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  • The central angles of two sectors of circles of radii 7 cm and 21 cm are respectively 120° and 40°. Find the areas of the two sectors as well as the lengths of the corresponding arcs. What do you observe?

The central angles of two sectors of circles of radii 7 cm and 21 cm are respectively 120° and 40°. Find the areas of the two sectors as well as the lengths of the corresponding arcs. What do you observe?

Answers (1)

\because Area of sector =\frac{\pi r^{2} \theta}{360}

Radius of first sector (r1) = 7 cm

Angle (\theta_{1} ) = 120°

Area of first sector(A1) =\frac{\pi r_{1}^{2} \theta}{360}

    \\=\frac{22 \times 7 \times 7 \times 120^{\circ} }{360^{\circ}}\\ =\frac{154}{3}cm^{2}

Radius of second sector (r2) = 21 cm

Angle (\theta_{2} ) = 40°

Area of sector of second circle (A2)=\frac{\pi r_{2}^{2} \theta}{360}
                                             \\=\frac{22}{7}\times \frac{21\times 21}{360}\times40\\\\ =154 cm^{2}

Corresponding arc length of first circle =\frac{2\pi r_{1} \theta}{360}

=\frac{\pi r_{1} \theta}{180}

\\=\frac{22}{7}\times \frac{7\times 120}{180}\\=\frac{44}{3}cm

Corresponding arc length of second circle =\frac{2\pi r_{2}\theta}{360}

=\frac{\pi r_{2}\theta}{180}

\\=\frac{22}{7}\times \frac{21\times 40}{180}\\=\frac{44}{3}cm

We observe that the length of the arc of both circles is equal.

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