Questions: Calcule el perímetro y área de los siguientes segmentos circulares

Solution Steps

The area of a sector is given by the formula: \[ \text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2 \] For the first segment (a):

• \(\theta = 60^\circ\)
• \(r = 3 \text{ cm}\)

\[ \text{Area of sector} = \frac{60^\circ}{360^\circ} \times \pi \times (3 \text{ cm})^2 \] \[ \text{Area of sector} = \frac{1}{6} \times \pi \times 9 \text{ cm}^2 \] \[ \text{Area of sector} = \frac{9\pi}{6} \text{ cm}^2 \] \[ \text{Area of sector} = 1.5\pi \text{ cm}^2 \]

The area of the triangle can be calculated using the formula for the area of an equilateral triangle: \[ \text{Area of triangle} = \frac{1}{2} \times r^2 \times \sin(\theta) \] For the first segment (a):

• \(\theta = 60^\circ\)
• \(r = 3 \text{ cm}\)

\[ \text{Area of triangle} = \frac{1}{2} \times (3 \text{ cm})^2 \times \sin(60^\circ) \] \[ \text{Area of triangle} = \frac{1}{2} \times 9 \text{ cm}^2 \times \frac{\sqrt{3}}{2} \] \[ \text{Area of triangle} = \frac{9\sqrt{3}}{4} \text{ cm}^2 \]

The area of the segment is the area of the sector minus the area of the triangle: \[ \text{Area of segment} = \text{Area of sector} - \text{Area of triangle} \] \[ \text{Area of segment} = 1.5\pi \text{ cm}^2 - \frac{9\sqrt{3}}{4} \text{ cm}^2 \]

Final Answer