Moment of inertia for thin uniform rectangle/hexagon

[rundream]

Homework Statement

the first problem is i have a rectangle split into four triangles the long side labeled a the short side labeled b and the inner angles of the triangles are \theta _{1} and \theta _{2}
(since the traingles are made from a rectangle their are 2 sets of triangles with different \theta)
also not \theta is the angle towards the center of the mass not an outer ridge angle(all together 4 \theta representing all 4 triangles)

Homework Equations

im given the eqaution
I=moment of inertia= (1/12)M(a^{2} + b^{2})
and also earlier in class worked out for isosceles trianlges
I=(1/2)M[1+(1/3)tan^{2}((1/2)\theta)]h^{2}

The Attempt at a Solution

given those 2 equations i have to prove that I=(1/12)M(a^{2}b^{2}) really gives the moment of inertia for full mass.
I started working backwards on the problems replacing M with \sigma h^{2} tan((1/2)\theta)
being that \sigma = M/A and A = h^{2}\ tan((1/2)\theta)
btw M=total mass of object, A=total area
(sorry trying my best to not confuse on problem)
since the height of one triangle is 1/2b or 1/2a depending on which triangle picked
i also replaced the terms for a^{2} and b^{2}

but I am at a lost at this point and don't know where to go past this
i figured if i reverse engineered the problem i may be able to figure the 4 triangles relationship to the total moment of inertia so then i could return the sumed up eqaution into the original eqaution for an isosceles triangle so i can understand how to use the triangles for a hexagon and octagon.

please help if possible with this problem.

 

[Gokul43201]

When you specify a moment of inertia for an object, you need to also specify the axis about which the moment is calculated.

You basically need to write down h and \theta in terms of the sides a,b, and use the fact that m(triangle) = (M/ab)*(area of triangle). If necessary you may need to use the Parallel Axis theorem to have all moments about the same point. Then it's just plugging and adding up the 4 moments to get the total.