2022-01-10

The top view of a circular table shown on the right has a radius of 120cm.find the area of the smaller segment of the table (shaded region) determined by 60° arc

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Area of segment (the shaded region) = Area of sector - area of triangle

Where:

Area of sector $=\left(\frac{\theta \pi }{360}\right){r}^{2}$

Area of segment $=\left(\frac{\mathrm{sin}\theta }{2}\right){r}^{2}$

Derive the equation:

Area of segment $=\left(\frac{\theta \pi }{360}\right){r}^{2}$$\left(\frac{\mathrm{sin}\theta }{2}\right){r}^{2}$

Area of segment $={r}^{2}\left(\frac{\theta \pi }{360}-\frac{\mathrm{sin}\theta }{2}\right)$

Given:

Central angle, $\theta =60°$

Radius, $r=120$ cm

pi, $\pi \approx 3.14$

Solve for the area of segment or shaded region:

Area of segment $={r}^{2}\left(\frac{\theta \pi }{360}-\frac{\mathrm{sin}\theta }{2}\right)$

Area = ($120$ cm)${}^{2}$ $\left[60×\frac{3.14}{360}-\frac{\mathrm{sin}60}{2}\right]$

Area = $14,400$ cm${}^{2}$ $\left[0.523-\frac{0.866}{2}\right]$

Area = $14,400$ cm${}^{2}$ $\left[0.523-0.433\right]$

Area = $14,400$ cm${}^{2}$ $\left(0.09\right)$

Area of segment or shaded region = $1,296$ cm${}^{2}$

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