A sector has central angle 90° in a circle of radius 4. What is its area?
A2π — using (90/360) times 2π times 4
B4π — using (90/360) times π times 4²
C8π — using (90/360) times π times 8
Dπ — using (90/360) times π times 4
Sector area = (θ/360) × πr² = (90/360) × π × 16 = 4π. Option A is the arc length formula, not sector area — a classic confusion. Arc length uses one factor of r (circumference = 2πr); area uses r² because area is two-dimensional. Option C uses diameter (8) instead of radius (4). Option D drops the squaring of r.
Question 2 Multiple Choice
A sector is cut from a circle of radius 6 with a central angle of 120°. A second sector is cut from a circle of radius 12 with the same 120° angle. How does the second sector's area compare to the first?
AIt is twice as large, because the radius doubled
BIt is four times as large, because area scales with r²
CIt is the same, because both have 120° angles
DIt is six times as large, because 12/6 = 2 and 120/360 doubles
Sector area = (θ/360) × πr². The angle fraction stays the same (120/360 = 1/3), so the area ratio is entirely determined by r²: (12)²/(6)² = 144/36 = 4. The sector area is four times larger. This illustrates why area formulas have r² — doubling a linear dimension multiplies area by 2² = 4, not by 2.
Question 3 True / False
A sector and an arc with the same central angle and radius always have a proportional relationship: as the central angle doubles, both the arc length and sector area double.
TTrue
FFalse
Answer: True
Both arc length (θ/360 × 2πr) and sector area (θ/360 × πr²) are linear functions of θ, so doubling the angle doubles both. This proportionality is the core insight — sector area and arc length are both just proportional fractions of their respective whole-circle measurements, with the fraction determined entirely by θ/360.
Question 4 True / False
If you know the arc length of a sector, you can compute its area using primarily that arc length value and hardly anything else.
TTrue
FFalse
Answer: False
Arc length L = (θ/360) × 2πr and sector area A = (θ/360) × πr². To convert between them you need the radius r, because A = (L × r) / 2. With only the arc length, you cannot determine the area — two sectors with the same arc length but different radii have different areas. The radius carries additional information that arc length alone does not.
Question 5 Short Answer
Why does the sector area formula use r² while the arc length formula uses only r? What does this reflect about the nature of each quantity?
Think about your answer, then reveal below.
Model answer: Arc length is a one-dimensional measurement (length along a curve), so it scales with r the same way circumference scales with radius. Sector area is a two-dimensional measurement (a filled region), so it scales with r² the same way the full circle's area scales with r². The extra factor of r reflects the step from one dimension to two — every point of the radius contributes to both the width and depth of the region.
This distinction is fundamental to dimensional analysis. Lengths scale linearly with size; areas scale as the square; volumes as the cube. The arc length formula (θ/360 × 2πr) and the sector area formula (θ/360 × πr²) are parallel — both multiply the whole-circle measurement by θ/360 — but the whole-circle measurements themselves differ by a factor of r because circumference is 1D and area is 2D.