Calculating Area

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1. What is Area?

• Definition: The amount of space inside a shape.
• Units: Measured in square units (e.g., cm², m², in²).
• Applications: Useful for practical tasks like painting walls, tiling floors, or landscaping lawns.

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2. Calculating Area Using Grids

• Grid Method: Count the number of squares inside a shape.
• Each grid square must represent a known size (e.g., 1 cm × 1 cm = 1 cm²).
• Example:
• A square with 16 grid squares = (16 \, {cm}^2).
• A shape with 25 full squares, 10 half squares, and 1 quarter square = (25 + 5 + 0.25 = 30.25 \, {cm}^2).
• Challenges: Works best for simpler shapes; less accurate for irregular or complex ones.

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3. Areas of Quadrilaterals

Squares and Rectangles

• Formula:
• Rectangle: ( {Area} = {height} * {width} ).
• Square: ( {Area} = {side}^2 ).
• Example:
• Rectangle: ( 5 \, {m} * 3 \, {m} = 15 \, {m}^2 ).
• Square: ( 4 \, {cm} * 4 \, {cm} = 16 \, {cm}^2 ).

Complex Shapes

• Split Method: Divide irregular shapes into smaller rectangles or squares, calculate their areas, and sum up the total.
• Border Problems:
• Find the total area of the larger shape.
• Subtract the area of the inner shape.

Parallelograms

• Formula: ( {Area} = {base} * {height} ).
• Height: The perpendicular distance between parallel sides (not the side length).

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4. Area of Triangles

• Formula: ( {Area} = \frac{{base} * {height}}{2} ).
• Example:
• A triangle with base (6 \, {m}) and height (4 \, {m}):
  ( \frac{6 * 4}{2} = 12 \, {m}^2 ).

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5. Areas of Circles

• Key Measurements:
• Radius (R): Distance from the center to the edge.
• Diameter (D): Distance across the circle through the center ((D = 2R)).
• Formula: ( {Area} = \pi R^2 ), where ( \pi = 3.142 ).
• Examples:
• Circle with ( R = 5 \, {cm} ):
  ( 3.142 * 5 * 5 = 78.55 \, {cm}^2 ).
• Circle with ( D = 3 \, {m} ):
  ( R = 1.5 \, {m} ), so ( 3.142 * 1.5 * 1.5 = 7.07 \, {m}^2 ).

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6. Real-Life Applications

Example: Painting a Gable End?

1. Split Shapes: Divide the wall into a rectangle and triangle.
2. Calculate Areas:
3. Rectangle: ( {Base} * {Height} ).
4. Triangle: ( \frac{{Base} * {Height}}{2} ).
5. Add Areas: Sum the rectangle and triangle areas.
6. Adjust for Coverage: Account for paint coverage and rounding.

Example: Complex House Front

• Break the shape into a rectangle and triangle.
• Subtract areas of windows and doors from the total area.

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Tips for Calculating Area

• Always use the correct units for measurements and results.
• Simplify complex shapes into manageable parts.
• Double-check formulas and calculations for accuracy.

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