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area of a shaded region — Definition, Formula & Examples

Area of a shaded region is the area of a specific portion of a figure, found by subtracting the area of the unshaded part from the area of the whole shape.

Given a composite figure where one region is visually distinguished (shaded), the area of the shaded region equals the area of the outer (larger) figure minus the total area of any inner (unshaded) figures removed from it.

Key Formula

Ashaded=Aouter−AinnerA_{\text{shaded}} = A_{\text{outer}} - A_{\text{inner}}
Where:
  • AshadedA_{\text{shaded}} = Area of the shaded region
  • AouterA_{\text{outer}} = Area of the larger (outer) shape
  • AinnerA_{\text{inner}} = Total area of all unshaded (inner) shapes removed

How It Works

Start by identifying every shape in the figure — both the outer boundary and any inner shapes that are cut out or left unshaded. Calculate the area of the entire outer shape. Then calculate the area of each unshaded inner shape. Finally, subtract the total unshaded area from the outer area. The result is the shaded region's area.

Worked Example

Problem: A square has side length 10 cm. A circle with radius 3 cm is cut out of the center. The remaining region is shaded. Find the shaded area.
Find the outer area: The outer shape is a square with side 10 cm.
Asquare=102=100 cm2A_{\text{square}} = 10^2 = 100 \text{ cm}^2
Find the inner area: The inner shape is a circle with radius 3 cm.
Acircle=π(3)2=9π≈28.27 cm2A_{\text{circle}} = \pi(3)^2 = 9\pi \approx 28.27 \text{ cm}^2
Subtract: Subtract the circle's area from the square's area.
Ashaded=100−9π≈71.73 cm2A_{\text{shaded}} = 100 - 9\pi \approx 71.73 \text{ cm}^2
Answer: The shaded area is 100−9π≈71.73100 - 9\pi \approx 71.73 cm².

Why It Matters

Shaded-region problems appear frequently on standardized tests and in geometry courses. They train you to decompose complex figures into simpler shapes — a skill used in architecture, engineering, and any field that involves calculating material or surface coverage.

Common Mistakes

Mistake: Forgetting to subtract all inner shapes when more than one is removed.
Correction: Count every unshaded region inside the figure. Add their areas together, then subtract the total from the outer area.

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