Question

what is the area of a sector formed by a central angle of π/3 radians in a circle with radius 6 cm?
a. 18π square cm
b. 12π square cm
c. 6π square cm
d. 3π square cm

which of the following is a property of the radian measure?
a. radians are primarily used for measuring volume.
b. radians are independent of the circles radius.
c. radians directly relate the arc length to the radius
d. radians measure the area of a sector

what is the sector area of a circle with radius 12 cm if the central angle is π/6 radians?
a. 12π square cm
b. 18π square cm
c. 24π square cm
d. 6π square cm

which of the following is a valid method for calculating the sector area?
a. sector area = 0.5 × radius² × central angle (in radians)
b. sector area = radius × central angle (in radians)
c. sector area = 2 × π × radius
d. sector area = π × radius²

what is the correct formula for calculating the central angle in radians if the arc length and radius are known?
a. central angle = arc length / radius
b. central angle = radius / arc length
c. central angle = arc length + radius
d. central angle = arc length × radius

if the central angle of a sector is 2π/3 radians and the radius is 6 meters, what is the sector area?
a. 18π square meters
b. 12π square meters
c. 36π/3 square meters
d. 64π/2 square meters

a bicycle wheel has a radius of 30 cm. if the wheel rotates through an angle of π/8 radians, what is the approximate distance traveled by a point on the outer edge of the wheel?

Explanation:

Step1: Recall the sector - area formula

The formula for the area of a sector is \(A=\frac{1}{2}r^{2}\theta\), where \(r\) is the radius of the circle and \(\theta\) is the central angle in radians.

Step2: Substitute the values for the first question

For the first question, \(r = 6\) cm and \(\theta=\frac{\pi}{3}\). Then \(A=\frac{1}{2}\times6^{2}\times\frac{\pi}{3}\).
First, calculate \(6^{2}=36\). Then \(\frac{1}{2}\times36\times\frac{\pi}{3}\). \(\frac{1}{2}\times36 = 18\), and \(18\times\frac{\pi}{3}=6\pi\). So the area of the sector is \(6\pi\) square cm.

Step3: Analyze the property of radian measure

Radians directly relate the arc length \(s\) to the radius \(r\) by the formula \(s = r\theta\) (where \(\theta\) is in radians). Radians are used for measuring angles (not volume or area). And they are independent of the circle's radius in the sense that \(\theta=\frac{s}{r}\) (a ratio).

Step4: Calculate the sector area for the second - sector - area question

For the sector with \(r = 12\) cm and \(\theta=\frac{\pi}{6}\), use \(A=\frac{1}{2}r^{2}\theta\). Substitute \(r = 12\) and \(\theta=\frac{\pi}{6}\): \(A=\frac{1}{2}\times12^{2}\times\frac{\pi}{6}\). \(12^{2}=144\), \(\frac{1}{2}\times144 = 72\), \(72\times\frac{\pi}{6}=12\pi\).

Step5: Verify the sector - area formula

The formula \(A=\frac{1}{2}r^{2}\theta\) (where \(\theta\) is in radians) can be written as \(A = 0.5\times r^{2}\times\theta\).

Step6: Recall the central - angle formula

From \(s=r\theta\) (arc length formula, where \(s\) is arc length, \(r\) is radius, and \(\theta\) is central angle in radians), we can solve for \(\theta\): \(\theta=\frac{s}{r}\).

Step7: Calculate the sector area for the given central angle and radius

For a sector with \(r = 6\) meters and \(\theta=\frac{2\pi}{3}\), use \(A=\frac{1}{2}r^{2}\theta\). Substitute \(r = 6\) and \(\theta=\frac{2\pi}{3}\): \(A=\frac{1}{2}\times6^{2}\times\frac{2\pi}{3}\). \(6^{2}=36\), \(\frac{1}{2}\times36 = 18\), \(18\times\frac{2\pi}{3}=12\pi\).

Answer:

1. c. \(6\pi\) square cm
2. b. Radians are independent of the circle's radius; c. Radians directly relate the arc length to the radius
3. a. \(12\pi\) square cm
4. a. Sector Area \(=0.5\times\) Radius\(^{2}\times\) Central Angle (in radians)
5. a. Central Angle \(=\) Arc Length / Radius
6. b. \(12\pi\) square meters