This lesson focuses on finding the perimeter of complex shapes, especially those involving parts of a circle.

1. Core Concepts (Short Notes)

• Perimeter: The total distance around the outer boundary of a plane figure.

• Sector of a Circle: A "slice" of a circle bounded by two radii and an arc. Think of it like a slice of pizza.

• Arc Length: The length of the curved part of a sector's boundary.

• Perimeter of a Sector: The arc length plus the lengths of the two straight sides (the two radii).

• Composite Figures: Shapes made by combining simpler shapes (e.g., a rectangle with a semi-circle attached). The perimeter is the total length of the outer boundary only.

2. Key Formulas to Memorize

Revision Formulas:

• Rectangle: Perimeter = 2 * (length + breadth)

• Square: Perimeter = 4 * length of a side

• Circle (Circumference): C = 2πr

New & Essential Formulas:

• Arc Length of a Sector:

  • Arc Length = (θ / 360) * 2πr

  • Where:

    • θ (theta) = the angle at the center of the sector (in degrees).

    • r = the radius of the circle.

    • π (pi) = usually 22/7 for exams, unless stated otherwise.

• Perimeter of a Sector:

  • Perimeter = Arc Length + 2r

  • Perimeter = ((θ / 360) * 2πr) + 2r

3. Tips & Tricks for Exams

• Use π = 22/7: In Sri Lankan G.C.E. O/L exams, you should almost always use π = 22/7. Using 3.14 might lead to a different answer and loss of marks.

• Identify the Outer Boundary: For composite shapes, trace the outer edge with your finger. Do not include the lengths of lines where two shapes are joined together. This is the most common mistake!

• Break It Down: When you see a complex shape, break it into simple parts: "This shape has one straight side, another straight side, and one semi-circular arc." Calculate the length of each part separately and then add them all up at the end.

• Semi-Circles & Quarter-Circles:

  • A semi-circle is a sector with θ = 180°. The arc length formula simplifies to πr.

  • A quarter-circle is a sector with θ = 90°. The arc length formula simplifies to (1/2)πr.

  • Memorizing these shortcuts can save valuable time in an exam.

• Check Your Units: Make sure all your measurements are in the same unit (e.g., cm) before you start calculating. Write the correct unit in your final answer (e.g., cm, m).

4. Important Points to Remember

• Read the Question Carefully: Does it ask for the arc length or the perimeter of the sector? These are different. The perimeter includes the two straight sides (radii).

• Look for Hidden Information: In composite figures, the side of a square or rectangle might also be the diameter or radius of an attached circular part.

• Pythagoras May Appear: In some complex shapes, you might need to use Pythagoras' Theorem (a² + b² = c²) to find the length of a slanted straight side.

Show Your Steps: Write down the formula you are using, show the values you are substituting, and then write the final answer. This helps you get partial marks even if there's a small calculation error.