Question

if ( bd = 18 mathrm{cm} ), find the area of the shaded region in the circle.

Explanation:

Step1: Find the radius of the circle

Since \(BD = 18\) cm is the diameter of the circle, the radius \(r=\frac{BD}{2}=\frac{18}{2} = 9\) cm.

Step2: Calculate the total area of the circle

The formula for the area of a circle is \(A=\pi r^{2}\). Substituting \(r = 9\) cm, we get \(A=\pi\times9^{2}=81\pi\) \(cm^{2}\).

Step3: Find the central angle of the shaded region

The total central angle of a circle is \(360^{\circ}\). The un - shaded region (two equal parts) and the known shaded part (\(136^{\circ}\)): Let the central angle of each un - shaded part be \(x\). Since \(BD\) is a diameter, the sum of the central angles of the two un - shaded parts is \(180^{\circ}\) (semicircle). The central angle of the shaded region \(\theta=360-(180 - 136)=256^{\circ}\)

Step4: Calculate the area of the shaded region

The formula for the area of a sector is \(A_{sector}=\frac{\theta}{360}\times\pi r^{2}\). Substituting \(\theta = 256^{\circ}\) and \(r = 9\) cm, we have \(A_{shaded}=\frac{256}{360}\times81\pi\)

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Answer:

\(180.864\) \(cm^{2}\)