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Door 14 · Grade 8

📏 Perimeter & Area

Rectangle area: \(A = l × w\) | Perimeter: \(P = 2(l + w)\)

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1: Area of Rectangles, Squares, Triangles, Parallelograms

What is Perimeter and Area?

• Perimeter is the total distance around the outside of a figure. Think of it as the length of fence needed to enclose a garden.
• Area is the amount of surface inside a figure. Think of it as the number of tiles needed to cover a floor.

Area Formulas for Basic Shapes:

ShapeFormulaVariablesRectangle\(A = l × w\)\(l\) = length, \(w\) = widthSquare\(A = s²\)\(s\) = side lengthTriangle\(A = 1/2 × b × h\)\(b\) = base, \(h\) = heightParallelogram\(A = b × h\)\(b\) = base, \(h\) = height (perpendicular)
Perimeter Formulas:

ShapeFormulaRectangle\(P = 2(l + w)\)Square\(P = 4s\)Triangle\(P = a + b + c\) (sum of all sides)Parallelogram\(P = 2(a + b)\) (where \(a\) and \(b\) are adjacent sides)
Important Notes:

• In a triangle, height must be perpendicular to the base (forms a right angle)
• In a parallelogram, the height is the perpendicular…

Worked example

Problem: Example 1: Find the area and perimeter of a rectangle with length 12 cm and width 7 cm.

  1. - Step 1: Area of rectangle = \(l × w\)
  2. - Step 2: \(A = 12 × 7 = 84\) cm²
  3. - Step 3: Perimeter of rectangle = \(2(l + w)\)
  4. - Step 4: \(P = 2(12 + 7) = 2 × 19 = 38\) cm

Answer: - Step 4: \(P = 2(12 + 7) = 2 × 19 = 38\) cm

2: Area of Trapezium and Perimeter Calculations

What is a Trapezium (Trapezoid)?

A trapezium (also called trapezoid) is a quadrilateral with one pair of parallel sides. The parallel sides are called bases (\(a\) and \(b\)), and the perpendicular distance between them is the height (\(h\)).

Area of a Trapezium Formula:

\(A = 1/2 × (a + b) × h\)

Where:

• \(a\) and \(b\) = lengths of the parallel sides (bases)
• \(h\) = perpendicular height (distance between the bases)

Why this formula works: A trapezium can be divided into two triangles or transformed into a parallelogram to derive the formula.

Perimeter of a Trapezium:

\(P = a + b + c + d\) (sum of all four sides)

Perimeter of Other Shapes (Review):

ShapePerimeter FormulaRectangle\(P = 2(l + w)\)Square\(P = 4s\)Triangle\(P = a + b + c\)Parallelogram\(P = 2(a + b)\)Trapezium\(P = a + b + c + d\)
Special Case - Isosceles Trapezium: Non-parallel sides are…

Worked example

Problem: Example 1: Find the area of a trapezium with parallel sides 10 cm and 14 cm, and height 6 cm.

  1. - Step 1: Identify \(a = 10\) cm, \(b = 14\) cm, \(h = 6\) cm
  2. - Step 2: Use formula \(A = 1/2 × (a + b) × h\)
  3. - Step 3: \(A = 1/2 × (10 + 14) × 6\)
  4. - Step 4: \(A = 1/2 × 24 × 6\)
  5. - Step 5: \(A = 12 × 6 = 72\) cm²

Answer: - Step 5: \(A = 12 × 6 = 72\) cm²

3: Composite Figures and Mixed Problems

What are Composite Figures?

Composite figures (or compound shapes) are shapes made up of two or more basic geometric shapes (rectangles, squares, triangles, trapeziums, etc.) combined together.

Strategy for Finding Area of Composite Figures:

StepAction1Divide the composite figure into simpler shapes2Calculate the area of each simple shape separately3Add areas for shapes that are combined4Subtract areas for holes/shapes that are cut out5Write final answer with correct units
Strategy for Finding Perimeter of Composite Figures:

• Trace the outer boundary of the figure carefully
• Add the lengths of all outer sides
• Do NOT include internal lines where shapes join

Common Composite Figure Types:

TypeApproachL-shape (two rectangles)Split into two rectangles, add areasRectangle with triangle on topArea of rectangle + area of triangleCircle cut out of rectangleArea of…

Worked example

Problem: Example 1: Find the area of an L-shaped figure formed by two rectangles: one rectangle is 12 cm by 5 cm, the other is 8 cm by 4 cm attached to its side. (Assume they share a 5 cm side)

  1. - Step 1: Draw and label: Rectangle A: 12 × 5, Rectangle B: 8 × 4 attached along a 5 cm side
  2. - Step 2: Area of Rectangle A = \(12 × 5 = 60\) cm²
  3. - Step 3: Area of Rectangle B = \(8 × 4 = 32\) cm²
  4. - Step 4: Total area = \(60 + 32 = 92\) cm²
  5. - Step 5: For perimeter, trace outer boundary carefully

Answer: - Step 5: For perimeter, trace outer boundary carefully