I will solve this problem by calculating the area of each shaded region in the three figures provided. The task is to find the area of the shaded parts in a square with a circle, a rectangle with a triangle, and a ring (annulus).
I'll tackle each figure one by one.
Figure 1: Square with an Inscribed Circle
First, I need to calculate the area of the square and subtract the area of the circle to find the shaded region.
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Square Area: The side of the square is 8 feet.
- Area = side² = 8 ft × 8 ft = 64 sq ft.
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Circle Area: The circle has a radius of 4 feet.
- Area = π × radius² = π × (4 ft)² = 16π sq ft.
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Shaded Area: Area of square - Area of circle = 64 - 16π sq ft.
Figure 2: Rectangle with a Triangle Cut Out
Next, I'll calculate the area of the rectangle and subtract the area of the white triangle.
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Rectangle Area: The dimensions are 12 cm by 4 cm.
- Area = length × width = 12 cm × 4 cm = 48 sq cm.
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Triangle Area: The base of the triangle is 9 cm and the height is 4 cm.
- Area = (1/2) × base × height = (1/2) × 9 cm × 4 cm = 18 sq cm.
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Shaded Area: Area of rectangle - Area of triangle = 48 - 18 = 30 sq cm.
Figure 3: Annulus (Ring)
Finally, I'll calculate the area of the ring by subtracting the area of the smaller circle from the area of the larger circle.
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Outer Circle Area: The radius is 50 ft.
- Area = π × (50 ft)² = 2500π sq ft.
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Inner Circle Area: The radius is 36 ft.
- Area = π × (36 ft)² = 1296π sq ft.
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Shaded Area (Annulus): Area of outer circle - Area of inner circle = 2500π - 1296π = 1204π sq ft.
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Final Answer:
The areas of the shaded regions are as follows:
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Figure 1 (Square with Circle): $64 - 16\pi$ square feet.
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Figure 2 (Rectangle with Triangle): $30$ square centimeters.
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Figure 3 (Annulus/Ring): $1204\pi$ square feet.
Parent Tip: Review the logic above to help your child master the concept of find the area of the shaded region worksheet.
