solve this---**Image Description:**
The image contains a diagram of a circle with center O. Two points A and B are located on the circumference of the circle. Lines OA and OB are drawn from the center to the points on the circumference. A line segment AB connects points A and B, forming a triangle OAB. The angle AOB is labeled as 60 degrees. The length of the line segment OB is labeled as 4. The region bounded by the arc AB and the line segment AB is shaded. This shaded region is a circular segment.
**Question Stem:**
In the figure above, the circle has center O. What is the area of the shaded region?
**Options:**
A) $\frac{8\pi}{3}$
B) $\frac{1}{3}(12\sqrt{3}-4\pi)$
C) $\frac{1}{3}(8\pi-8\sqrt{3})$
D) $\frac{1}{3}(8\pi-12\sqrt{3})$
视频信息
答案文本
视频字幕
In this problem, we have a circle with center O. Points A and B lie on the circumference of the circle. The angle AOB is 60 degrees, and the radius OB equals 4 units. We need to find the area of the shaded region, which is a circular segment formed by the arc AB and the chord AB.
To find the area of the shaded region, we first need to calculate the area of the sector OAB. The formula for the area of a sector is the central angle divided by 360 degrees, multiplied by pi and the radius squared. In our case, the central angle is 60 degrees and the radius is 4 units. Substituting these values, we get 60 divided by 360, multiplied by pi and 16, which simplifies to 8 pi divided by 3.
Next, we need to calculate the area of triangle OAB. Since OA and OB are both radii of the circle, they are equal to 4 units. The angle between them is 60 degrees. To find the area of the triangle, we can use the formula: one-half times the product of two sides times the sine of the included angle. Substituting our values, we get one-half times 4 times 4 times sine of 60 degrees. Since sine of 60 degrees equals square root of 3 divided by 2, our calculation gives us 4 times square root of 3 as the area of the triangle.
Now we can find the area of the shaded region by subtracting the area of the triangle from the area of the sector. The area of the sector is 8 pi divided by 3, and the area of the triangle is 4 times square root of 3. So the area of the shaded region equals 8 pi divided by 3 minus 4 times square root of 3, which can be rewritten as 1 over 3 times the quantity 8 pi minus 12 times square root of 3. Looking at the given options, this matches option D.
To summarize what we've learned: The area of a circular segment can be found by subtracting the area of the triangle from the area of the sector. For a sector with central angle θ and radius r, the area is calculated as θ divided by 360 degrees, multiplied by pi and r squared. The area of a triangle can be found using the formula one-half times the product of two sides times the sine of the included angle. In our problem, we calculated the area of the sector as 8 pi divided by 3, and the area of the triangle as 4 times square root of 3. Subtracting these values gave us the area of the shaded region as one-third times the quantity 8 pi minus 12 times square root of 3, which corresponds to option D.