This lesson builds on your knowledge of basic shapes by introducing the area of a sector of a circle. You will learn the formula for calculating this area and apply it to find the area of complex shapes (compound figures) made from sectors and other figures like squares and rectangles.

1. Core Concepts (Short Notes)

• Sector of a Circle: A "slice" of a circle, bounded by two radii and the arc connecting them. Its size is determined by the angle at the centre.

• Area of a Sector: It is a fraction of the full circle's area. The angle θ (theta) at the centre tells you what that fraction is.

• Compound Figure: A shape created by joining basic geometric figures (like a square with a semi-circle attached) or by cutting one shape out of another (like a quarter-circle removed from a rectangle).

2. Key Formulas You MUST Memorize

1. Area of a full Circle: A = πr²

   • r is the radius.

   • In Sri Lankan exams, π is almost always given as 22/7 to make calculations with multiples of 7 easier.

2. Area of a Sector: A = (θ / 360) × πr²

   • θ is the angle at the centre of the sector, in degrees.

   • This formula simply means: (Fraction of the circle) × (Area of the full circle).

3. Exam Tips & Tricks

Tip 1: Look for Shortcut Angles!

Examiners frequently use angles that are simple fractions of 360°. Recognizing these saves a lot of time.

• θ = 180° (Semi-circle): Fraction is 180/360 = 1/2. Area = (1/2)πr²

• θ = 90° (Quarter-circle): Fraction is 90/360 = 1/4. Area = (1/4)πr²

• θ = 120°: Fraction is 120/360 = 1/3. Area = (1/3)πr²

• θ = 60°: Fraction is 60/360 = 1/6. Area = (1/6)πr²

• θ = 45°: Fraction is 45/360 = 1/8. Area = (1/8)πr²

Tip 2: Strategy for Compound Shapes: "ADD or SUBTRACT?"

1. Identify the basic shapes that make up the figure (e.g., a rectangle and two semi-circles).

2. Decide the operation:

   • If the final shape is made by joining figures, you ADD their individual areas.

   • If the final shape is made by removing a smaller piece from a larger one, you SUBTRACT the smaller area from the larger area.

3. Calculate each area separately before you add or subtract.

Worked Example (Subtraction): Question: A quarter-circle of radius 7 cm is cut from a rectangle of 20 cm by 7 cm. Find the area of the remaining (shaded) portion.

• Step 1: Area of the large shape (Rectangle) = length × breadth = 20 × 7 = 140 cm².

• Step 2: Area of the piece cut out (Quarter-circle) = (1/4) × πr² = (1/4) × (22/7) × 7 × 7 = 38.5 cm².

• Step 3: Remaining Area = Area of Rectangle - Area of Quarter-circle = 140 - 38.5 = 101.5 cm².

Tip 3: Working Backwards

If a question gives you the area and asks for the radius (r) or angle (θ), you must rearrange the formula.

• To find r: r² = (Area × 360) / (θ × π)

• To find θ: θ = (Area × 360) / (πr²)

4. Important Points to Remember

• Radius vs. Diameter: The formula requires the radius (r). Exam questions often give the diameter. Your first step should always be to divide the diameter by 2.

• Units: Area is always measured in square units (e.g., cm², m²). Don't forget to include them in your final answer.

• Show Your Method: Always write the formula you are using first, then substitute the values. You can earn marks for the correct method even if there's a small mistake in your final calculation.