + 10 cm² + 6 cm²) = 2(31 cm²) = 62 cm² Therefore, the surface area of the rectangular prism is 62 cm². 2.3 Volume of Cubes and Rectangular Prisms: Volume is the amount of space a 3D object occupies.

Cube:* Volume = side x side x side = s³ Rectangular Prism:* Volume = length x width x height = lwh The volume is measured in cubic units (cm³, m³). 1 m³ = 1000 litres Example 5: Find the volume of a cube with a side length of 3m.

Solution: Volume = s³ = (3m)³ = 3m x 3m x 3m = 27 m³ Therefore, the volume of the cube is 27 m³.

Example 6: Find the volume of a rectangular prism with length 6cm, width 4cm, and height 2cm.

Solution: Volume = lwh = 6cm x 4cm x 2cm = 48 cm³ Therefore, the volume of the rectangular prism is 48 cm³. 2.4 Unit Conversions: Area: 1 cm = 10 mm, therefore 1 cm² = (10 mm)² = 100 mm²; 1 m = 100 cm, therefore 1 m² = (100 cm)² = 10 000 cm² Volume: 1 m = 100 cm, therefore 1 m³ = (100 cm)³ = 1 000 000 cm³; 1 litre = 1000 cm³, 1 m³ = 1000 litres Guided Practice (With Solutions)

Question 1: A garden consists of a rectangle measuring 7m by 4m with a square flower bed of side 2m inside the rectangle. What is the area of the garden excluding the flower bed?

Solution: Area of Rectangle = 7m x 4m = 28 m² Area of Square = 2m x 2m = 4 m² Area of Garden (excluding flower bed) = Area of Rectangle - Area of Square = 28 m² - 4 m² = 24 m² Therefore, the area of the garden excluding the flower bed is 24 m². The keyword 'excluding' means we need to subtract.

Question 2: Find the surface area of a rectangular prism with length 8cm, width 5cm, and height 3cm.

Solution: Surface Area = 2(lw + lh + wh) = 2((8cm x 5cm) + (8cm x 3cm) + (5cm x 3cm)) = 2(40 cm² + 24 cm² + 15 cm²) = 2(79 cm²) = 158 cm² Therefore, the surface area of the rectangular prism is 158 cm². Remember to use the correct formula.

Question 3: A water tank is in the shape of a rectangular prism. It is 2m long, 1.5m wide and 1m high. How many litres of water can it hold?

Solution: Volume = lwh = 2m x 1.5m x 1m = 3 m³ Since 1 m³ = 1000 litres, Volume = 3 m³ x 1000 litres/m³ = 3000 litres Therefore, the water tank can hold 3000 litres of water. Remember to convert cubic meters to litres.

Question 4: A square has an area of 144 cm². What is the length of one side of the square?

Solution: Area of Square = s² = 144 cm² Therefore, s = √144 cm² = 12 cm Therefore, the side length of the square is 12 cm.