By Joshua Bush
Apr 22, 2023 1:03 pm EST
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When a rectangular yard contains a circular pool, the fertilizer required depends on the area that actually needs fertilizing. The shaded portion of the yard is the region outside the pool, and calculating its area involves subtracting the pool’s area from the yard’s total area. This article walks you through that process using clear, expert‑level explanations.
1. Identify the shapes. In most problems you’ll encounter basic polygons or circles. For this example, the yard is a rectangle and the pool is a circle.
2. Compute each area.
For a rectangle:
\(A_{\text{rectangle}} = l \times w\)
For a circle:
\(A_{\text{circle}} = \pi r^2\)
3. Subtract to find the shaded area. The shaded region’s area equals the rectangle’s area minus the circle’s area. This gives the exact acreage that must be fertilized.
4. Verify units. Ensure the result is expressed in square units—such as m², ft², or yd²—to confirm the calculation’s validity.
Shaded‑region problems combine basic shapes—circles, triangles, rectangles—into composite figures. Break complex shapes into familiar components, calculate each area, and subtract the interior shape(s) from the exterior shape to get the desired area.
High‑school math covers many regular polygons. Here are the essential formulas:
\(A_{\text{triangle}} = \tfrac{1}{2}\times\text{base}\times\text{height}\)
Use the perpendicular height relative to the chosen base. For right triangles, the legs themselves serve as base and height.
Since a square is a special rectangle, the same formula applies:
\(A_{\text{rectangle}} = \text{length}\times\text{width}\)
For a full circle:
\(A_{\text{circle}} = \pi r^2\)
When working with sectors or segments, multiply the full‑circle area by the ratio of the sector’s central angle to 360° (or 2π radians).
For more on sector area, see our in‑depth guide on circle geometry.
