Questions: If m ∠AB=105, find the area of the shaded sector. Round to the nearest tenth.

Solution Steps

The length of segment AB is given as 16 cm. Since AB is a radius of the circle, the radius is 16 cm.

The area of a circle is given by the formula \(A = \pi r^2\). In this case, the radius \(r = 16\) cm. So, the area of the entire circle is: \(A = \pi (16^2) = 256\pi \text{ cm}^2\)

The measure of arc AB is given as 105 degrees. A full circle has 360 degrees. The shaded sector represents a fraction of the circle equal to the ratio of the arc measure to the full circle measure: \(\text{Fraction} = \frac{105}{360} = \frac{7}{24}\)

The area of the shaded sector is the fraction of the circle multiplied by the total area: \(\text{Area of sector} = \frac{7}{24} \times 256\pi = \frac{1792\pi}{24} = \frac{224\pi}{3} \approx 234.572 \text{ cm}^2\)

Final Answer The final answer is