Mathematics · plane geometry
Geometry shapes that prove themselves
Four identical triangles, one square and a moment of rearranging — and suddenly you see why a² + b² = c². No memorizing formulas here: you cut, slide and rotate and watch what stays the same. You’ll build a hexagon with nothing but a compass, find an area by counting dots on a grid, locate four centers of a triangle and lay out a floor like Escher.
Step-by-step proof
Drag the vertices
When the angle is not a right angle, the difference is described by the law of cosines: c² = a² + b² − 2ab cos γ — see Trigonometry.
Areas of the squares
Pythagorean triples
Whole numbers for which a² + b² = c². Euclid’s formula gives all of them (primitive ones when m and n are coprime and one of them is even):
3² + 4² = 5²
The Babylonian clay tablet Plimpton 322 (c. 1800 BC) contains columns of numbers corresponding to such triples, e.g. 119 and 169 from the triple (119, 120, 169) — more than a thousand years before Pythagoras. More on the history in Trigonometry.
Pythagoras on a map
Drag the pins. The straight-line distance is the hypotenuse of a triangle whose legs run along the streets.
Transformation
Method
Draw dot to dot
Click the dots in order, and finally click the first vertex to close the polygon. With the keyboard: arrows and Enter.
A string 12 m long
Scale calculator
Steps
Three problems that cannot be solved
The ancient Greeks searched for these constructions for more than 2,000 years. The answer came only from algebra: a compass and straightedge can only add, subtract, multiply, divide and take square roots.
- Angle trisection — dividing any angle into three equal parts. Some angles can be trisected (e.g. 90°), but a 60° angle cannot, because that would require solving a cubic equation. This was proved by Pierre Wantzel in 1837.
- Doubling the cube — constructing the edge of a cube with twice the volume, i.e. a segment of length ∛2. A cube root cannot be obtained this way (also Wantzel, 1837).
- Squaring the circle — a square with the same area as a circle would require a segment √π. Ferdinand von Lindemann in 1882 proved that π is transcendental, so this is impossible. See The number π.
“Impossible” here means: impossible to do exactly in finitely many steps. Approximations, as well as tricks with a marked straightedge, have existed since antiquity.
Which regular polygons can you construct?
Show
Triangle
A triangle from three sticks
Set the lengths and try to put a triangle together. The compass draws arcs from the ends of the base — the third vertex must lie on both.
Where are the centers?
- Centroid S — always inside. It divides each median in the ratio 2 : 1, counting from the vertex. A triangle cut out of cardboard balances on a pencil tip at exactly this point.
- Circumcenter O — inside an acute triangle, exactly at the midpoint of the hypotenuse in a right triangle (this is Thales’s theorem on the angle in a semicircle — see Trigonometry), and outside an obtuse one.
- Incenter I — always inside; it is equally far from all three sides.
- Orthocenter H — inside an acute triangle, at the right-angle vertex of a right triangle, and outside an obtuse one.
- Euler line — O, S and H always lie on one line (Leonhard Euler, 1765), and |SH| = 2 · |OS|. In an equilateral triangle all these points coincide.
Shape
Apply a symmetry
Type of tiling
Tile
Drag the points on the bottom and left edges — the top and right edges change along with them, so whatever you cut out on one side grows back on the opposite side.
MathematicsGeometry you can see
1. The Pythagorean theorem
In a right triangle, the square built on the hypotenuse c has the same area as the two squares on the legs a and b together. It also works the other way round: if a² + b² = c², the triangle is a right triangle (this is how you can lay out a right angle with a rope with 12 evenly spaced knots: 3 + 4 + 5).
2. Areas without memorizing
Area does not change when you cut a shape up and rearrange it. A parallelogram is a “tilted” rectangle, a triangle is half a parallelogram, a trapezoid is half a parallelogram with base a + b, and a circle cut into ever thinner slices arranges itself into a rectangle πr × r.
On a grid, counting dots is enough: Pick’s theorem (Georg Pick, 1899) says that a polygon with vertices at grid points has area A = I + B/2 − 1, where I is the number of dots inside and B the number on the boundary. For any polygon, the shoelace formula (Gauss’s area formula) works:
Among all shapes with the same perimeter, the circle has the largest area (the isoperimetric inequality). The same principle in space is why soap bubbles are round.
3. Similarity and scale
When you enlarge a shape k times, every segment grows k times, the area k² times, and the volume of a solid k³ times (see Solids). A map scale of 1 : 25,000 means that 1 cm on the map is 25,000 cm = 250 m on the ground.
4. Compass and straightedge
An unmarked straightedge draws a line through two points; a compass draws a circle with a given center through a given point. New points arise only as intersections. That is enough to find the midpoint of a segment, bisect an angle or build a regular hexagon — but not to trisect an angle.
5. Special points of a triangle
Three medians, three perpendicular bisectors, three angle bisectors and three altitudes — each trio meets at a single point. The interior angle of a regular polygon with n sides is (n − 2) · 180° / n, so only triangles (6 × 60°), squares (4 × 90°) and hexagons (3 × 120°) fit around a vertex without gaps.
Simplifications: the “water” proof of Pythagoras is a diagram (real models rotate and the water flows through channels); the Escher tile uses only translations, without rotations; the giant ant is a scaling calculation, not a biological model.
Did you know…?
- One of the proofs of the Pythagorean theorem was invented by James Garfield, a congressman from Ohio — he published it in 1876, and in 1881 he became President of the United States. It is the “trapezoid” proof in the Pythagoras tab.
- In 1796 the 18-year-old Carl Friedrich Gauss discovered how to construct a regular 17-gon with compass and straightedge — the first new such polygon since Euclid. Reportedly, this decided that he became a mathematician rather than a philologist.
- In 2023 David Smith, Joseph Myers, Craig Kaplan and Chaim Goodman-Strauss announced the “hat” — a single tile that can cover the plane only in a way that never repeats. Before that, nobody knew whether such an “einstein” (German for “one stone”) existed at all.
Challenges
- Assemble the Pythagoras proof step by stepIn “Pythagoras” → “Rearranging”, use the “Next step” button to go through to the end of the proof.not completed
- A polygon of area 10 on the gridIn “Areas” → “Pick’s theorem”, draw a closed polygon with area exactly 10 and check the formula.not completed
- Construct a regular hexagonIn “Compass and straightedge”, choose the hexagon, turn on “Do it yourself” and complete all the steps.not completed
- Orthocenter outside the triangleIn “Special points”, turn on the altitudes and shape the triangle so that H ends up outside.not completed
- Pentagons won’t fill the planeIn “Tilings”, arrange regular pentagons around one vertex and see what happens.not completed
- Your own Escher-style tileBend both edges of the square — the bottom and the left — by dragging the points.not completed
Challenges check themselves. Your progress is saved in this browser.