Find the area of the figure.
The Correct Answer and Explanation is:
To find the area of the given figure, we need to break it into two simpler shapes:
- A rectangle in the middle.
- A circle, formed by combining the two semicircles on each end of the rectangle.
🔹 Step 1: Area of the Rectangle
The rectangle in the center has:
- Length = 12 ft
- Width = 3 ft
Area of rectangle=length×width=12×3=36 ft2\text{Area of rectangle} = \text{length} \times \text{width} = 12 \times 3 = 36 \text{ ft}^2Area of rectangle=length×width=12×3=36 ft2
🔹 Step 2: Area of the Circle
There are two semicircles on each end of the rectangle. Together, they make one full circle.
From the image:
- Each semicircle has a radius of 3 ft.
Area of full circle=πr2=π(3)2=9π≈28.27 ft2\text{Area of full circle} = \pi r^2 = \pi (3)^2 = 9\pi \approx 28.27 \text{ ft}^2Area of full circle=πr2=π(3)2=9π≈28.27 ft2
🔹 Step 3: Total Area
Now, add the area of the rectangle and the circle:Total Area=36+9π≈36+28.27=64.27 ft2\text{Total Area} = 36 + 9\pi \approx 36 + 28.27 = \boxed{64.27 \text{ ft}^2}Total Area=36+9π≈36+28.27=64.27 ft2
✅ Final Answer: 64.27 square feet
🔍 Explanation
The composite shape in the image combines a rectangle and two semicircles to form a more complex figure. To find the area accurately, we apply the strategy of breaking down the shape into simpler components.
The central rectangle is easy to calculate. From the image, its length runs horizontally and measures 12 feet. The height or width is 3 feet. By multiplying these two dimensions, we get the area of the rectangle as 12×3=3612 \times 3 = 3612×3=36 square feet.
The curved ends are not full circles but two half circles attached on each side. Because the diameter of each semicircle is 6 feet (the same as the height of the full shape), the radius is half of that, or 3 feet. Adding both halves together forms a full circle with radius 3 feet. Using the circle area formula πr2\pi r^2πr2, we substitute r=3r = 3r=3 to get π×9=9π≈28.27\pi \times 9 = 9\pi \approx 28.27π×9=9π≈28.27 square feet.
After calculating both parts—the rectangle and the circle—we add them together to find the full area of the shape: 36+28.27=64.2736 + 28.27 = 64.2736+28.27=64.27 square feet.
This process shows the importance of visual decomposition in geometry. Recognizing standard shapes (like rectangles and circles) within a compound figure simplifies the calculation and ensures precision.
