Question

a sidewalk in the shape of two triangles, a rectangle, and a square was built around the edge of a building as shown.what is the area of the sidewalk in square feet?a $108\\ \text{ft}^2$$162\\ \text{ft}^2$$144\\ \text{ft}^2$$180\\ \text{ft}^2$

Explanation:

Step1: Calculate square area

$A_{square} = 6 \times 6 = 36$ $\text{ft}^2$

Step2: Calculate rectangle area

First find rectangle length: $30 - 6 = 24$ ft
$A_{rectangle} = 24 \times 6 = 144$ $\text{ft}^2$

Step3: Calculate two triangles area

Combine triangles to 1 trapezoid:
$A_{trapezoid} = \frac{1}{2} \times 6 \times (18 + 24) = 126$ $\text{ft}^2$

Step4: Sum all areas

$A_{total} = 36 + 144 + 126 = 306$
Wait, correction: Reinterpret shape: The top trapezoid is $\frac{1}{2} \times 6 \times (18+30)=144$, bottom rectangle/square: $6 \times 18=108$. Total $144+108=252$? No, correct breakdown: The figure is a composite: left square ($6×6$), middle rectangle ($18×6$), right two triangles (base $30-18=12$, height 6: area $\frac{1}{2}×12×6=36$), top trapezoid? No, re-read: sidewalk is 2 triangles, rectangle, square.
Square: $6×6=36$
Rectangle: $18×6=108$
Two triangles: each has base $\frac{30-18}{2}=6$, height 6. Area per triangle: $\frac{1}{2}×6×6=18$, two triangles: $36$
Top trapezoid? No, the vertical 6 is square height, top dashed 6 is triangle height.
Total area: $36 + 108 + 36 = 180$

Answer:

180 $\text{ft}^2$ (matches the option: 180 ft²)