A regular hexagon circumscribes circle O and another regular hexagon is inscribed in circle O. If the radius of circle O is r, show that 3 < π < 2√3.
Three congruent circles are pairwise tangent and each has a radius equal to 2. A circle circumscribes the three circles. Calculate the total area shaded in blue. The blue region is comprised of two parts. One region is the three circular sectors of the small circles enclosed by the line segments connecting the three small circle’s centers. The other region is outside the three small circles and bound by the large circumscribing circle (exclude the area in between the three small circles).
Square ABCD is inscribed in a quarter circle O such that B and C are on the arc of the quarter circle. If the quarter circle has a radius equal to 1, what is the area of the square?
In a square with a side length of 1, two quarter circles are drawn and a circle is inscribed between the quarter circles, as shown in the diagram. What is the radius of the inscribed circle?
In a circle of radius 1, an equilateral triangle is inscribed in the circle as shown. What is the area of the blue region?
A cube is sliced into halves in such a way that the cut is a regular hexagon. What is the angle (in degrees) between the plane of the cut and the base of the cube?
