Mathematics from the ground up

Every concept, connected.

From fractions to Spivak calculus.

Principia teaches mathematics the way it actually works — unified, visual, and grounded in the why behind every formula. 6th grade through college. Free to start.

Free forever for Tiers 1–2 · No account required to start

The Unit Circle — sin²θ + cos²θ = 1

8 tiers · 60+ concepts · 6th grade through Spivak

Tier 1
Numbers & Fractions
6th grade
Free
Tier 2
Proportions & Equations
7th grade
Free
Tier 3
Algebra I
8th–9th grade
Tier 4
Geometry & Proof
9th–10th grade
Tier 5
Algebra II
10th–11th grade
Tier 6
Pre-Calc & Trig
11th–12th grade
Tier 7
Calculus I
College
Tier 8
Calculus II & Proofs
College+

How it teaches

The why before
the formula.

Every concept starts with a plain-English explanation and a real-world connection — before a single symbol appears. The formula follows understanding, not the other way around.

👁

Live visualizations

Every concept has an animated diagram that responds to what's being taught — not a static image you ignore.

Step-by-step guided walkthroughs

Worked problems where each step is hidden until you're ready. You control the pace.

🔗

Cross-discipline connections

Pythagorean theorem connects to trig identities connects to calculus derivatives. Every module shows where its idea lives in the larger structure of mathematics.

Concept
Guided
Practice
Connections
Problem
Find the derivative of f(x) = x² at x = 3 using the limit definition.
f(x) = x², find f′(3)
1
Identify the structure
The average rate of change between two points approximates the instantaneous slope.
[f(3+h) − f(3)] / h
2
Expand algebraically
Substituting and simplifying reveals the pattern.
(9 + 6h + h² − 9) / h = 6 + h
I understand — show step 3 →

Mathematics is unified

One idea,
many forms.

Principia shows how algebra lives in geometry, how the Pythagorean theorem is secretly a trig identity, and how a parabola and a derivative are the same conversation.

Tier 4 Tier 6
Pythagorean Theorem → Trig Identity
The unit circle is a right triangle with hypotenuse 1. Substituting gives you sin²θ + cos²θ = 1 directly — no memorization required.
a² + b² = c² → cos²θ + sin²θ = 1
Tier 5 Tier 7
Vertex of a Parabola → Derivative = 0
The vertex formula x = −b/2a is derived by setting the derivative of ax² + bx + c equal to zero. Algebra and calculus are the same optimization.
f′(x) = 2ax + b = 0 → x = −b/2a
Tier 1 Tier 8
Fractions → Limits → Derivatives
The intuition behind [f(x+h) − f(x)] / h as h → 0 is just a fraction where the denominator approaches zero. Every calculus student already knows fractions.
rise / run → Δy / Δx → dy/dx
"Mathematics is not a collection of formulas to memorize — it is a way of thinking. Principia teaches the thinking."

Built for every student, from first fractions to epsilon-delta proofs.

Simple pricing

Free to start.
Unlock everything.

No ads. No data sold. Cancel anytime.

Free Forever
Explorer
$0

always free

  • Tier 1 — Numbers, Fractions & Ratios (8 concepts)
  • Tier 2 — Proportions & Equations (7 concepts)
  • Guided walkthroughs for all free concepts
  • Live visualizations
  • Progress saved on your device
  • · Tiers 3–8 locked
  • · Cross-device sync
Download Free
Monthly
Principia Pro
$6.99

per month

  • All 8 tiers — 60+ concepts
  • Algebra through Calculus II
  • Every guided walkthrough
  • Progress syncs across all devices
  • Printable study aids
  • All future content included
  • Cancel anytime
Subscribe Monthly

Subscriptions managed through Apple App Store and Google Play. Auto-renews until cancelled.
Terms of Service  ·  Privacy Policy