Perpendicular axis theorem — I_z = I_x + I_y — proof, restrictions: Foundations

Before you can prove anything, you must be able to read everything. Below is every symbol, picture, and idea the parent note leans on, ordered so each one is built from the one before it.

1. A point in space: coordinates $(x, y, z)$

The picture: three arrows meeting at a corner, like the corner of a room where two walls meet the floor.

Why the topic needs it: the whole theorem is about where each bit of mass sits. Without a way to name positions, we cannot write down any distance, and distance is the heart of everything here.

Figure — Perpendicular axis theorem — I_z = I_x + I_y — proof, restrictions

2. A tiny piece of mass: $d m$

The picture: a solid plate dissolved into a cloud of dots, each dot one $d m$ .

Why the topic needs it: real objects are spread out, not concentrated at one point. To handle the whole body we treat each crumb separately, then add them all up.

3. Adding up crumbs: the integral $\int \dots d m$

The picture: the cloud of dots, each carrying a little value, all funnelled into one running total.

Why the topic needs it: the final line of the proof adds two integrals into one. That step is only allowed because of this splitting rule.

4. Distance from a point to an axis (Pythagoras)

This is the most important picture on the page. Spend time here.

Why does this need Pythagoras? Because to find that shortest distance, we build a right triangle out of the point's coordinates and read off the hypotenuse.

WHY this tool and not another: we want a distance built from horizontal and vertical offsets that meet at a right angle. Pythagoras is exactly the machine that turns two perpendicular legs into one straight distance — no other rule does this job.

Now apply it to each axis. To measure how far a point is from an axis, you use the other two coordinates (the ones across from that axis):

Why the topic needs it: these three expressions are the entire skeleton of the proof. Everything after Step 2 is just plugging these in.

5. Moment of inertia: $I$

Read it in plain words: "for every crumb, take its distance to the axis, square it, multiply by the crumb's mass, and add all these up."

WHY the distance is squared, and WHY perpendicular: a crumb far from the axis sweeps a big circle when it spins, so it resists much more — and the resistance grows with the square of the radius, not just the radius. We use the perpendicular distance because only the sideways reach from the spin-line matters; sliding along the axis doesn't change the circle a crumb travels.

Why the topic needs it: the theorem is a statement about three moments of inertia — $I_{z}$ , $I_{x}$ , $I_{y}$ . This is the object we compute. See Moment of inertia — definition for the fuller story.

6. The three characters: $I_{z}$ , $I_{x}$ , $I_{y}$

The picture: the coin analogy — $I_{z}$ is a fidget-top spin, $I_{x}$ and $I_{y}$ are two end-over-end flips at right angles to each other.

Why the topic needs it: these are the exact three symbols in $I_{z} = I_{x} + I_{y}$ . Once you know each one is "hardness to spin about that axis," the boxed formula reads as a plain English sentence.

7. Symmetry — the free shortcut

The picture: a disc — rotate it and every diameter looks the same as every other, so no diameter can be "harder" than another.

Why the topic needs it: in nearly every worked example the theorem is combined with $I_{x} = I_{y}$ to get an answer in one line, with no integration. See Symmetry arguments in MOI. But beware — symmetry is a convenience, never a requirement; the theorem holds even for a rectangle where $I_{x} \neq = I_{y}$ .

How the foundations feed the topic

Read top to bottom: positions let us name crumbs; Pythagoras turns positions into distances; the integral adds crumbs; together they define $I$ ; three copies of $I$ plus the flat-body fact ( $z = 0$ ) plus the summing rule give the theorem.

Equipment checklist

Cover the right-hand side and answer each before revealing.

What do the three numbers

(x, y, z)

tell you?

How far to walk along each of the three mutually perpendicular axes to reach a point.

What does

d m

mean?

A tiny crumb of mass, small enough to sit at a single point.

What does

\int f d m

mean in plain words?

Add up the value

f

over every mass crumb in the body.

State the linearity rule that lets us combine integrals.

\int (a + b) d m = \int a d m + \int b d m

State Pythagoras' theorem.

For legs

p, q

and hypotenuse

h

h^{2} = p^{2} + q^{2}

Perpendicular distance² of point

(x, y, z)

from the

z

-axis?

x^{2} + y^{2}

Perpendicular distance² of point

(x, y, z)

from the

x

-axis?

y^{2} + z^{2}

Define moment of inertia as an integral.

I = \int r_{⊥}^{2} d m

, with

r_{⊥}

the perpendicular distance to the axis.

Why is the distance squared in

I

Resistance to spinning grows with the square of the radius a crumb sweeps.

What does "planar body" mean in coordinates?

Every mass point has

z = 0

; the body lies flat in the

x y

-plane.

When can you claim

I_{x} = I_{y}

When the body looks identical about both in-plane axes (a symmetry argument) — never as an automatic assumption.

Connections

1.5.07 Perpendicular axis theorem — I_z = I_x + I_y — proof, restrictions (Hinglish) — the parent topic these foundations serve.
Moment of inertia — definition — the $\int r^{2} d m$ built here in full.
Symmetry arguments in MOI — the $I_{x} = I_{y}$ shortcut.
Parallel axis theorem — the sister theorem; needs the same distance-and-integral toolkit.
Radius of gyration — repackages $I$ once you have it.