The Right Triangle

A right triangle has one angle measuring exactly $90°$ — a right angle. It's the foundation of trigonometry and satisfies the famous Pythagorean theorem: $a^2 + b^2 = c^2$.

Keywords: right triangle, right angle, Pythagorean theorem, hypotenuse, legs, trigonometry, 90 degrees

Difficulty: beginner

About Right Triangles

A right triangle (also called a right-angled triangle) is a triangle with exactly one interior angle measuring $90°$ — a right angle. That little square symbol in the corner tells you it's a right angle!

The Parts of a Right Triangle

The Right Angle

The $90°$ angle is what makes this triangle special. The vertex where the right angle occurs is called the right angle vertex. In our diagram, that's point $C$.

The Sides Have Special Names

Unlike regular triangles, right triangles have special names for their sides:

• Legs: The two shorter sides that form the right angle (sides $a$ and $b$). These are perpendicular to each other — they meet at exactly $90°$.
• Hypotenuse: The longest side (side $c$), always opposite the right angle. It never touches the right angle vertex.

Fun fact: The hypotenuse is always the longest side. This isn't just a coincidence — it follows mathematically from the Pythagorean theorem!

The Pythagorean Theorem

Right triangles obey one of the most famous equations in all of mathematics:

a^2 + b^2 = c^2
    

This says: if you square the lengths of the two legs and add them together, you get the square of the hypotenuse.

Example: A triangle with legs $a = 3$ and $b = 4$ has hypotenuse:

c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
    

This $3$-$4$-$5$ triangle is one of the most famous right triangles in history!

A Bit of History

Right triangles have been studied for thousands of years. Ancient civilizations including the Egyptians, Babylonians, and Indians knew about the special properties of right triangles long before the Greeks formalized them.

The relationship $a^2 + b^2 = c^2$ is named after the Greek mathematician Pythagoras (c. 570–495 BCE), though evidence suggests this relationship was known even earlier in other cultures.

Ancient construction trick: The Egyptians used a rope with 12 equally spaced knots to create a $3$-$4$-$5$ right triangle. This helped them build perfect right angles for the pyramids!

Special Right Triangles

Some right triangles have side ratios that make calculations much easier:

Cool Properties

• The two acute angles always add up to $90°$ (since the right angle is $90°$ and all three angles sum to $180°$).
• The circumcenter (center of the circle through all vertices) lies exactly at the midpoint of the hypotenuse.
• The orthocenter (where the altitudes meet) is at the right angle vertex itself!
• The area is simply: $\text{Area} = \frac{1}{2} \times \text{leg}_1 \times \text{leg}_2$

Where You'll See Right Triangles

Right triangles are everywhere:

• Architecture: Ensuring walls are perpendicular
• Navigation: Calculating distances and bearings
• Trigonometry: The basis for sine, cosine, and tangent
• Engineering: Structural design and force calculations
• Video games: 3D graphics and collision detection

Right triangles are the foundation of trigonometry — the entire subject is built on the ratios of sides in right triangles!