Final answer:
To find the perimeter and area of the sector of a circle with a central angle π/6 and radius 8 cm, we use the formulas for the arc length and the area of a sector. The arc length is 4π/3 cm, which helps to calculate the perimeter as 4π/3 + 16 cm, while the area is 8π/3 cm².
Step-by-step explanation:
The student asks to find the perimeter and area of a sector of a circle with a central angle of π/6 and a radius of 8 cm. The perimeter of a sector consists of the length of the arc plus twice the radius (since a sector includes two radii). The arc length can be found by using the formula L = rθ, where L is the arc length, r is the radius, and θ is the central angle in radians. The area of a sector can be calculated by the formula A = 0.5r²θ.
The calculation of the perimeter (P) is:
- Arc length (L) = 8 cm × (π/6) = 8π/6 = 4π/3 cm
- Perimeter of the sector (P) = Arc length (L) + 2 × radius (r) = 4π/3 + 2× 8 = 4π/3 + 16 cm
The calculation of the area (A) of the sector is:
- Area (A) = 0.5 × (8 cm)² × (π/6) = 32 × (π/6)/2 = 16π/6 = 8π/3 cm²
Since the circle is not divided into a simple shape like a square, we use the formulas specific to circles and sectors to find the perimeter and area. It's important to note that when finding the calculated values, the significant figures of the given radius limit the precision of our results.