F-RED: Finding the Area of the Shaded Region Between the Circle and Triangle

Among all the geometric problems, determining the area of the shaded region between a circle and a triangle is frequently provided for both a circumscribed circle and an inscribed circle in a triangle are often given. Having limited time in competitive settings, calculating its area can be pretty time-consuming. Thus, the formula, referred to as F-RED (Formula RED), was derived to establish a consistent relationship between the area of the triangle to the inscribed circle and the circumscribing circle. With the help of the constant value (5.19615) that has been calculated, it is easier to get the area of the inscribed circle in a triangle with the given measurement of the radius, even without the help of other given numbers, by just multiplying it with radius squared and subtracted to the area of the circle. Area of the shaded region = (5.19615) (radius)2 - (area of a circle) Likewise, with the help of the constant value (1.299), it is easier to get the area of the circumscribing circle to a triangle using only the radius, even without the help of other given numbers, by just subtracting the area of the circle to the constant value multiplied by the radius squared. Area of the shaded region = (area of a circle) - (1.299) (radius)2 The researcher discovered that the area of the shaded region can now be determined just using the radius and constant values. Thus, all assumptions are proven correct.