A circular wheel is divided into 8 equal sectors. If the area of one sector is 45 cm², what is the radius of the wheel?

Correct — C. Eight equal sectors make up the whole wheel, so start from the total area.

Whole wheel = 8 × 45 = 360 cm²
Circle area: πr² = 360
Divide by π: r² = 360 ÷ 3.1416 ≈ 114.6
Square root: r ≈ 10.70 cm

Check: π × 10.7² ≈ 359.7 cm², and one-eighth of that is ≈ 45 cm² ✓
Radius ≈ 10.7 cm → option (c)

Why the others are wrong

• (a)
  8.6 cm — 8.6 cm is too small: π × 8.6² ≈ 232.4 cm² for the whole wheel, so one sector would be about 29 cm², not 45 cm².
• (b)
  6.5 cm — 6.5 cm gives π × 6.5² ≈ 132.7 cm² for the whole wheel and about 16.6 cm² per sector, far below the 45 cm² stated.
• (d)
  7.5 cm — 7.5 cm gives π × 7.5² ≈ 176.7 cm² for the wheel, about 22.1 cm² per sector — roughly half of the 45 cm² a sector must have.

Equal sectors split a circle's area equally. With 8 of them, each has a central angle of 360° ÷ 8 = 45° and one-eighth of the area.

So the sector formula (θ⁄360°) × πr² = 45 becomes πr²⁄8 = 45, the same equation as πr² = 360.

Once the whole area A is known, the radius is r = √(A⁄π).

Key facts

• Area of a sector with central angle θ = (θ⁄360°) × πr².
• A circle cut into n equal sectors gives each a central angle of 360°⁄n and an area of πr²⁄n.
• Radius from a known circle area A: r = √(A⁄π).

Common traps

• Solving πr² = 45 with one sector's area, which gives r ≈ 3.8 cm, the radius of a circle one-eighth the size.
• Dividing 45 by 8 instead of multiplying: each sector is one-eighth of the wheel, so the wheel is 8 times a sector.

No directly related past PYQ was found.