Polygon Angles B476F9

1. Problem: Find each interior angle of a regular dodecagon. Formula: Each interior angle of a regular polygon with $n$ sides is given by $$\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}$$ For a dodecagon, $n=12$. Calculation: $$\text{Interior angle} = \frac{(12-2) \times 180^\circ}{12} = \frac{10 \times 180^\circ}{12} = \frac{1800^\circ}{12} = 150^\circ$$ Answer: Each interior angle of a regular dodecagon is $150^\circ$. 2. Problem: Find the number of sides of a polygon if the sum of its interior angles is $1080^\circ$. Formula: Sum of interior angles of a polygon with $n$ sides is $$S = (n-2) \times 180^\circ$$ Calculation: $$1080^\circ = (n-2) \times 180^\circ$$ Divide both sides by 180: $$\cancel{180^\circ} \times 6 = (n-2) \times \cancel{180^\circ}$$ So, $$6 = n - 2$$ Add 2 to both sides: $$n = 8$$ Answer: The polygon has 8 sides. 3. Problem: Find the number of diagonals in a hexagon. Formula: Number of diagonals in a polygon with $n$ sides is $$D = \frac{n(n-3)}{2}$$ For a hexagon, $n=6$. Calculation: $$D = \frac{6(6-3)}{2} = \frac{6 \times 3}{2} = \frac{18}{2} = 9$$ Answer: A hexagon has 9 diagonals. 4. Problem: The altitude of a triangle is half the base. Find the base if the area is 64 sq. units. Formula: Area of a triangle is $$A = \frac{1}{2} \times \text{base} \times \text{altitude}$$ Let base = $b$, then altitude = $\frac{b}{2}$. Calculation: $$64 = \frac{1}{2} \times b \times \frac{b}{2} = \frac{1}{2} \times \frac{b^2}{2} = \frac{b^2}{4}$$ Multiply both sides by 4: $$4 \times 64 = b^2$$ $$256 = b^2$$ Take square root: $$b = \sqrt{256} = 16$$ Answer: The base is 16 units. 5. Problem: Find the number of sides of a regular polygon if an exterior angle is $45^\circ$. Formula: Exterior angle of a regular polygon is $$E = \frac{360^\circ}{n}$$ Calculation: $$45^\circ = \frac{360^\circ}{n}$$ Multiply both sides by $n$: $$45^\circ n = 360^\circ$$ Divide both sides by 45: $$\cancel{45^\circ} n = \frac{360^\circ}{\cancel{45^\circ}}$$ $$n = 8$$ Answer: The polygon has 8 sides. 6. Problem: Find the number of sides of a regular polygon if an interior angle is $162^\circ$. Formula: Interior angle of a regular polygon is $$I = \frac{(n-2) \times 180^\circ}{n}$$ Calculation: $$162^\circ = \frac{(n-2) \times 180^\circ}{n}$$ Multiply both sides by $n$: $$162^\circ n = (n-2) \times 180^\circ$$ Expand right side: $$162^\circ n = 180^\circ n - 360^\circ$$ Bring terms to one side: $$180^\circ n - 162^\circ n = 360^\circ$$ $$18^\circ n = 360^\circ$$ Divide both sides by 18: $$\cancel{18^\circ} n = \frac{360^\circ}{\cancel{18^\circ}}$$ $$n = 20$$ Answer: The polygon has 20 sides. 7. Problem: Find the area and perimeter of a rectangle with length 15 cm and width 8 cm. Formulas: - Area: $$A = \text{length} \times \text{width}$$ - Perimeter: $$P = 2(\text{length} + \text{width})$$ Calculation: $$A = 15 \times 8 = 120 \text{ cm}^2$$ $$P = 2(15 + 8) = 2 \times 23 = 46 \text{ cm}$$ Answer: Area is $120$ cm$^2$ and perimeter is $46$ cm.