Find the area (in cm 2 ) of the sector, where the radius of the circle is 24 cm and length of the arc is 7 cm.

Correct — B. The arc length is given, so the central angle is never needed.

Area of a sector = (1⁄2) × r × l
It is the circle's version of (1⁄2) × base × height, with the arc as the base.

= (1⁄2) × 24 × 7
= 12 × 7 = 84

So the area is 84 cm² → option (b).

Why the others are wrong

• (a)
  48 — 48 is 2 × 24 — the radius handled on its own. It uses the arc length nowhere, and the area needs both r and l.
• (c)
  72 — 72 is what (1⁄2) × 24 × 6 gives. The method is right but the arc is read as 6, not the 7 cm the question prints.
• (d)
  168 — 168 is 24 × 7, the product with the one-half dropped. The sector is half the rectangle built on r and l, so that factor cannot go.

A sector has two area formulas and they are the same statement written from different data.

With the angle: area = (θ⁄360) × πr².
With the arc: area = (1⁄2) × r × l, since l = (θ⁄360) × 2πr.

When a paper hands you the arc length instead of the angle, the second form finishes in one line. Reaching for the first form here means solving for θ that you then cancel out again.

Key facts

• Area of a sector = (1⁄2) × radius × arc length.
• Arc length l = rθ when θ is measured in radians.
• Area of a sector = (θ⁄360) × πr² when θ is measured in degrees.
• With r = 24 cm and l = 7 cm the area is 84 cm².

Common traps

• Hunting for the central angle because the (θ⁄360)πr² form is the one you memorised.
• Dropping the one-half and answering 168.
• Reading the 7 cm as a chord rather than as the arc.

No directly related past PYQ was found.