Ratio of area of one circle to the equilateral triangle when three equal circles are placed inside an equilateral triangle Last Updated : 29 Oct, 2020 Comments Improve Suggest changes Like Article Like Report Given three equal circles that are placed inside an equilateral triangle such that every circle is tangential to the sides of the equilateral triangle and to other circles. The task is to find the ratio of the area of one circle to the area of the equilateral triangle. Solution: Below is the image of how circles are inscribed in a triangle: Now since AB and BC are tangents to the circle with center P, PQ will be perpendicular to BC and PB will bisect angle ABC. Hence angle PBQ=30° since ABC is an equilateral triangle and angle ABC=60°. Consider the triangle PBQ, tan30°= PQ/BQ = 1/√3 BQ = PQ*√3 = R*√3 (R is radius one circle). Similarly RC = R*√3 Now BC = BQ+QR+CR = R√3 + 2R + R√3 = 2R(√3 +1)Therefore, the ratio of the area of the circle to the area of the triangle is given by: Ratio = \frac{area(circle)}{area(triangle)} Since, area(circle) = \pi*r^{2} area(triangle) = \frac{\sqrt{3}(2r(\sqrt{3}+1))^{2}}{4} Therefore, the ratio is given by: Ratio = \frac{\pi r^{2}}{\frac{\sqrt{3}(2r(\sqrt{3}+1))^{2}}{4}} Ratio = \frac{\pi}{\sqrt{3}(\sqrt{3}+1)^{2}} Comment More infoAdvertise with us Next Article Ratio of area of one circle to the equilateral triangle when three equal circles are placed inside an equilateral triangle S saif_ahmad_khan Follow Improve Article Tags : Aptitude Aptitude-Puzzles Similar Reads Area of Equilateral triangle inscribed in a Circle of radius R Given an integer R which denotes the radius of a circle, the task is to find the area of an equilateral triangle inscribed in this circle. Examples: Input: R = 4 Output: 20.784 Explanation: Area of equilateral triangle inscribed in a circle of radius R will be 20.784, whereas side of the triangle wi 4 min read Area of circle inscribed in a Isosceles Trapezoid Given two bases of the isosceles trapezoid ABCD as a and b, the task is to find the area of circle inscribed in this trapezoid Examples:  Input: a = 10, b = 30 Output: Area = 235.57Input: a = 20, b = 36 Output: Area = 565.38Derivation: Given a circle inscribed in trapezium ABCD (sides AB = n and C 5 min read GRE Geometry | Circles A Circle is a 2-dimensional figure. It is a closed figure in which boundary is equidistant from the center and distance from the center to the boundary is called radius and it remains the same throughout the figure. C is the center of circle. Perimeter of a circle is called circumference. Important 4 min read Puzzle 67 | Fit Triangle Sameer asks his friend Kartik a question to test his intelligence. Sameer – You have been given a right-angled triangle ABC, right-angled at B. The hypotenuse length is 10m and the altitude to the hypotenuse from B is 6m. Kartik – Go on. Sameer – Tell me the area of the triangle ABC. Kartik – 30 sq. 1 min read Area of the circle that has a square and a circle inscribed in it Given the side of a square that is kept inside a circle. It keeps expanding until all four of its vertices touch the circumference of the circle. Another smaller circle is kept inside the square now and it keeps expanding until its circumference touches all four sides of the square. The outer and th 3 min read Like