Visual Geometry
Before reaching for a complicated formula, cut · rotate · draw an auxiliary line. No integration or memorized formula: the answer appears as soon as the pieces move.
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.
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?
Wherever 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.
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.
The 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.
Cut 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.
Doubling 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.
The 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.
Area 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.
Trapezoid 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.
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.
Odd 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.
Rectangle 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.
Triangular 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.
Three Medians, Six Equal Areas
A triangle’s three medians split it into six small triangles. Different shapes — yet all six have equal area. Why?
A 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.
The 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.
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.
Varignon’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.
The 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.
Inradius 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.