Question

a circle having a diameter of 8 cm is inscribed in a sector of a circle whose central angle is 80°. find the area of the sector.

Explanation:

Step1: Find the radius of the inscribed circle

The diameter of the inscribed circle is \(d = 8\mathrm{cm}\), so the radius \(r=4\mathrm{cm}\).

Step2: Relate the radius of the sector and the inscribed circle

Let the radius of the sector be \(R\).
We know that \(R - r=r\csc\frac{\theta}{2}\), where \(\theta = 80^{\circ}\), so \(\frac{\theta}{2}=40^{\circ}\) and \(\csc40^{\circ}=\frac{1}{\sin40^{\circ}}\approx\frac{1}{0.6428}\approx1.5557\)
Substitute \(r = 4\) into \(R - r=r\csc\frac{\theta}{2}\), we get \(R-4 = 4\times1.5557\)
\(R=4 + 4\times1.5557=4\times(1 + 1.5557)=4\times2.5557 = 10.2228\mathrm{cm}\)

Step3: Calculate the area of the sector

The formula for the area of a sector is \(A=\frac{\theta}{360}\times\pi R^{2}\), where \(\theta = 80^{\circ}\) and \(R\approx10.2228\)
\(A=\frac{80}{360}\times\pi\times(10.2228)^{2}\)
\(A=\frac{2}{9}\times\pi\times104.506\)
\(A=\frac{209.012\pi}{9}\approx\frac{209.012\times3.1416}{9}\)
\(A=\frac{656.677}{9}\approx72.96\mathrm{cm}^{2}\)

Answer:

The area of the sector is approximately \(73\mathrm{cm}^{2}\)