Right angle

In geometry and trigonometry, a right angle is an angle that measures exactly 90 degrees or π/2 radians. This special angle appears when two lines meet to form a perfect "L" shape, like the corner of a square. Right angles are very important in many areas of math and science.

When two lines cross each other at a right angle, they are called perpendicular lines. This idea of being at right angles is also called orthogonality, especially when talking about directions or vectors.

Right angles are also key in triangles. When a triangle has one right angle, it is known as a right triangle. This special type of triangle is the foundation for the study of trigonometry, which helps us understand angles and their relationships.

Etymology

The word "right" in "right angle" may come from the Latin word rectus, which means "upright" or "perpendicular". A similar Greek word is orthos, meaning "straight" or "perpendicular". This idea connects to the concept of orthogonality.

Latin Greek

In elementary geometry

A rectangle is a four-sided shape with four right angles, which are angles that measure exactly 90 degrees. A square also has four right angles and its sides are all the same length. The Pythagorean theorem helps us know when a triangle has a right angle.

Symbols

In special characters, there is a sign for a right angle. It looks like a small square in the corner of an angle in drawings. Some places use a circle with a dot to show a right angle instead.

Euclid

Right angles are very important in Euclid's Elements. In Book 1, definition 10, Euclid describes a right angle as two straight lines crossing to make two equal angles next to each other. These lines are called perpendicular. Euclid also uses right angles to help define smaller angles, called acute angles, and larger angles, called obtuse angles. Two angles that add up to a right angle are called complementary.

Euclid’s Postulate 4 says that all right angles are the same size. This helps us use a right angle as a basic unit to measure other angles. Later mathematicians like Proclus and Saccheri tried to prove this idea, and Hilbert included it in his axiomatization of geometry after building up many other ideas first.

Conversion to other units

A right angle can be shown in many different ways:

⁠1/4⁠ turn
90° (degrees)
⁠π/2⁠ radians
100 grad (also called grade, gradian, or gon)
8 points (of a 32-point compass rose)

Rule of 3-4-5

Carpenters and masons have used a simple way to check if an angle is a true right angle. This method uses the numbers 3, 4, and 5. If you measure three units along one side of the angle and four units along the other side, the distance between the ends of these lines will be exactly five units. This shows that the angle is a right angle.

Thales' theorem

Main article: Thales' theorem

Thales' theorem tells us that if you have an angle drawn inside a semicircle, where the point of the angle is on the curve and the sides of the angle touch the ends of the semicircle, then that angle will always be a right angle. This helps us understand and prove that certain angles are exactly 90 degrees.

Generalizations

The space angle covered by one part of a sphere with three right angles is equal to π/2 sr. This shows how right angles can be used in three dimensions, just like they are used in flat shapes.