The area of a sector of a circle of radius 12 cm, formed by an arc of length 7 cm is:

Correct — A. The paper gives you an arc, not an angle, so use the arc-length form of the sector area.

Area = (1⁄2) × r × l, where l is the arc length
= (1⁄2) × 12 × 7
= (1⁄2) × 84 = 42

The sector measures 42 → option (a).

Why the others are wrong

• (b)
  84 cm — 84 is r × l, the product before you halve it. The formula is (1⁄2)rl, so 84 is exactly double the sector's area.
• (c)
  21 cm — 21 is a quarter of 12 × 7 — the halving done twice. Halve rl once and only once.
• (d)
  28 cm — 28 is 84 ÷ 3. It is the answer you reach by dividing the product rl by 3 instead of by 2.

A sector has two area formulas and the question decides which one you can use.

With a central angle θ you use (θ⁄360) × πr². With an arc length l you use (1⁄2) × r × l, and no π and no angle are needed.

The two agree because the arc itself is l = (θ⁄360) × 2πr. Substitute that into (1⁄2)rl and the θ⁄360 and πr² reappear. Here only r = 12 and l = 7 are given, so the second form finishes in one line.

Key facts

• Sector area = (1⁄2) × radius × arc length, the form to use when an arc is given instead of an angle.
• Sector area = (θ⁄360) × πr² when the central angle is given.
• The two forms agree because arc length l = (θ⁄360) × 2πr.
• The perimeter of a sector is 2r + l, a different quantity from its area.

Common traps

• Multiplying r × l and forgetting the 1⁄2, which lands on the printed option 84
• Hunting for a central angle that the question never supplies
• Reporting an area in cm when the quantity is in cm²

No directly related past PYQ was found.