Area Irregular Polygon 3F35C5

1. **State the problem:** We need to find the area of an irregular polygon with given side lengths and a semicircular arc on the left side. 2. **Analyze the figure:** The figure consists of a large semicircle on the left with radius $5$ cm (since the vertical side adjacent to the arc is $5$ cm), and a step-like polygon on the right. 3. **Calculate the area of the semicircle:** $$\text{Area}_{semicircle} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (5)^2 = \frac{25\pi}{2} \approx 39.27 \text{ cm}^2$$ 4. **Calculate the area of the step-like polygon:** Break the polygon into rectangles and squares using the given side lengths. - Bottom rectangle: width $8$ cm, height $2$ cm, area = $8 \times 2 = 16$ cm$^2$ - Next step up: width $4$ cm, height $3$ cm, area = $4 \times 3 = 12$ cm$^2$ - Next step up: width $3$ cm, height $3$ cm, area = $3 \times 3 = 9$ cm$^2$ - Top rectangle: width $3$ cm, height $3$ cm, area = $3 \times 3 = 9$ cm$^2$ 5. **Sum the areas of the steps:** $$16 + 12 + 9 + 9 = 46 \text{ cm}^2$$ 6. **Total area:** $$\text{Area}_{total} = \text{Area}_{semicircle} + \text{Area}_{steps} = 39.27 + 46 = 85.27 \text{ cm}^2$$ 7. **Final answer:** The area of the figure is approximately **85.27 square centimeters**.