Question

1 compound
fill in the blanks: (fractions are not allowed in the answer.)

1. the perimeter of the sector is ; the area of the sector is . (π≈3, and the two radii are perpendicular to each other.)
2. the perimeter of the sector is ; the area of the sector is . (π≈3)

Explanation:

1.

Step1: Calculate the perimeter of the sector (quarter - circle)

The formula for the perimeter of a sector is \(P = 2r+\frac{\theta}{360}\times2\pi r\). For a quarter - circle (\(\theta = 90^{\circ}\), \(r = 4\)), \(\frac{\theta}{360}=\frac{90}{360}=\frac{1}{4}\).
\(P=2\times4+\frac{1}{4}\times2\times3\times4\)
\(P = 8 + 6\)

Step2: Calculate the area of the sector (quarter - circle)

The formula for the area of a sector is \(A=\frac{\theta}{360}\times\pi r^{2}\). For \(\theta = 90^{\circ}\), \(r = 4\), \(\frac{\theta}{360}=\frac{90}{360}=\frac{1}{4}\)
\(A=\frac{1}{4}\times3\times4^{2}\)
\(A=\frac{1}{4}\times3\times16\)
\(A = 12\)

2.

Step1: Calculate the perimeter of the sector (\(\theta=120^{\circ}\), \(r = 3\))

The formula for the perimeter of a sector is \(P = 2r+\frac{\theta}{360}\times2\pi r\). \(\frac{\theta}{360}=\frac{120}{360}=\frac{1}{3}\)
\(P=2\times3+\frac{1}{3}\times2\times3\times3\)
\(P = 6+6\)

Step2: Calculate the area of the sector (\(\theta = 120^{\circ}\), \(r = 3\))

The formula for the area of a sector is \(A=\frac{\theta}{360}\times\pi r^{2}\). \(\frac{\theta}{360}=\frac{120}{360}=\frac{1}{3}\)
\(A=\frac{1}{3}\times3\times3^{2}\)
\(A=9\)

Answer:

1. The perimeter of the sector is \(14\); the area of the sector is \(12\).
2. The perimeter of the sector is \(12\); the area of the sector is \(9\).