Consider a thin lamina of area A and its differential area dA is shown in Fig. 7.1

Suppose dA is at a distance of x from y-axis and y distance from x-axis

Its distance from point O is r

For calculating are a moment of Inertia about x-axis for dA is given by,

dI_{x} = y^{2}dA

Total moment of inertia can be given by,

I_{x} = _{x} = ^{2} dA

Similarly, for y-axis, total moment of inertia is given by,

I_{y} = _{y} = ^{2} dA

For z-axis,

Moment of inertia can be written as-

dI_{z} = r^{2}dA

Total moment of inertia can be given by,

I_{z} = _{z} = ^{2} dA

Thus, we get

I_{x} = ^{2} dA

I_{y} = ^{2} dA

I_{z} = ^{2} dA

= ^{2} + y^{2}) dA

= ^{2}dA + ^{2} dA

= I_{x} + I_{y}

This can also be written as

i.e.,

I_{z} = I_{x} + I_{y}

**Note:**

S.I. unit of area moment of inertia is L^{4}

Where,

L = Length (m) = m^{4}