1) A certain sector has arc length pi and radius 10. What is the central angle of the sector in degrees?
2)A regular pentagon has side length 2. A circle with radius 1 is drawn at each vertex of the pentagon. The overlap of the pentagon with the circles is shaded blue. What is the total shaded area?
3)A square circumscribes a circle and is inscribed in another circle. The region between the two circles is shaded gray. If the area of the smaller circle is 12, what is the area of the gray region?
4)A regular hexagon with perimeter 36 is inscribed in a circle. What is the circumference of the circle in inches?
5)A 55° arc on circle A has the same length as a 40° arc on circle B. What is the ratio of the area of circle A to the area of circle B?
The total circumference is 10 * 2pi = 20pi, and the whole circle is 360 degrees, so the answer is pi / 20 pi x 360 = 1/20 x 360 = 18 degrees.
Keep in mind that the measure of each degree is 108. Because the circles have radius 1, they will not overlap with each other, and each circle's blue region is 108/360 of its whole area, or 3/10. Since there are 5 circles and each one has area pi, the answer is 3pi/2.
Let's call the radius of the small circle r. Then, the side length of the square is 2r, and the diameter of the large circle is 2rsqrt2, and its radius is rsqrt2. The gray region is the large circle's area minus the small circle's. We know that pir^2 = 12, and subtracting the areas gives pir^2, so our answer is 12.
Draw lines that split the hexagon in two equal congruent pieces and go through two vertices. You will end up with 6 equaliteral triangles, so the radius of the circle is 36 / 6 = 6. This means that the circumference is 6 * 2pi = 12pi.
Call the radius of circle A r and the radius of circle B s. Then, 55r/360 = 40s/360 => 55r = 40s => 11r = 8s.
We want the ratio of the areas, and since it is 2D, we just want r^2/s^2, or (8/11)^2 = 64 / 121.
Hoping this helped,