Lesson Notes By Weeks and Term v5 - Grade 9

Lesson Video

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Performance objectives

• Calculate area and perimeter of composite shapes.
• Calculate surface area and volume of prisms and cylinders.
• Identify hypotenuse, opposite, and adjacent sides of a right-angled triangle.
• Define sine, cosine, and tangent ratios.
• Use trig ratios to find unknown sides and angles.

Lesson summary

This week, we delve into the fascinating world of measurement and trigonometry. We will be focusing on applying our knowledge of area, perimeter, volume, and surface area to more complex shapes, and we'll be introduced to the basics of trigonometry, specifically focusing on right-angled triangles. These skills are crucial in various fields, from construction and architecture to navigation and even everyday problem-solving. Imagine planning a braai area, calculating the amount of paint needed for a wall, or figuring out the angle of a ramp for accessibility – all these require an understanding of measurement and, in some cases, trigonometry.

Lesson notes

2.1 Area and Perimeter of Composite Shapes Composite shapes are shapes made up of two or more basic shapes like rectangles, squares, triangles, and circles. To find the area or perimeter of a composite shape, we need to break it down into its constituent shapes and then calculate the area or perimeter of each part.

Area: The area of the composite shape is the sum of the areas of its individual parts.

Perimeter: The perimeter is the total length of the boundary of the shape. Be careful not to include internal lines!

Example 1: Consider a shape formed by a rectangle (length = 8cm, breadth = 4cm) with a semi-circle on top of it (diameter = 4cm).

Area: Area of rectangle = length × breadth = 8cm × 4cm = 32 cm² Area of semi-circle = (1/2) × πr², where r = radius = diameter/2 = 4cm/2 = 2cm Area of semi-circle = (1/2) × π × (2cm)² = (1/2) × π × 4 cm² ≈ 6.28 cm² Total area = Area of rectangle + Area of semi-circle = 32 cm² + 6.28 cm² = 38.28 cm² Perimeter: Perimeter of rectangle (excluding top side) = 8cm + 4cm + 8cm = 20 cm Circumference of semi-circle = (1/2) × 2πr = πr = π × 2cm ≈ 6.28 cm Total perimeter = Perimeter of rectangle (excluding top side) + Circumference of semi-circle = 20cm + 6.28cm = 26.28 cm Example 2: A square (side length 6m) has a right-angled triangle cut out from one corner. The base of the triangle is 2m, and the height is 3m. Find the area. Area of square = side × side = 6m × 6m = 36 m² Area of triangle = (1/2) × base × height = (1/2) × 2m × 3m = 3 m² Area of composite shape = Area of square – Area of triangle = 36 m² – 3 m² = 33 m² 2.2 Surface Area and Volume of Prisms and Cylinders Prism: A prism is a 3D shape with two identical ends (bases) and flat rectangular sides.

The volume of a prism is found by: Volume = Area of base × height Surface Area = 2 × (Area of base) + (Perimeter of base) × height Cylinder: A cylinder is a 3D shape with two identical circular ends and a curved surface. Volume = Area of base × height = πr²h (where r is the radius and h is the height) Surface Area = 2 × (Area of base) + (Circumference of base) × height = 2πr² + 2πrh Example 3: A rectangular prism has a length of 10cm, a width of 5cm, and a height of 3cm. Find its volume and surface area. Volume = length × width × height = 10cm × 5cm × 3cm = 150 cm³ Surface Area = 2(length × width + length × height + width × height) = 2(10cm × 5cm + 10cm × 3cm + 5cm × 3cm) = 2(50 cm² + 30 cm² + 15 cm²) = 2(95 cm²) = 190 cm² Example 4: A cylinder has a radius of 4cm and a height of 7cm. Find its volume and surface area. Volume = πr²h = π × (4cm)² × 7cm = π × 16 cm² × 7cm ≈ 351.86 cm³ Surface Area = 2πr² + 2πrh = 2π × (4cm)² + 2π × 4cm × 7cm = 2π × 16 cm² + 2π × 28 cm² ≈ 100.53 cm² + 175.93 cm² ≈ 276.46 cm² 2.3 Introduction to Trigonometry: Right-Angled Triangles Trigonometry deals with the relationships between the sides and angles of triangles, particularly right-angled triangles.

Right-Angled Triangle: A triangle with one angle equal to 90 degrees.

Hypotenuse: The side opposite the right angle (the longest side).

Opposite: The side opposite to the angle we are considering (other than the right angle).

Adjacent: The side next to the angle we are considering (other than the hypotenuse). Imagine you're standing on a hill. The 'angle of elevation' is the angle between the horizontal line of sight and your line of sight upwards to the top of a building. Similarly, the 'angle of depression' is the angle between the horizontal line of sight and your line of sight downwards to a point on the ground. 2.4 Trigonometric Ratios: Sine, Cosine, and Tangent For an acute angle (less than 90 degrees) in a right-angled triangle, we define three basic trigonometric ratios: Sine (sin): sin(angle) = Opposite / Hypotenuse Cosine (cos): cos(angle) = Adjacent / Hypotenuse Tangent (tan): tan(angle) = Opposite / Adjacent A useful mnemonic to remember these ratios is SOH CAH TOA: Sine = Opposite / Hypotenuse Cosine = Adjacent / Hypotenuse Tangent = Opposite / Adjacent Example 5: In a right-angled triangle ABC, where angle B = 90°, angle A = 30°, and the hypotenuse AC = 10cm, find the length of the side opposite angle A (BC). We know sin(A) = Opposite / Hypotenuse Therefore, sin(30°) = BC / 10cm sin(30°) = 0.5 (This value is often known or can be found using a calculator) 5 = BC / 10cm BC = 0.5 × 10cm = 5cm Example 6: In a right-angled triangle PQR, where angle Q = 90°, PQ = 8cm, and QR = 6cm, find the angle at P. We know tan(P) = Opposite / Adjacent = QR / PQ = 6cm / 8cm = 0.75 To find angle P, we use the inverse tangent function (arctan or tan⁻¹) on our calculator: P = tan⁻¹(0.75) ≈ 36.87° Guided Practice (With Solutions)

Question 1: A rectangular garden is 12m long and 8m wide. A circular flower bed with a diameter of 4m is in the center of the garden. What is the area of the garden that is not covered by the flower bed?