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Calculate the areas of the shapes.

Mathematics

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2 parts
a

22.5 mm2\text{22.5 mm}^2

b

54 m2\text{54 m}^2

Step-by-step solution

8 steps

Here are the areas of the shapes:

Part a· 4 steps

  1. Decompose the shape into a rectangle and a triangle.

    The rectangle has a width of 4 mm4 \text{ mm} and a height of 3 mm3 \text{ mm}.
    The triangle has a base of 11 mm−4 mm=7 mm11 \text{ mm} - 4 \text{ mm} = 7 \text{ mm} and a height of 3 mm3 \text{ mm}.

  2. Calculate the area of the rectangle.

    Arectangle=width×height=4 mm×3 mm=12 mm2A_{\text{rectangle}} = \text{width} \times \text{height} = 4 \text{ mm} \times 3 \text{ mm} = 12 \text{ mm}^2

  3. Calculate the area of the triangle.

    Atriangle=12×base×height=12×7 mm×3 mm=212 mm2=10.5 mm2A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 7 \text{ mm} \times 3 \text{ mm} = \frac{21}{2} \text{ mm}^2 = 10.5 \text{ mm}^2

  4. Calculate the total area of shape (a).

    A(a)=Arectangle+Atriangle=12 mm2+10.5 mm2=22.5 mm2A_{(a)} = A_{\text{rectangle}} + A_{\text{triangle}} = 12 \text{ mm}^2 + 10.5 \text{ mm}^2 = \boxed{\text{22.5 mm}^2}

Part b

54 m2\text{54 m}^2

  1. Decompose the shape into a large trapezoid and a small rectangular cut-out.

    The large trapezoid has parallel sides of 7 m7 \text{ m} and 17 m17 \text{ m}, and a height of 5 m5 \text{ m}.
    The cut-out rectangle has a width of 2 m2 \text{ m} and a height of 3 m3 \text{ m}.

  2. Calculate the area of the large trapezoid.

    Atrapezoid=12×(sum of parallel sides)×heightA_{\text{trapezoid}} = \frac{1}{2} \times (\text{sum of parallel sides}) \times \text{height} Atrapezoid=12×(7 m+17 m)×5 mA_{\text{trapezoid}} = \frac{1}{2} \times (7 \text{ m} + 17 \text{ m}) \times 5 \text{ m} Atrapezoid=12×24 m×5 m=12 m×5 m=60 m2A_{\text{trapezoid}} = \frac{1}{2} \times 24 \text{ m} \times 5 \text{ m} = 12 \text{ m} \times 5 \text{ m} = 60 \text{ m}^2

  3. Calculate the area of the cut-out rectangle.

    Acut-out=width×height=2 m×3 m=6 m2A_{\text{cut-out}} = \text{width} \times \text{height} = 2 \text{ m} \times 3 \text{ m} = 6 \text{ m}^2

  4. Calculate the total area of shape (b).

    A(b)=Atrapezoid−Acut-outA_{(b)} = A_{\text{trapezoid}} - A_{\text{cut-out}} A(b)=60 m2−6 m2=54 m2A_{(b)} = 60 \text{ m}^2 - 6 \text{ m}^2 = \boxed{\text{54 m}^2}
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