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Calculate the area of the composite shapes shown below. Round your answers to two decimal places.

Mathematics

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Here are the areas for each figure:

1)

  1. Divide the shape into a rectangle and a semicircle.

    The rectangle has a width of 8 ft8 \text{ ft} and a height of 12 ft12 \text{ ft}.
    The semicircle has a diameter of 8 ft8 \text{ ft}, so its radius is r=82=4 ftr = \frac{8}{2} = 4 \text{ ft}.

  2. Calculate the area of the rectangle.

    Arectangle=length×width=12 ft×8 ft=96 ft2A_{\text{rectangle}} = \text{length} \times \text{width} = 12 \text{ ft} \times 8 \text{ ft} = 96 \text{ ft}^2

  3. Calculate the area of the semicircle.

    The area of a full circle is πr2\pi r^2. The area of a semicircle is 12πr2\frac{1}{2} \pi r^2.

    Asemicircle=12π(4 ft)2=12π(16 ft2)=8π ft2≈8×3.14159 ft2≈25.1327 ft2A_{\text{semicircle}} = \frac{1}{2} \pi (4 \text{ ft})^2 = \frac{1}{2} \pi (16 \text{ ft}^2) = 8\pi \text{ ft}^2 \approx 8 \times 3.14159 \text{ ft}^2 \approx 25.1327 \text{ ft}^2

  4. Add the areas to find the total area.

    Atotal=Arectangle+Asemicircle=96 ft2+25.1327 ft2=121.1327 ft2A_{\text{total}} = A_{\text{rectangle}} + A_{\text{semicircle}} = 96 \text{ ft}^2 + 25.1327 \text{ ft}^2 = 121.1327 \text{ ft}^2

  5. Round to two decimal places.

    Atotal≈121.13 ft2A_{\text{total}} \approx 121.13 \text{ ft}^2

    The area is 121.13 ft2\boxed{\text{121.13 ft}^2}.

    2)

  6. Divide the shape into a rectangle and a triangle.

    The rectangle has a length of 13 in13 \text{ in} and a width of 8 in8 \text{ in}.
    The total length of the figure is 19 in19 \text{ in}. The length of the rectangular part is 13 in13 \text{ in}.
    The base of the triangle is 8 in8 \text{ in} (same as the width of the rectangle).
    The height of the triangle is 19 in−13 in=6 in19 \text{ in} - 13 \text{ in} = 6 \text{ in}.

  7. Calculate the area of the rectangle.

    Arectangle=length×width=13 in×8 in=104 in2A_{\text{rectangle}} = \text{length} \times \text{width} = 13 \text{ in} \times 8 \text{ in} = 104 \text{ in}^2

  8. Calculate the area of the triangle.

    Atriangle=12×base×height=12×8 in×6 in=24 in2A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \text{ in} \times 6 \text{ in} = 24 \text{ in}^2

  9. Add the areas to find the total area.

    Atotal=Arectangle+Atriangle=104 in2+24 in2=128 in2A_{\text{total}} = A_{\text{rectangle}} + A_{\text{triangle}} = 104 \text{ in}^2 + 24 \text{ in}^2 = 128 \text{ in}^2

    The area is 128.00 in2\boxed{\text{128.00 in}^2}.

    3)

  10. Divide the T-shaped figure into two rectangles.

    Rectangle 1 (top horizontal part): length 11 cm11 \text{ cm}, height 4 cm4 \text{ cm}.
    Rectangle 2 (bottom vertical part): length 5 cm5 \text{ cm}, height 5 cm5 \text{ cm}.

  11. Calculate the area of Rectangle 1.

    Arectangle 1=11 cm×4 cm=44 cm2A_{\text{rectangle 1}} = 11 \text{ cm} \times 4 \text{ cm} = 44 \text{ cm}^2

  12. Calculate the area of Rectangle 2.

    Arectangle 2=5 cm×5 cm=25 cm2A_{\text{rectangle 2}} = 5 \text{ cm} \times 5 \text{ cm} = 25 \text{ cm}^2

  13. Add the areas to find the total area.

    Atotal=Arectangle 1+Arectangle 2=44 cm2+25 cm2=69 cm2A_{\text{total}} = A_{\text{rectangle 1}} + A_{\text{rectangle 2}} = 44 \text{ cm}^2 + 25 \text{ cm}^2 = 69 \text{ cm}^2

    The area is 69.00 cm2\boxed{\text{69.00 cm}^2}.

    4)

  14. Divide the shape into a semicircle and a triangle.

    The semicircle has a diameter of 8 ft8 \text{ ft}, so its radius is r=82=4 ftr = \frac{8}{2} = 4 \text{ ft}.
    The triangle has a base of 8 ft8 \text{ ft} and a height of 7 ft7 \text{ ft}.

  15. Calculate the area of the semicircle.

    Asemicircle=12πr2=12π(4 ft)2=12π(16 ft2)=8π ft2≈25.1327 ft2A_{\text{semicircle}} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (4 \text{ ft})^2 = \frac{1}{2} \pi (16 \text{ ft}^2) = 8\pi \text{ ft}^2 \approx 25.1327 \text{ ft}^2

  16. Calculate the area of the triangle.

    Atriangle=12×base×height=12×8 ft×7 ft=28 ft2A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \text{ ft} \times 7 \text{ ft} = 28 \text{ ft}^2

  17. Add the areas to find the total area.

    Atotal=Asemicircle+Atriangle=25.1327 ft2+28 ft2=53.1327 ft2A_{\text{total}} = A_{\text{semicircle}} + A_{\text{triangle}} = 25.1327 \text{ ft}^2 + 28 \text{ ft}^2 = 53.1327 \text{ ft}^2

  18. Round to two decimal places.

    Atotal≈53.13 ft2A_{\text{total}} \approx 53.13 \text{ ft}^2

    The area is 53.13 ft2\boxed{\text{53.13 ft}^2}.

    5)

  19. Divide the shape into a rectangle and a triangle.

    The rectangle has a length of 7 m7 \text{ m} and a width of 7 m7 \text{ m}.
    The total length of the figure is 11 m11 \text{ m}. The length of the rectangular part is 7 m7 \text{ m}.
    The base of the triangle is 7 m7 \text{ m} (same as the width of the rectangle).
    The height of the triangle is 11 m−7 m=4 m11 \text{ m} - 7 \text{ m} = 4 \text{ m}.

  20. Calculate the area of the rectangle.

    Arectangle=length×width=7 m×7 m=49 m2A_{\text{rectangle}} = \text{length} \times \text{width} = 7 \text{ m} \times 7 \text{ m} = 49 \text{ m}^2

  21. Calculate the area of the triangle.

    Atriangle=12×base×height=12×7 m×4 m=14 m2A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 7 \text{ m} \times 4 \text{ m} = 14 \text{ m}^2

  22. Add the areas to find the total area.

    Atotal=Arectangle+Atriangle=49 m2+14 m2=63 m2A_{\text{total}} = A_{\text{rectangle}} + A_{\text{triangle}} = 49 \text{ m}^2 + 14 \text{ m}^2 = 63 \text{ m}^2

    The area is 63.00 m2\boxed{\text{63.00 m}^2}.

    6)

  23. Divide the L-shaped figure into two rectangles.

    Rectangle 1 (bottom right): length 12 yd12 \text{ yd}, height 6 yd6 \text{ yd}.
    Rectangle 2 (top left): length 6 yd6 \text{ yd}, height 6 yd6 \text{ yd} (total height on the left is 6+6=126+6=12 yd, and the right side is 6 yd, so the top left part has a height of 12−6=612-6=6 yd).

  24. Calculate the area of Rectangle 1.

    Arectangle 1=12 yd×6 yd=72 yd2A_{\text{rectangle 1}} = 12 \text{ yd} \times 6 \text{ yd} = 72 \text{ yd}^2

  25. Calculate the area of Rectangle 2.

    Arectangle 2=6 yd×6 yd=36 yd2A_{\text{rectangle 2}} = 6 \text{ yd} \times 6 \text{ yd} = 36 \text{ yd}^2

  26. Add the areas to find the total area.

    Atotal=Arectangle 1+Arectangle 2=72 yd2+36 yd2=108 yd2A_{\text{total}} = A_{\text{rectangle 1}} + A_{\text{rectangle 2}} = 72 \text{ yd}^2 + 36 \text{ yd}^2 = 108 \text{ yd}^2

    The area is 108.00 yd2\boxed{\text{108.00 yd}^2}.

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