Break the shape before you calculate the answer.
Move a point, deform a quadrilateral, and divide without end. Something survives every disturbance. Four voyages reveal the mathematics you can only see after you shake the diagram.
What survives a shake
Viviani’s Theorem — Three Distances Always Sum to the Height
Pick any point inside an equilateral triangle: the three distances to the sides always add to the same value — exactly the triangle’s height. Move the point and nothing changes.
Wake up the first invariantWhat survives a shake
Deform the figure and hunt the invariant
Move a point or wreck a quadrilateral: one half, parallelism, and a distance sum refuse to disappear.
Connect the Midpoints — Exactly Half
Join the midpoints of a rectangle to form a rhombus. No matter the proportions, the rhombus is always exactly half. Why?
Reveal the hidden structureWherever the Point Lands — Opposite Triangles Match
Pick any point inside a rectangle and join it to the four corners. The two pairs of opposite triangles always have equal area — one auxiliary line proves it.
Reveal the hidden structureThree Medians, Six Equal Areas
A triangle’s three medians split it into six small triangles. Different shapes — yet all six have equal area. Why?
Reveal the hidden structureA Regular Hexagon = Six Equilateral Triangles
Area of a regular hexagon? Draw lines from the center to each vertex and it splits into six equilateral triangles — related to why bees build hexagons.
Reveal the hidden structureViviani’s Theorem — Three Distances Always Sum to the Height
Pick any point inside an equilateral triangle: the three distances to the sides always add to the same value — exactly the triangle’s height. Move the point and nothing changes.
Reveal the hidden structureVarignon’s Theorem — Midpoints of Any Quadrilateral Form a Parallelogram
Take any lopsided quadrilateral. Join the midpoints of its four sides and you always get a parallelogram — no matter how skewed.
Reveal the hidden structureAccounting for curves
Count round areas as if they were straight
Overlap, cut, and unfold leaves, lunes, and circles until their hidden area ledger becomes visible.
The Leaf Inside a Square — Area Without Calculus
A square of side a. Draw two quarter-circles from opposite corners and a leaf appears in the middle. Its area? No calculus, no memorized formula.
Reveal the hidden structureThe Lune of Hippocrates — A Curved Area That Equals a Triangle
A crescent bounded by two arcs — yet its area exactly equals that of a triangle. It is among the earliest surviving exact quadratures of a curvilinear figure.
Reveal the hidden structureArea of a Circle = πr² — Unrolling Sectors into a Rectangle
Slice a circle like a pizza and lay the sectors alternately: it approaches a rectangle of base πr and height r — area πr².
Reveal the hidden structureThe Arbelos — A Curved Sliver Equal to a Circle
Carve two small semicircles out of a big one and a knife-shaped “arbelos” remains. Bounded by three arcs, its area equals exactly one round circle — a figure Archimedes loved.
Reveal the hidden structureScenes that never finish
Fold infinity into a single picture
Series, square numbers, and self-similarity stop being strings of symbols and start filling space.
¼ + 1/16 + 1/64 + … = ⅓ — Infinity at a Glance
An infinite sum that lands exactly on ⅓? Color one of four squares at each scale: each colored square always has two uncolored twins — so exactly one third.
Reveal the hidden structureOdd Numbers Sum to a Square (1+3+5+…=n²)
Add the odd numbers 1, 3, 5, 7, … and you always get a perfect square. Wrapping L-shaped gnomons around a square shows why.
Reveal the hidden structureTriangular Numbers: 1+2+…+n = n(n+1)/2
A fast way to add 1+2+…+n? Stack it as a staircase, mate an upside-down copy, and you get a rectangle — the answer is half.
Reveal the hidden structureThe Golden Rectangle — Cut a Square, Get Another (φ)
Cut a square off a golden rectangle and what remains is another golden rectangle. Repeat and a golden spiral winds inward — the ratio φ that nature and art adore.
Reveal the hidden structureMotion becomes proof
Cut and move until the formula assembles itself
Area formulas and the Pythagorean theorem reappear as two astonishing arrangements of the same pieces.
A Slanted Parallelogram — Cut and Slide into a Rectangle
A slanted parallelogram looks awkward — until you cut the triangle off one end and slide it to the other, making a clean rectangle: base × height at a glance.
Reveal the hidden structureCut a Triangle into a Rectangle — ½ base × height
Why is a triangle ½ base × height? Cut along the midline, rotate the top pieces, and a clean rectangle appears — the formula made visible.
Reveal the hidden structureDoubling the Square (Plato’s Meno)
To double a square’s area, how much longer should the side be? The answer lies in the diagonal — the puzzle Plato posed to a slave boy in the Meno.
Reveal the hidden structureThe Pythagorean Theorem — Rearranging Pieces to a²+b²=c²
Fill one big square two ways with four identical right triangles: once leaving a² and b², once leaving c². Equal leftovers ⟹ a²+b²=c².
Reveal the hidden structureTrapezoid Area — Double It into a Parallelogram
Why is a trapezoid ½(a+b)·h? Flip a copy onto it and you get a parallelogram — so one trapezoid is half.
Reveal the hidden structureRectangle into a Square — Equal Area (Quadrature)
A rectangle and a square of equal area. Two cuts turn the rectangle into the square — the heart of “quadrature,” measuring area by a square for two millennia.
Reveal the hidden structureInradius of a Right Triangle — r = (a+b−c)/2
The inscribed circle of a right triangle — its radius? Just (a+b−c)/2 from the three sides. One tangent fact does it.
Reveal the hidden structure