Krishna
0

Step 1:  Inspect the given figure and change the figure according to the given hints in the question.

                equilateral triangle made with centers of three tangent congruent circles


            NOTE :Figure above shows that the centers of the circles make an

           equilateral triangle of side 2r where r = 15 (given) is the radius of one circle. 


Step 2:  Set up a formula to calculate the shaded region

             NOTE: The area of the red shape = area of the equilateral triangle - areas

                        of three congruent sectors (each sector has 60° angle) 


Step 3:  Find the area of the equilateral triangle

             FORMULA:   a^2 \frac{\sqrt{3}}{4}

                                = (2r)^2 \frac{\sqrt{3}}{4}

                                = (30)^2 \frac{\sqrt{3}}{4}


Step 4: Calculate the areas of three congruent sectors

             NOTE: Areas of three congruent sectors = 3 × area of one sector

                              Area of the sector  = \frac{\theta}{360} * \pi r^2

                                                            = \frac{60}{360} \pi (15)^2

                                                            = \frac{1}{6} 15^2 \pi

       Areas of three congruent sectors = 3 * \frac{1}{6} 15^2 \pi

Step 5: Find the shaded region in the figure

             The area of the red shape =   (30)^2 \frac{\sqrt{3}}{4} - \frac{15^2 \pi}{2}


Step 6: Simplify the equation

              NOTE: (30)^2 \frac{\sqrt{3}}{4} - \frac{15^2 \pi}{2}

                         = [(900 * 0.433 ) - (112.5 * 3.14)

                         = 389.711 - 353.25

                         = 36.46 unit^2