🔺Triangles

Camera tripods stand firm, bike frames do not wobble, and the pyramids have held for millennia — all these dependable structures hide the same shape: the triangle. It is the polygon with the fewest sides, and it props up half the engineered world.

Three sides, three angles

A triangle has 3 sides, 3 corners and 3 interior angles. Tear the three corners off a paper triangle and push them together: they fit perfectly into a straight line,

$\angle A + \angle B + \angle C = 180^\circ$

Fat or skinny, huge or tiny, the angles of every triangle add up to exactly 180° — half a turn, never more, never less.

One angle missing? Fill it in

The 180° rule has one great use: know two angles, and the third cannot escape. Say a triangle has angles of $50^\circ$ and $60^\circ$. Then

$\angle C = 180^\circ - 50^\circ - 60^\circ = 70^\circ$

With all three angles on the table, the triangle's "personality" is fixed — and the acute, right and obtuse angles from Angles are just what you need to classify it.

Sorting by sides, sorting by angles

By side length, three types:

• Equilateral: all sides equal, every angle $60^\circ$;
• Isosceles: two sides equal, and the two base angles equal;
• Scalene: no two sides alike.

By angle size, also three: acute (all angles under $90^\circ$), right (one angle exactly $90^\circ$), obtuse (one angle over $90^\circ$). Right triangles carry a famous theorem of their own — the Pythagorean theorem.

Leg a3Leg b4

3² + 4² = 9 + 16 = 25 = c²

Hypotenuse c c = √25 ≈ 5.00整数勾股数 · Pythagorean triple!

Drag the two legs around: a square grows on every side, and the square on the hypotenuse always equals the two leg squares combined. That is a secret only right triangles keep.

The most rigid shape

Nail four sticks into a quadrilateral frame and a light push skews it into a parallelogram. Nail three into a triangle and nothing will budge it. Fix the three side lengths and the shape is locked — this is called rigidity. That is why roof trusses, bridges and basketball hoops bristle with triangles, while folding clothes racks and scissor gates exploit the very flexibility of quadrilaterals.

Check yourself

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Quick quiz

↺ Try again

1. 1. Two angles are 50° and 60°. What is the third?

   60°70°80°

2. 2. What is each angle of an equilateral triangle?

   30°60°90°

3. 3. Why are bridges and roof trusses full of triangles?

   They look goodThey save materialTriangles cannot deform