Question

Three circles with radii 3, 6, and 9 ft are externally tangent to one another, as shown in the figure. Find the area of the sector of the circle of radius 3 that is cut off by the line segments joining the center of that circle to the centers of the other two circles.

Answer 1

The area of the sector of the circle of radius 3 that is cut off by the line segments joining the centers of the other two circles is approximately 14.1372 square units.

Step-by-step explanation:

To find the area of the sector of the circle of radius 3 that is cut off by the line segments joining the centers of the other two circles, we first need to find the angle of the sector. Since the three circles are externally tangent to each other, the radii drawn from the centers of the circles to the points of tangency form an equilateral triangle. Each angle of an equilateral triangle is 60 degrees. So, the angle of the sector is 60 degrees.

The area of a sector can be found using the formula ∡ / 360 * π * r², where ∡ is the angle of the sector and r is the radius of the circle. Substituting the values, we have: 60 / 360 * 3.14159 * 3² = 0.5236 * 28.2743 = 14.1372 square units.

Therefore, the area of the sector of the circle of radius 3 that is cut off by the line segments joining the centers of the other two circles is approximately 14.1372 square units.