Triangles – The Strongest Shape in Geometry
Triangles are the simplest polygons and one of the strongest shapes in nature and engineering. Every other polygon can be divided into triangles, making them the foundation of geometry.
The relationship a² + b² = c² is named after the Greek philosopher Pythagoras (c. 570–495 BCE), but the idea was known long before him: a Babylonian clay tablet called Plimpton 322, dated to around 1800 BCE, already lists numerous sets of whole-number side lengths that satisfy the rule — over a thousand years before Pythagoras is credited with proving it in general. Ancient Egyptian surveyors are also believed to have used a 3-4-5 triangle, formed with a simple knotted rope, to mark out perfect right angles when rebuilding field boundaries. That same triangle rigidity — the fact that a triangle's shape cannot change once its three side lengths are fixed, unlike a rectangle which can be pushed into a slanted parallelogram — is exactly why engineers brace bridges, roof trusses, and pylons with triangular frameworks, and why land surveyors and GPS satellites both use triangulation to fix an exact position from a network of triangles.
What Is a Triangle?
A triangle is a closed shape with three straight sides and three angles. The sum of the interior angles of any triangle is always 180°. This is one of the most important facts in all of geometry.
Types of Triangles by Sides
| Type | Sides | Angles |
|---|---|---|
| Equilateral | All three sides equal | All angles = 60° |
| Isosceles | Two sides equal | Base angles equal |
| Scalene | No sides equal | No angles equal |
Types of Triangles by Angles
| Type | Defining Angle |
|---|---|
| Acute triangle | All three angles less than 90° |
| Right triangle | One angle exactly 90° |
| Obtuse triangle | One angle greater than 90° |
The Angle Sum Property
Third angle = 180 − 50 − 70 = 60°.
x + 2x + 3x = 180. 6x = 180. x = 30. Angles: 30°, 60°, 90°. This is a right triangle.
Pythagoras' Theorem
In a right-angled triangle, the square of the longest side (hypotenuse) equals the sum of the squares of the other two sides. See our dedicated Pythagoras' Theorem page for the full proof, its converse, and Pythagorean triples.
a² + b² = c², where c is the hypotenuse (the side opposite the right angle).
c² = 3² + 4² = 9 + 16 = 25. c = √25 = 5.
b² = 13² − 5² = 169 − 25 = 144. b = √144 = 12.
Perimeter and Area of a Triangle
| Measurement | Formula |
|---|---|
| Perimeter | Sum of all three sides: P = a + b + c |
| Area | Half base times height: A = ½ × b × h |
Key Takeaways
- All triangles have interior angles summing to 180°.
- Equilateral: all equal. Isosceles: two equal. Scalene: none equal.
- Pythagoras: a² + b² = c² applies only to right triangles.
- Area = ½ × base × height.
Practice: Angle Sum & Pythagoras’ Theorem
Related Topics
Continue exploring related topics:
- Pythagoras' Theorem – The Rule Behind Every Right Triangle
- Area – How Much Space a Shape Covers
- Circles – The Perfect Shape
- Congruence – Identical Shapes in Every Way
- Coordinate Geometry – Algebra Meets Geometry
- Lines – Paths That Define Shape and Direction
- Geometry – The Mathematics of Shape and Space
- Finding Missing Angles – Putting the Rules Together