We will learn the formula for finding the area of a sector. The idea is the same as for arc length: use “what fraction of the entire circle is represented by the central angle.”
Formula for Area (Using the Central Angle)
For a sector with radius r and central angle a degrees, the area S is:
S ＝ πr² × (a/360)
Meaning: A sector is the part of the area of a circle with radius r (πr²) represented by the fraction a/360.
Find the area of a sector with radius 6 cm and central angle 60°. Leave π as it is.
Solution:
S = π × 6² × (60/360) = 36π × (1/6) = 6π (cm²)
Area Formula Using the Arc Length
A useful formula when the arc length ℓ is known is:
S ＝ (1/2) × r × ℓ
Meaning: A sector has a shape similar to a triangle whose “base = arc length” and “height = radius” (the actual side is curved, but the area ratio is the same).
Find the area of a sector with radius 8 cm and arc length 6π cm.
Solution:
S = (1/2) × 8 × 6π = 24π (cm²)
Verification Calculation
Let us also verify Example ② using the formula with the central angle.
Arc 6π = 2π × 8 × (a/360) → 6π = 16π × (a/360) → a/360 = 6/16 = 3/8 → a = 135°.
Area S = π × 8² × (135/360) = 64π × (3/8) = 24π. The result is indeed 24π.
- Area S = πr² × (a/360)
- When the arc length is known, S = (1/2)rℓ is useful.
- Remember: “From the arc and radius, use (1/2) × radius × arc.”
- Leave π as it is during calculations and in the final answer.
Practice Problems
- Find the area of a sector with radius 10 cm and central angle 72° (leave π as it is).
- Find the area of a sector with radius 5 cm and arc length 4π cm.
- A sector with radius 6 cm has area 9π cm². Find its central angle.
Answers and Explanations
- Answer: 20π cm²
Explanation: S = π×100×72/360 = 100π × 1/5 = 20π. - Answer: 10π cm²
Explanation: S = (1/2)×5×4π = 10π. - Answer: 90°
Explanation: S = π×36×a/360 = 9π → 36a/360 = 9 → a/10 = 9 → a = 90°.