Question

the radius of circle c is 6 units and the measure of central angle acb is \\(\frac{\pi}{2}\\) radians. what is the approximate area of the entire circle? \\(\square\\) square units what is the approximate area of the entire sector created by central angle acb? \\(\square\\) square units what is the approximate area of the shaded region only? \\(\square\\) square units

Explanation:

First Question: Area of the entire circle

Step1: Recall the formula for the area of a circle

The formula for the area of a circle is \( A = \pi r^2 \), where \( r \) is the radius of the circle.

Step2: Substitute the given radius into the formula

Given that the radius \( r = 6 \) units. Substituting into the formula, we get \( A=\pi\times(6)^2 = 36\pi\).

Step3: Approximate the value

Using \( \pi\approx3.14 \), we have \( 36\times3.14 = 113.04 \) square units.

Step1: Recall the formula for the area of a sector (in radians)

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

Step2: Substitute the given values

We know that \( r = 6 \) units and \( \theta=\frac{\pi}{2} \) radians. Substituting these values, we get \( A=\frac{1}{2}\times(6)^2\times\frac{\pi}{2} \).

Step3: Simplify the expression

First, calculate \( (6)^2 = 36 \). Then, \( \frac{1}{2}\times36\times\frac{\pi}{2}=9\pi \).

Step4: Approximate the value

Using \( \pi\approx3.14 \), we have \( 9\times3.14 = 28.26 \) square units.

Step1: Analyze the shaded region

The shaded region is the area of the larger sector (with radius \( 6 \)) minus the area of the smaller sector (with radius \( 3 \)), both with the same central angle \( \frac{\pi}{2} \) radians.

Step2: Recall the sector area formula

The formula for the area of a sector is \( A=\frac{1}{2}r^2\theta \).

Step3: Calculate the area of the larger sector

For the larger sector, \( r = 6 \) and \( \theta=\frac{\pi}{2} \). So, \( A_{1}=\frac{1}{2}\times(6)^2\times\frac{\pi}{2}=9\pi \) (as calculated in the second question).

Step4: Calculate the area of the smaller sector

For the smaller sector, \( r = 3 \) and \( \theta=\frac{\pi}{2} \). Substituting into the formula, we get \( A_{2}=\frac{1}{2}\times(3)^2\times\frac{\pi}{2}=\frac{1}{2}\times9\times\frac{\pi}{2}=\frac{9\pi}{4} \).

Step5: Calculate the area of the shaded region

The area of the shaded region \( A = A_{1}-A_{2}=9\pi-\frac{9\pi}{4}=\frac{36\pi - 9\pi}{4}=\frac{27\pi}{4} \).

Step6: Approximate the value

Using \( \pi\approx3.14 \), we have \( \frac{27\times3.14}{4}=\frac{84.78}{4}=21.195\approx21.20 \) square units. (Alternatively, since we know \( A_{1}\approx28.26 \) and \( A_{2}=\frac{1}{2}\times3^2\times\frac{\pi}{2}=\frac{9\times3.14}{4}=7.065 \), then \( A = 28.26 - 7.065 = 21.195\approx21.20 \))

Answer:

\( 113.04 \)

Second Question: Area of the sector created by central angle \( \angle ACB \)