Find out the Moment of Inertia of a triangular area:

Find out the Moment of Inertia of a triangular area ABC with base b and height d around its base BC. Therefore or otherwise determine the Moment of Inertia around centroidal axis parallel to the base

Solution

Figure illustrates the triangular area ABC with its centroid G, where perpendicular distance from G to BC is = (d/3).

Let a thin strip LM of thickness dy at distance y from the base BC. Letting similar triangles, ALM and ABC,

LM/(d -  y)      =  BC/ d =  b/ d

∴ LM = (b/ d) (d - y)

= b (1 -   (y/d))

∴ Elemental Area dA = b (1 -   (y/d))dy

= bd ³  ((1 /3)-  (1/4 ))=  bd³/12

By using theorem of parallel axes,

IGX = I _BC  -  A  (d/3)²

b d ³  /12 -(bd/2)(d²/9)