Q.265. Minimum area of a triangle given its in-radius and minimum area of the circumcircle given the perimeter of the triangle

Question 265.
1 ) What is the minimum area of a triangle for a given in-radius ?
2 ) What is the minimum area of a circumcircle given the perimeter of the triangle ?

Answer 265.
1 )
Of all the triangles drawn for the incircle with radius, r, the equilateral triangle will have the least area.
Let a/2 = half the length of side of the equilateral triangle
=> (a/2) / r = cot30°
=> a/2 = r√3 cm
=> minimum area of the triangle
= 3 * (a/2) * r
= 3 * r√3 * r
= 3√3 r^2 sq. units.

2)
For minimum area of circumcircle, we have to find the minimum value of circumradius, R
Minimum R
= (p/3) / 2sin60°, where p = perimeter ... [Using the formula a/sinA = 2R]
= p / (3√3)
=> minimum area of a circumscribing circle for a given perimeter, p
= π(Rmin)^2
= πp^2/(27) sq. units.

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