Find the area of the shaded region of each circle Answer the questions that follow. 1. A 90° 3. Q 45° 135° X C 6 cm B S 12 cm R Z 8 cm Y
The Correct Answer and Explanation is:
To find the area of the shaded regions in each circle, we use the formula for the area of a sector. The formula is given as
Area of sector = (θ / 360) × πr²
where θ is the central angle in degrees and r is the radius of the circle.
For the first circle:
θ = 90° r = 6 cm
Area = (90 / 360) × π × 6² = (1/4) × π × 36 = 9π cm²
For the second circle:
θ = 45° r = 12 cm
Area = (45 / 360) × π × 12² = (1/8) × π × 144 = 18π cm²
For the third circle:
θ = 135° r = 8 cm
Area = (135 / 360) × π × 8² = (3/8) × π × 64 = 24π cm²
Explanation:
Each shaded region in the image represents a sector of a circle. To find the area of a sector, we determine what fraction of the total circle the sector occupies based on its central angle. Since the total circle has an area of πr², we multiply this by the fraction θ/360 to get the sector’s area.
For the first circle, the shaded region is one-fourth of the total circle because the central angle is 90°. Multiplying one-fourth by the total area π × 36 gives 9π cm². For the second circle, the shaded region is one-eighth of the total circle because the central angle is 45°. One-eighth of π × 144 gives 18π cm². For the third circle, the sector is three-eighths of the circle since 135° is three-eighths of 360°. Multiplying three-eighths by π × 64 yields 24π cm².
These calculations illustrate the relationship between angle size and sector area. A larger angle results in a larger fraction of the circle being shaded. This principle is useful in fields like engineering and architecture where circular sectors are commonly used.
