The Triangle
Three is the minimum. Two points can't close space. Three can. And from that one fact, all of geometry grows.
Why does this exist?
Take two points. Connect them. You have a segment — a distance, a direction. But nothing is enclosed. The space on either side of the segment is still open, infinite, free.
Now add a third point, not on that segment. Connect all three.
Something changes. For the first time, a region is trapped inside. It cannot escape. You have enclosed space.
Three is the minimum number of points that can do this. Not two, not one. Three — and no fewer. That is why the triangle is not just another shape. It is the first shape. Every polygon you will ever meet is secretly made of triangles underneath.
Feel the enclosure
Drag any vertex. The enclosed region moves with it — it stretches, tilts, squashes — but it never opens up. The space inside is always captured.
Canvas 400 420
Scale 50
Grid 1
Origin at 200 210
Point A at -3, -2
Point B at 3, -2
Point C at 0, 3
Triangle T vertices A B C
Fill T 0.12 blue
Color T blue
Label A "A"
Label B "B"
Label C "C"
Drag A
Drag B
Drag C
Three points. Three segments. One enclosed region.
That region has an area — a quantity of space that changes as you drag. The triangle is the simplest object in geometry that has an area. A segment has length. A point has neither. Three connected points have both.
The angle sum — find it yourself
Every triangle has three interior angles — one at each vertex. Here they are, shown live as you drag.
Canvas 400 440
Scale 50
Grid 1
Origin at 200 220
Point A at -3, -2
Point B at 3, -2
Point C at 0, 3
Triangle T vertices A B C
Fill T 0.08 blue
Color T blue
Angle alpha at A from B to C
Angle beta at B from C to A
Angle gamma at C from A to B
Color alpha orange
Color beta orange
Color gamma orange
Label alpha "α"
Label beta "β"
Label gamma "γ"
Show angle alpha
Show angle beta
Show angle gamma
Label A "A"
Label B "B"
Label C "C"
Drag A
Drag B
Drag C
Add the three values in the panel. Drag a vertex. Add them again. Drag to something flat, something tall, something lopsided. Add them every time.
The sum is always the same number.
Why is it always 180°?
This is not a coincidence. There is a reason, and it is surprisingly clean.
Draw a line through C parallel to AB. Now look at what happens to the three angles. The angle at A and the angle at B each reappear on that parallel line — one to the left of C, one to the right — as alternate interior angles. Together with the angle at C itself, they sit side by side along a straight line.
A straight line is 180°. So the three angles must sum to 180°. Always.
This proof works for every triangle — flat, tall, obtuse, nearly degenerate. The parallel line always exists. The alternate angles always match. The straight line is always 180°.
Three families by their sides
Drag the triangle into different shapes. Watch the label in the panel change.
Canvas 400 400
Scale 50
Grid 1
Origin at 200 210
Point A at -3, -2
Point B at 3, -2
Point C at 0, 3
Triangle T vertices A B C
Fill T 0.08 blue
Color T blue
Show classification T
Show area T
Show perimeter T
Label A "A"
Label B "B"
Label C "C"
Drag A
Drag B
Drag C
Scalene — all three sides are different lengths. The general case. Most triangles you draw at random will be scalene.
Isosceles — exactly two sides are equal. Try to make one: drag C to sit directly above the midpoint of AB. The two equal sides create a line of symmetry.
Equilateral — all three sides equal. This is the hardest to find by dragging. All three angles are also equal — each exactly 60°. It is the most constrained, most symmetric triangle possible.
The names
A triangle is a polygon with three vertices, three sides, and three interior angles. The angle sum is always 180°.
Triangles are also classified by their angles:
An acute triangle has all three angles less than 90°.
A right triangle has one angle exactly equal to 90°. The side opposite the right angle is called the hypotenuse — the longest side. This triangle is where Pythagoras lives.
An obtuse triangle has one angle greater than 90°. Only one — because if two angles were each greater than 90°, their sum alone would exceed 180°, leaving nothing for the third.
The seed planted here
One triangle above all others changes everything.
Take any right triangle — one with a 90° angle. The three sides have lengths that obey a precise relationship: the square of the hypotenuse equals the sum of the squares of the other two sides.
That relationship is the Pythagorean theorem. It connects triangles to distances, distances to coordinates, and coordinates to every branch of mathematics that follows.
It is the most important single fact in all of geometry. We will reach it soon.