The Geometry Hidden Inside the Taylor Series

A dozen cuts to carve a circle from a square — and a dozen hops to trace its arc

Most people who encounter the formula

π/4  =  1 − 1/3 + 1/5 − 1/7 + 1/9 − …

are taught it as an algebraic fact. You take the Taylor series for arctan(x), set x = 1, and out falls π. It appears in textbooks as a curiosity of infinite series — symbol manipulation that happens to produce a circle constant. The geometry is stripped out. The meaning is lost.

But π is geometry. It is the ratio of a circle's circumference to its diameter — a statement about perimeter. Everything else flows from this basic inception. If an infinite series produces π, that series must be doing geometry.

And it is — in two distinct ways. First, the series is an arc length. On a unit circle, arctan(1) = π/4 radians = 45°. Each term of the series is a hop along the circumference: go forward 1 radian (overshoot), come back 1/3 radian (undershoot), go forward 1/5 radian (overshoot again). The hops get shorter. The position oscillates around 45° and converges. You are literally tracing the perimeter of the circle.

Second, the same series is a sequence of area slabs. Inside a unit square, the curve y = 1/(1+x²) encloses exactly π/4 worth of area — one quarter of a unit circle. The Taylor expansion breaks that area into layers: add the full square, subtract the region under x², add back the sliver under x⁴, subtract x⁶. Each term is a geometric cut. You are sculpting a circle's worth of area from a square.

Around 250 BC, Archimedes inscribed polygons in a circle and doubled their sides, watching the perimeter converge on the circumference. The Taylor series does the same thing — once via perimeter, once via area — in the language of algebra. The algebra was never anything other than geometry in disguise.

To the best of our knowledge, this is the first interactive visualisation that presents the Taylor-series derivation of π as an explicitly geometric construction — both as arc-length hops along a circle's perimeter and as area slabs carved from a unit square — placed alongside Archimedes' classical polygon method, making visible what the formalism hides.
The Series — Term by Term
You can also use ← → arrow keys to navigate steps

I. Perimeter — Where π Begins

Arc-Length Hops Along the Circumference
Unit Circle — Arc Being Traced
Archimedes — Polygon Perimeter
Leibniz (1 term)
π/40.785398
Archimedes (4-gon)
Perimeter Convergence
● Archimedes perimeter● Leibniz arc hops--- π/4
Step 1 of 12

II. Area — Same Number, Different Shape

Unit Square — Area Being Carved
Quarter Circle — Same Area
Area Convergence
● Archimedes area● Leibniz area--- π/4
Step 1 of 12

Archimedes approached π by inscribing polygons — adding sides until the perimeter converged. The Taylor series does the same thing twice: once by hopping along the arc (perimeter), once by carving inside the square (area). Both roads lead to π/4. Both are finite geometric operations, iterated toward the infinite. The algebra was never anything other than geometry in disguise.