Digital SATMath: Geometry and TrigonometryHard
A graphic designer is creating a logo that involves a sector of a circle. The circle has a radius of 10 cm, and the central angle of the sector is 72°. What is the area of this sector?
- A5π cm²
- B10π cm²
- C50π cm²
- D20π cm²
Show answer & explanationAnswer & explanation
Correct answer: D. 20π cm²
The area of a sector can be found using the formula: Area = (θ/360°) * πr², where θ is the central angle in degrees and r is the radius. Given θ = 72° and r = 10 cm: Area = (72/360) * π * (10 cm)² = (1/5) * π * 100 cm² = 20π cm². Alternatively, convert 72° to radians: 72° * (π/180°) = 2π/5 radians. Then use Area = (1/2)r²θ (where θ is in radians): (1/2) * (10)² * (2π/5) = (1/2) * 100 * (2π/5) = 50 * (2π/5) = 20π cm².
Why the other options are wrong
- A. This is 1/10th of the full circle area, or an incorrect fraction.
- B. This is 1/5th of 50π, implying an incorrect radius or fraction.
- C. This is half the area of the full circle, not the sector.
Area of a Circular Sector
The area of a region bounded by two radii and the intercepted arc of a circle.
- Formula (degrees): A = (θ/360°) * πr²
- Formula (radians): A = (1/2)r²θ
- Represents a 'slice' of the circle.
Memory trick: A sector is a pizza slice – its area is a fraction of the whole pizza.