The angle between two radii of a circle is (60^). Find the area of a sector (in sq.cm.) of radius 7 cm.
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Always carefully scan the answer options before fully resolving constants like (). Many geometry problems in examinations leave the answers written in terms of () directly instead of expanding out to (227) or (3.14), which saves valuable mechanical calculation steps!
Step 1: Picture the sector. A sector is a slice of a circle bounded by two radii and the arc between them. Its area is a fraction of the whole circle, and that fraction is the central angle divided by $360^\circ$.
Step 2: Write the sector area formula. \[ \text{Area} = \frac{\theta}{360^\circ}\times \pi r^2 \] Here $\theta = 60^\circ$ and $r = 7$ cm.
Step 3: Simplify the angle fraction. \[ \frac{60}{360} = \frac{1}{6} \] So the sector is one sixth of the full circle.
Step 4: Put in the radius. Using $\pi = \frac{22}{7}$, the full area is $\frac{22}{7}\times 7^2 = \frac{22}{7}\times 49 = 154$.
Step 5: Take one sixth. \[ \text{Area} = \frac{1}{6}\times 154 = \frac{154}{6} = \frac{77}{3} \]
Step 6: State the result. This equals $\frac{49}{6}\times \frac{22}{7}$, which is the value shown by the option. So the sector area is \[ \boxed{\dfrac{49}{6}\ \text{sq.cm}} \]