Right angle

In geometry and trigonometry, a right angle is an angle that measures exactly 90 degrees or π/2 radians. This special angle looks like 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 important in triangles. When a triangle has one right angle, it is known as a right triangle. This special type of triangle helps us learn about angles and how they relate to each other.

Etymology

The word "right" in "right angle" may come from the Latin word rectus, meaning "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. Right angles measure exactly 90 degrees. A square also has four right angles, and all its sides are the same length. The Pythagorean theorem helps us find out 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 important in Euclid's Elements. Euclid says a right angle is when two straight lines cross 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, Saccheri, and Hilbert included this idea in their work on geometry.

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 a simple way to check if an angle is a right angle. They use 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 draw an angle inside a semicircle, with the point of the angle on the curve and the sides touching the ends of the semicircle, that angle will always be a right angle. This helps us know when an angle is exactly 90 degrees.

Generalizations

A part of a sphere with three right angles covers a space angle of π/2 sr. This shows how right angles work in three dimensions, like they do in flat shapes.