Question

A circle of radius 2cm is inscribed in an equilateral triangle. Find the area of the triangle

Answer 1

12×3^1/2 is the qnswer of the

Answer 2

Let the triangle be ABC
Let the circle be DEF (the 3 points of contact) (D lies on AB, E on BC, F on AC)
O will be the centre

Join OD , OE, OF and OB , OA , OC
 In Triangle OBE and Triangle OBD

OD is perpendicular to AB and OE is perpendicular to BC, So 900
OE = OD (Radii)
OB = OB (Common) 
Therefore Triangle OBE is congruent to Triangle OBD
Hence, Angle OBE = Angle OBD

Similarly, in the other triangles , we get
Angle OCE = Angle OCA
Angle OAF = Angle OAD

Now Angle CAB, ABC, BCA = 60 degrees (eq. triangle)

Which means Angle OBD = OBE = 30 degrees (half)
Do Tan (Angle OBE)
Tan30 = 1/√3

1/√3 = radius (2 cm) / BE
Therefore BE = 2√3
Similarly, CE = 2√3

So BC = 4√3

Ar(eq. Triangle) = [(Side) 2 √3] / 4
= 48√3 / 4 
= 12√3 cm2