Angle

In geometry, an angle is formed when two lines meet at a point. Each of these lines is called a side of the angle, and the point where they meet is called the vertex. Angles help us describe how things are sloped or turned, and they are important in many areas of math and science.

The term angle can refer to both the shape made by the lines and the amount of turn between them. To talk about the amount of turn, people use words like angular measure or measure of angle. This helps us tell the difference between the shape itself and how much it has turned.

Measuring angles is closely connected to circles and turning. We often imagine or define angles using a part of a circle called an arc. This arc is centered at the vertex and lies between the two sides of the angle. Understanding angles helps us work with shapes, buildings, maps, and many other parts of the world around us.

Fundamentals

Angles are shapes made when two lines meet at a point. Each line is called a side of the angle, and the point where they meet is called the vertex. Angles can be thought of in different ways: as 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 line segments meet, like at the corners of triangles or where two planes cross. The sides of the angle split the space around them into two parts: the inside and the outside of the angle. The inside part is sometimes called the angular sector.

We use special symbols to name angles, like ∠, along with points to show which lines we mean. For example, if point A is where lines AB and AC meet, we can call this angle ∠A or ∠BAC. We can also use letters or Greek letters to show how big an angle is.

Angles can be measured in different ways, the most common being degrees (°), radians (rad), and turns. A full angle, where one line turns all the way around back to its start, is 360 degrees, 1 turn, or about 6.28 radians.

The angle addition postulate tells us that if a point D lies inside angle BAC, then the size of angle BAC is the sum of the sizes of angles BAD and DAC. This helps us understand how to add angles together by putting their vertices together and sharing a side.

Types

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

Common angles

• An angle equal to 0° or not turned 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 ("obtuse" meaning "blunt").
• 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 indicate equality between pairs.

Adjacent angles (abbreviated adj. ∠s), are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles side by side or adjacent, sharing an "arm".

Vertical angles are formed when two straight lines intersect at a point producing four angles. A pair of angles opposite each other are called vertical angles, opposite angles or vertically opposite angles (abbreviated vert. opp. ∠s), where "vertical" refers to the sharing of a vertex, rather than an up-down orientation. A theorem states that vertical angles are always congruent or equal to each other. A transversal is a line that intersects a pair of (often parallel) lines and is associated with exterior angles, interior angles, alternate exterior angles, alternate interior angles, corresponding angles, and consecutive interior angles.

Combining angle pairs

When summing two angles (either adjacent or separated in space), three special cases are named complementary, supplementary, and explementary angles.

Complementary angles are angle pairs whose measures sum to a right angle (⁠1/4⁠ turn, 90°, or ⁠π/2⁠ rad). If the two complementary angles are adjacent, their non-shared sides form a right angle. In a right-angle triangle the two acute angles are complementary as the sum of the internal angles of a triangle is 180°. The difference between an angle and a right angle is termed the complement of the angle.

Supplementary angles sum to a straight angle (⁠1/2⁠ turn, 180°, or π rad). If the two supplementary angles are adjacent, their non-shared 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 termed the supplement of the angle.

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

Examples of non-adjacent supplementary angles include the consecutive angles of a parallelogram and opposite angles of a cyclic quadrilateral. For a circle with center O, and tangent lines from an exterior point P touching the circle at points T and Q, the resulting angles ∠TPQ and ∠TOQ are supplementary.

• 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 lies on the inside of that simple polygon. A simple concave polygon has at least one interior angle, that is, a reflex angle.

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

• The supplement of an interior angle is called an exterior angle; that is, an interior angle and an exterior angle form a linear pair of angles. There are two exterior angles at each vertex of the polygon, each determined by extending one of the two sides of the polygon that meet at the vertex; these two angles are vertical and hence are equal. An exterior angle measures the amount of rotation one must make at a vertex to trace the polygon. If the corresponding interior angle is a reflex angle, the exterior angle should be considered negative. Even in a non-simple polygon, it may be possible to define the exterior angle. Still, one will have to pick an orientation of the plane (or surface) to decide the sign of the exterior angle measure.

  In Euclidean geometry, the sum of the exterior angles of a simple convex polygon, if only one of the two exterior angles is assumed at each vertex, will be one full turn (360°). The exterior angle here could be called a supplementary exterior angle. Exterior angles are commonly used in Logo Turtle programs when drawing regular polygons.

• In a triangle, the bisectors of two exterior angles and the bisector of the other interior angle are concurrent (meet at a single point).^: 149 

• In a triangle, three intersection points, each of an external angle bisector with the opposite extended side, are collinear.^: 149 

• In a triangle, three intersection points, two between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended are collinear.^: 149 

• Some authors use the name exterior angle of a simple polygon to mean the explement exterior angle (not supplement!) of the interior angle. This conflicts with the above usage.

Plane-related angles

• The angle between two planes (such as two adjacent faces of a polyhedron) is called a dihedral angle. It may be defined as the acute angle between two lines normal to the planes.
• The angle between a plane and an intersecting straight line is complementary to the angle between the intersecting line and the normal to the plane.

Measuring angles

See also: List of measuring instruments § Angle

Angles are measured using tools like a protractor or by calculating their size from other known values. An angle is formed when two lines meet at a point, called the vertex. We can measure angles in two main ways: using a reference angle or using circular measurement.

One common way to measure angles is by using a reference angle, like a right angle. A right angle is divided into 90 equal parts called degrees. Another system divides a right angle into 100 parts called gradians.

With circular measurement, we imagine placing the angle inside a circle. The vertex is at the center, and the sides of the angle meet the circle’s edge. The space between these points is called an arc. The length of this arc helps us measure the angle. By comparing the arc’s length to the circle’s radius or total distance around, we get a special unit called a radian. This method works for circles of any size because the ratios stay the same.

The ratio of the arc length to the radius gives us the angle in radians. This is a key way to measure angles in math and science.

The value of the angle in radians does not change with the circle’s size. If the circle gets bigger, both the arc and the radius grow equally, keeping the ratio the same.

In math, angles are seen as having no physical dimension, like length or time. This means when we measure an angle in radians, we are simply comparing two lengths, which cancel out. This can sometimes feel odd in equations, but it is the standard way angles are handled.

Angles can also be described as positive or negative to show direction. In a flat, two-dimensional space, we usually start from the right side (positive x-axis) and measure angles from there. Positive angles move upward, while negative angles move downward. This helps us describe rotations clearly.

Angles that have the same size are called equal or congruent. Two angles that end up in the same place but started from different spots are called coterminal angles.

Some related ideas include slope, which tells us how steep a line is, and spread, which measures the difference between two lines using the sine of the angle between them.

Angles between curves

An angle between a straight line and a curve or between two curves that cross each other is the angle between their tangents at the spot where they meet. These angles used to have special names, but we don't use them much anymore.

Bisecting and trisecting angles

Long ago, ancient Greek mathematicians learned how to split an angle into two smaller angles of the same size using just a compass and a straightedge. However, they could only split some angles into three equal parts and not most of them. In 1837, a mathematician named Pierre Wantzel proved that splitting most angles into three equal parts with these tools is not possible.

Dot product and generalisations

In Euclidean space, the angle between two Euclidean vectors is connected to their dot product and their lengths. This gives a way to find the angle between two planes or curved surfaces using their normal vectors, and between skew lines using their vector equations.

To define angles in more general spaces, we use a concept called the inner product instead of the Euclidean dot product. In complex spaces, we adjust the formula to make sure we get real number results. This helps us understand angles between directions defined by vectors, even in more complicated mathematical settings.

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 think of angles as shapes made 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 toward each other when they cross but are not straight. Different thinkers had different ideas about what angles really are.

Vertical angle theorem

The vertical angle theorem tells us that vertically opposite angles are equal. Long ago, a thinker named Thales noticed that when two lines cross, the angles opposite each other are always the same size. He used simple ideas, like all straight angles being equal, to show why this works. When two angles next to each other make a straight line, they add up to 180°. Using this, we can see that the opposite angles must be equal.

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 to show where a place is, with the equator and the Greenwich meridian as important guides.

In astronomy, we use angles to describe where stars and other objects appear in the sky. Astronomers imagine lines from the center of the Earth to these objects and measure the angle between the lines.

We can also describe directions using angles. For example, we can measure how high something is above the horizon, called altitude angle or elevation, and which way it is from north, called azimuth.

Astronomers can even measure how big objects look in the sky using angles. The full moon, for example, looks about half a degree wide from Earth. There are useful rules of thumb, like a closed fist at arm's length looks about 10° wide.