Angle

An angle is a shape made when two lines meet at a point. In geometry, each of these lines is called a side of the angle, and the point where they meet is called the vertex.

Angles are important because we use them in building, art, and sports. We measure angles using degrees. A full circle has 360 degrees, which shows how much something can turn.

The size of an angle can be shown using parts of a circle. By using the arc of a circle, we can see exactly how big the angle is. This helps us solve problems in math and science.

Fundamentals

Angles are shapes made when two lines meet at a point. The point where they meet is called the vertex, and each line is a side of the angle. Angles can be the space between the lines, the area between them, or how much you need to turn one line to match the other.

Angles also form when two lines meet, like at the corners of triangles or where two flat surfaces cross. Special symbols and letters are used to name and measure angles, helping us describe their size in geometry. Different units, like degrees and radians, are used to measure how big an angle is.

Types

"Oblique angle" redirects here. For the cinematographic technique, see Dutch angle.

Common angles

• An angle equal to 0° is called a zero angle.
• An angle smaller than a right angle (less than 90°) is called an acute angle.
• An angle equal to 1/4 turn (90° or π/2 rad) is called a right angle. Two lines that form a right angle are said to be normal, orthogonal, or perpendicular.
• An angle larger than a right angle and smaller than a straight angle (between 90° and 180°) is called an obtuse angle.
• An angle equal to 1/2 turn (180° or π rad) is called a straight angle.
• An angle larger than a straight angle but less than 1 turn (between 180° and 360°) is called a reflex angle.
• An angle equal to 1 turn (360° or 2π rad) is called a full angle, complete angle, round angle or perigon.
• An angle that is not a multiple of a right angle is called an oblique angle.

Adjacent and vertical angles

"Vertical angle" redirects here; not to be confused with Zenith angle.

Angles A and B are adjacent.

Angles A and B, and pair C and D are two pairs of vertical angles. [Hatch marks](/wiki/Hatch_mark#Congruency_notation) show that the pairs are equal.

Adjacent angles (abbreviated adj. ∠s), are angles that share a common point and edge but do not overlap. They are angles next to each other, sharing one side.

Vertical angles are formed when two straight lines cross at a point, making four angles. Angles opposite each other are called vertical angles, opposite angles or vertically opposite angles (abbreviated vert. opp. ∠s). A rule says that vertical angles are always the same size. A transversal is a line that crosses two (often parallel) lines and is linked with exterior angles, interior angles, alternate exterior angles, alternate interior angles, corresponding angles, and consecutive interior angles.

Combining angle pairs

When adding two angles, three special cases are named complementary, supplementary, and explementary angles.

Complementary angles are pairs of angles that add up to a right angle (1/4 turn, 90°, or π/2 rad). If the two complementary angles are adjacent, their other sides form a right angle. In a right-angle triangle the two smaller angles are complementary because the angles in a triangle add up to 180°. The difference between an angle and a right angle is called the complement of the angle.

Supplementary angles add up to a straight angle (1/2 turn, 180°, or π rad). If the two supplementary angles are adjacent, their other sides form a straight angle or straight line and are called a linear pair of angles. The difference between an angle and a straight angle is called the supplement of the angle.

Explementary angles or conjugate angles add up to a full angle (1 turn, 360°, or 2π radians). The difference between an angle and a full angle is called the explement or conjugate of the angle.

Examples of non-adjacent supplementary angles include angles next to each other in a parallelogram and angles across from each other in a cyclic quadrilateral. For a circle with center O, and tangent lines from a point outside the circle touching the circle at points T and Q, the angles ∠TPQ and ∠TOQ add up to 180°.

• Angles a and b are complementary angles

• Angles a and b are supplementary angles

Angles `AOB` and `COD` are **explementary** **or conjugate** angles

Polygon-related angles

• An angle that is part of a simple polygon is called an interior angle if it is inside that polygon. A simple concave polygon has at least one interior angle that is a reflex angle.

  In Euclidean geometry, the angles inside a triangle add up to π radians, 180°, or 1/2 turn; the angles inside a simple convex quadrilateral add up to 2π radians, 360°, or 1 turn. For a simple convex polygon with n sides, the interior angles add up to (n − 2)π radians, or (n − 2)180 degrees.

• The amount that makes up an interior angle’s “missing” part is called an exterior angle; an interior angle and an exterior angle together form a linear pair of angles. There are two exterior angles at each corner of the polygon. These angles are equal. An exterior angle shows how much you turn at a corner to follow the polygon’s shape. If the interior angle is a reflex angle, the exterior angle is negative. Even in a complex polygon, it is possible to talk about exterior angles. You need to choose an orientation to decide if the exterior angle is positive or negative.

  In Euclidean geometry, the exterior angles of a simple convex polygon, if you take just one at each corner, add up to one full turn (360°). Exterior angles are often used in Logo Turtle programs when drawing regular polygons.

• In a triangle, the lines that split two exterior angles in half and the line that splits the other interior angle in half all meet at one point.^: 149 

• In a triangle, three points where lines split exterior angles meet the opposite extended side, and these points line up.^: 149 

• In a triangle, three points where lines split angles meet the opposite side, and these points line up.^: 149 

• Some authors use the term exterior angle of a simple polygon to mean the explement exterior angle (not supplement!) of the interior angle. This differs from the meaning above.

Plane-related angles

• The angle between two planes (such as two nearby faces of a polyhedron) is called a dihedral angle. It can be described as the smaller angle between two lines normal to the planes.
• The angle between a plane and a line that cuts through it is related to the angle between that line and the normal to the plane.

Measuring angles

See also: List of measuring instruments § Angle

Angles are measured by tools like a protractor or by using other known values. An angle is made when two lines meet at a point, called the vertex.

There are two main ways to measure angles. One way is to use a reference angle, like a right angle, and split it into equal parts. For example, a right angle can be split into 90 equal parts called degrees, or into 100 parts called gradians.

Another way is to use a circle. The angle is placed inside the circle. The length of the curve between the two points where the lines meet the circle helps find the angle’s size. This is often measured in radians, a special unit for angles.

The size of an angle stays the same no matter the size of the circle used.

In math, angles are seen as having no physical size, unlike length or time. This means when we measure an angle in radians, we are comparing lengths, and the units cancel out.

Angles can also be described as positive or negative to show the direction of turning. In a flat space, positive angles usually turn counter-clockwise, and negative angles turn clockwise.

Angles that are the same size are called equal or congruent. Angles that end up in the same place after full turns are called coterminal angles.

Some related ideas include slope, which is how steep a line is, and spread, which is about the space between two lines.

Angles between curves

The angle between a line and a curve, or between two curves that cross, is the angle between their tangent lines at the point where they meet. Special names for these cases are rarely used today.

Bisecting and trisecting angles

Long ago, mathematicians in ancient Greece learned how to split an angle into two equal parts using just a compass and straightedge. However, they found that they could not always split angles into three equal parts. Much later, in 1837, Pierre Wantzel showed why this method does not work for most angles.

Dot product and generalisations

In Euclidean space, an angle is the space between two lines that meet at a point. We can find this angle using a tool called the dot product. This helps us connect the angle to the lengths of the lines.

This idea can also help us understand angles between flat surfaces and curved lines. We use special vectors that show direction. In more complex spaces, we use something called an inner product to describe angles in similar ways.

History and etymology

The word angle comes from the Latin word angulus, meaning "corner". Related words include the Greek ἀγκύλος (ankylοs) meaning "crooked, curved" and the English word "ankle". These words all connect back to an ancient root meaning "to bend" or "bow".

People have talked about angles for thousands of years, wondering if they are a type of measurement, a shape, or both. Today, we usually think of angles as shapes formed by two lines meeting at a point, and we measure how big they are.

Euclid described a plane angle as the way two lines lean towards each other when they cross but aren’t straight. Different thinkers had different ideas about what angles really are. Some said they were like qualities, others said they were amounts of space, and Euclid chose to see them as relationships between lines.

Vertical angle theorem

The vertical angle theorem tells us that angles opposite each other at the crossing point of two lines are always equal. Long ago, a thinker named Thales noticed this when he saw people in Egypt checking that angles were equal when lines crossed. He used simple ideas—like all straight angles being the same—to show why vertical angles must always match up in size.

Angles in geography and astronomy

In geography, we can find any place on Earth using a special system called a geographic coordinate system. This system uses angles from the center of the Earth, with the equator and the Greenwich meridian as important lines for reference.

In astronomy, we use angles to describe where stars and other objects appear in the sky. By drawing lines from the center of the Earth to two stars, we can measure the angle between them. We can also measure how high something is in the sky using a vertical angle like altitude angle or elevation, and we can find direction using the azimuth from north.

Astronomers also talk about how big objects look in the sky using something called an angular diameter. For example, the full moon looks about half a degree wide from Earth. There are some easy ways to guess angles, like using your hand: a closed fist is about 10° wide, and a hand spread out is about 20° wide. These are just rough guesses.