1. **State the problem:** Find the area of each regular polygon given the side length or apothem. 2. **Formula for area of a regular polygon:** $$\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}$$ 3. **Important rules:** - The perimeter is the number of sides times the side length. - The apothem is the perpendicular distance from the center to a side. - For polygons where only side length is given, apothem can be found using trigonometry. 4. **Calculate each polygon's area:** **Polygon 1: Regular triangle (equilateral) with side length 5 m** - Number of sides $n=3$ - Side length $s=5$ - Apothem $a = \frac{s}{2\tan(\pi/n)} = \frac{5}{2\tan(\pi/3)} = \frac{5}{2\times \sqrt{3}/3} = \frac{5}{2\times 0.5774} \approx 4.33$ - Perimeter $P = n \times s = 3 \times 5 = 15$ - Area $= \frac{1}{2} \times 15 \times 4.33 = 32.5$ **Polygon 2: Regular pentagon with side length $hm$ (not specified, cannot calculate)** **Polygon 3: Regular octagon with side length 9 cm** - $n=8$, $s=9$ - Apothem $a = \frac{s}{2\tan(\pi/n)} = \frac{9}{2\tan(\pi/8)} = \frac{9}{2\times 0.4142} \approx 10.87$ - Perimeter $P = 8 \times 9 = 72$ - Area $= \frac{1}{2} \times 72 \times 10.87 = 391.3$ **Polygon 4: Regular hexagon with side length 18 yd** - $n=6$, $s=18$ - Apothem $a = \frac{s}{2\tan(\pi/n)} = \frac{18}{2\tan(\pi/6)} = \frac{18}{2\times 0.5774} = 15.59$ - Perimeter $P = 6 \times 18 = 108$ - Area $= \frac{1}{2} \times 108 \times 15.59 = 842.0$ **Polygon 5: Regular pentagon with side length 32 ft** - $n=5$, $s=32$ - Apothem $a = \frac{s}{2\tan(\pi/n)} = \frac{32}{2\tan(\pi/5)} = \frac{32}{2\times 0.7265} = 22.02$ - Perimeter $P = 5 \times 32 = 160$ - Area $= \frac{1}{2} \times 160 \times 22.02 = 1761.6$ **Polygon 6: Regular octagon with side length 24 mm** - $n=8$, $s=24$ - Apothem $a = \frac{24}{2\tan(\pi/8)} = \frac{24}{2\times 0.4142} = 28.97$ - Perimeter $P = 8 \times 24 = 192$ - Area $= \frac{1}{2} \times 192 \times 28.97 = 2779.0$ 5. **Summary of areas rounded to nearest tenth:** - Polygon 1: 32.5 - Polygon 3: 391.3 - Polygon 4: 842.0 - Polygon 5: 1761.6 - Polygon 6: 2779.0 Note: Polygon 2 cannot be calculated due to missing side length.