Question

The angle between two radii of a circle is (60^). Find the area of a sector (in sq.cm.) of radius 7 cm.

• A. (496)
• B. (493)
• C. (492)
• D. (49)

Hint:

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!

Answer 1

The Correct Option is A

Concept: A circle covers a total sweeping angular rotation of (360^) around its center point. A circular sector is a pie-like slice bounded by two distinct radii lines and an outer connecting circular arc. The total area enclosed within a full circular boundary is given by ( r^2). Therefore, the localized fractional area enclosed within any specific sector tracking an internal central subtended angle of () degrees is proportional to the full rotation: \[ \text{Area of a Sector} = \frac{\theta}{360^{\circ}} \times \pi r^2 \]

Step 1: Extract and clarify the geometric variables provided within the problem description.

• The radius of the circle, (r = 7 cm)

• The central angle subtended by the two bounding radii lines, ( = 60^)

Step 2: Substitute the geometric measurements directly into our mathematical sector identity. \[ \text{Area of Sector} = \frac{60^{\circ}}{360^{\circ}} \times \pi \times (7)^2 \]

Step 3: Reduce the arithmetic fractions down into their simplest simplified forms step-by-step.

• Simplify the rotational ratio factor: \[ \frac{60^{\circ}}{360^{\circ}} = \frac{60}{360} = \frac{1}{6} \]

• Compute the absolute square of our radial value: \[ (7)^2 = 7 \times 7 = 49 \]
Now combine these terms back into the product expression: \[ \text{Area of Sector} = \frac{1}{6} \times \pi \times 49 = \frac{49\pi}{6} = 49\frac{\pi}{6} \text{ sq. cm.} \] This calculated mathematical simplification corresponds exactly with the symbolic presentation given in Option (A).