Question

find the radius of incircle of an equilateral triangle ABC with side 8cm​

Answer 1

Step-by-step explanation:

The center of the inscribed circle = O = incenter = centroid of the triangle ABC, as it is equilateral. Given AB = 8 cm

Draw the median = perpendicular from A to BC to meet BC at D. Draw a perpendicular from O to AB to meet AB at E. OE is the radius of incircle.

Altitude = AD = √3/2 * AB

AO = 2/3 * AD = AB/√3

AE = AB/2

In the Right angle triangle AOE, AO^2 = AE^2 + OE^2

OE^2 = AB^2 /3 - AB^2/4 = AB^2 /12

Area of Incircle = π OE^2 = π AB^2/12

Area of Equilateral Triangle = 1/2 * √3/2 * AB^2

Unoccupied space inside the triangle = AB^2 [ √3/4 - π/12 ]

= 10.957 cm^2

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