# Angle

# Angle

Units of measurement of an angle

Geometric characteristics of angles

An angle is a geometric figure created by two line segments that extend from a single point or two planes which extend from a single line. The size of an angle, measured in units of degrees or radians, is related to the amount of rotation required to superimpose one of its sides on the other. First used by ancient civilizations, angles continue to be an important tool in science and industry today. Angle is derived from the Latin word angulus, which means corner.

The study of angles has been known since the time of the ancient Babylonians (4000–300 BC). These people used angles for measurement in many areas such as construction, commerce, and astronomy. The ancient Greeks developed the idea of an angle further and were even able to use it to calculate the circumference of the Earth and the distance to the moon.

A geometric angle is formed by two lines (rays) that intersect at a common endpoint called the vertex. The two rays are known as the sides of the angle. An angle can be specified in various ways. If the vertex of an angle is at point P, for example, then the angle could be denoted by ∠P. It can be further described by using a point from each ray. Thus, the angle ∠OPQ would have the point O on one ray, a vertex at point P, and the point Q on the remaining ray. A single number or character that is placed on it can also denote an angle. The most common character used is the Greek letter θ (theta).

## Units of measurement of an angle

An angle is commonly given an arithmetic value that describes its size. To specify this value, an angle is drawn in a standard position on a coordinate system, with its vertex at the center and one side, called the initial side, along the x axis. The value of the angle then represents the amount of rotation needed to move from the initial side to the other side, called the terminal side. The direction of rotation indicates the sign of the angle. Traditionally, a counterclockwise rotation gives a positive value and a clockwise rotation gives a negative value. The three terms that are typically used to express the value of an angle include revolutions, degrees, or radians.

The revolution is the most natural unit of measurement for an angle. It is defined as the amount of rotation required to go from the initial side of the angle all the way around back to the initial side. One way to visualize a revolution is to imagine spinning a wheel around one time. The distance traveled by any point on the wheel is equal to one revolution. An angle can then be given a value based on the fraction of the distance a point travels divided by the distance traveled in one rotation. For example, an angle represented by a quarter (one-fourth) turn of the wheel is equal to 0.25 rotations.

A more common unit of measurement for an angle is the degree. This unit was used by the Babylonians as early as 1000 BC. At that time, they used a number system based on the number 60, so it was natural for mathematicians of the day to divide the angles of an equilateral triangle into 60 individual units. These units became known as degrees. Since six equilateral triangles can be evenly arranged in a circle, the number of degrees in one revolution became 6 × 60 degrees = 360 degrees. The unit of degrees was subdivided into 60 smaller units called minutes, and in turn, these minutes were subdivided into 60 smaller units called seconds. Consequently, the notation for an angle that has a value, for example, of 44 degrees, 15 minutes, and 25 seconds would be 44° 15′ 25′.

An angle may be measured with a protractor, which is a flat instrument in the shape of a semi-circle. There are marks on its outer edges that subdivide it into 180 evenly spaced units, or degrees. Measurements are taken by placing the midpoint of the flat edge over the vertex of the angle and lining the 0° mark up with the initial side. The number of degrees can be read off at the point where the terminal side intersects the curve of the protractor.

Another unit of angle measurement, used extensively in trigonometry, is the radian. This unit relates a unique angle to each real number. Consider a circle with its center at the origin of a graph and its radius along the x axis. One radian is defined as the angle created by a counterclockwise rotation of the radius around the circle such that the length of the arc traveled is equal to the length of the radius. Using the formula for the circumference of a circle, it can be shown that the total number of radians in a complete revolution of 360° is 2π rad (short for radian), or approximately 6.2831852 rad where π equals 3.1415926. Given this relationship, it is possible to convert between a degree and a radian measurement.

## Geometric characteristics of angles

An angle is typically classified into four categories, including acute, right, obtuse, and straight. An acute angle is one that has a degree measurement greater than 0° but less than 90°. A right angle has a 90° angle measurement; an obtuse angle has a measurement greater than 90° but less than 180°; and a straight angle, which looks like a straight line, has a 180° angle measurement.

Two angles are known as congruent angles if they have the same measurement. If their sum is 90°, then they are said to be complementary angles. If their sum is 180°, they are supplementary angles. Angles can be bisected (divided in half) or trisected (divided in thirds) by rays protruding from the vertex.

When two lines intersect, they form four angles. The angles directly across from each other are known

### KEY TERMS

**Congruent angles—** Angles that have the same measurement.

**Degree—** A unit of measurement used to describe the amount of revolution of an angle denoted by the symbol°. There are 360° in a complete revolution.

**Initial side—** The ray of an angle that falls on the x axis when an angle is in its standard position.

**Radian—** A unit of angle measurement that relates the amount of revolution of an angle in terms of a the radius of a circle. There are 2π radians in a complete revolution.

**Terminal side—** The ray of an angle in its standard position that extends away from the x axis.

**Vertex—** The point at which the two sides of an angle meet.

**Vertical angles—** Angles created by the intersection of two lines that are directly across from each other and share a common vertex.

as vertical angles and are congruent. The neighboring angles are called adjacent because they share a common side. If the lines intersect such that each angle measures 90°, the lines are then considered perpendicular or orthogonal.

In addition to size, angles also have trigonometric values associated with them such as sine, cosine, and tangent. These values relate the size of an angle to a given length of its sides. These values are particularly important in areas such as navigation, astronomy, and architecture. For instance, astronomers measure the angular separation of two stars by drawing a line from the Earth to each star. The angle between these two lines is the angular separation of the two stars.

*See also* Geometry.

Perry Romanowski

# Angle

# Angle

An angle is a geometric figure created by two line segments that extend from a single **point** or two planes which extend from a single line. The size of an angle, measured in units of degrees or radians, is related to the amount of **rotation** required to superimpose one of its sides on the other. First used by ancient civilizations, angles continue to be an important tool to science and industry today.

The study of angles has been known since the time of the ancient Babylonians (4,000-300 b.c.). These people used angles for measurement in many areas such as construction, commerce, and **astronomy** . The ancient Greeks developed the idea of an angle further and were even able to use them to calculate the circumference of **Earth** and the **distance** to the **moon** .

A geometric angle is formed by two lines (rays) that intersect at a common endpoint called the vertex. The two rays are known as the sides of the angle. An angle can be specified in various ways. If the vertex of an angle is at point P, then the angle could be denoted by *L*P. It can be further described by using a point from each ray. For example, the angle *L*OPQ would have the point O on one ray, a vertex at point P, and the point Q on the remaining ray. An angle can also be denoted by a single number or character which is placed on it. The most common character used is the Greek letter θ (theta).

## Units of measurement of an angle

An angle is commonly given an **arithmetic** value which describes its size. To specify its this value, an angle is drawn in a standard position on a coordinate system, with its vertex at the center and one side, called the initial side, along the x axis. The value of the angle then represents the amount of rotation needed to get from the initial side to the other side, called the terminal side. The direction of rotation indicates the sign of the angle. Traditionally, a counterclockwise rotation gives a positive value and a clockwise rotation gives a **negative** value. The three terms which are typically used to express the value of an angle include revolutions, degrees, or radians.

The revolution is the most natural unit of measurement for an angle. It is defined as the amount of rotation required to go from the initial side of the angle all the way around back to the initial side. One way to visualize a revolution is to imagine spinning a wheel around one time. The distance traveled by any point on the wheel is equal to one revolution. An angle can then be given a value based on the fraction of the distance a point travels divided by the distance traveled in one rotation. For example, an angle represented by a quarter turn of the wheel is equal to .25 rotations.

A more common unit of measurement for an angle is the degree. This unit was used by the Babylonians as early as 1,000 b.c. At that time, they used a number system based on the number 60, so it was natural for mathematicians of the day to divide the angles of an equilateral triangle into 60 individual units. These units became known as degrees. Since six equilateral triangles can be evenly arranged in a **circle** , the number of degrees in one revolution became 6 × 60 = 360. The unit of degrees was subdivided into 60 smaller units called minutes and in turn, these minutes were subdivided into 60 smaller units called seconds. Consequently, the notation for an angle which has a value of 44 degrees, 15 minutes, and 25 seconds would be 44° 15' 25".

An angle may be measured with a protractor, which is a flat instrument in the shape of a semi-circle. There are marks on its outer edges which subdivide it into 180 evenly spaced units, or degrees. Measurements are taken by placing the midpoint of the flat edge over the vertex of the angle and lining the 0° mark up with the initial side. The number of degrees can be read off at the point where the terminal side intersects the **curve** of the protractor.

Another unit of angle measurement, used extensively in **trigonometry** , is the radian. This unit relates a unique angle to each real number. Consider a circle with its center at the origin of a graph and its radius along the x-axis. One radian is defined as the angle created by a counterclockwise rotation of the radius around the circle such that the length of the **arc** traveled is equal to the length of the radius. Using the formula for the circumference of a circle, it can be shown that the total number of radians in a complete revolution of 360° is 2π. Given this relationship, it is possible to convert between a degree and a radian measurement.

## Geometric characteristics of angles

An angle is typically classified into four categories including acute, right, obtuse, and straight. An acute angle is one which has a degree measurement greater than 0° but less than 90°. A right angle has a 90° angle measurement. An obtuse angle has a measurement greater than 90° but less than 180°, and a straight angle, which looks like a straight line, has a 180° angle measurement.

Two angles are known as congruent angles if they have the same measurement. If their sum is 90°, then they are said to be complementary angles. If their sum is 180°, they are supplementary angles. Angles can be bisected (divided in half) or trisected (divided in thirds) by rays protruding from the vertex.

When two lines intersect, they form four angles. The angles directly across from each other are known as vertical angles and are congruent. The neighboring angles are called adjacent because they share a common side. If the lines intersect such that each angle measures 90°, the lines are then considered **perpendicular** or orthogonal.

In addition to size, angles also have trigonometric values associated with them such as sine, cosine, and tangent. These values relate the size of an angle to a given length of its sides. These values are particularly important in areas such as navigation, astronomy, and architecture.

See also Geometry.

Perry Romanowski

## KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .**Congruent angles**—Angles which have the same measurement.

**Degree**—A unit of measurement used to describe the amount of revolution of an angle denoted by the symbol °. There are 360° in a complete revolution.

**Initial side**—The ray of an angle which falls on the x-axis when an angle is in its standard position.

**Radian**—A unit of angle measurement that relates the amount of revolution of an angle in terms of a the radius of a circle. There are 2π radians in a complete revolution.

**Terminal side**—The ray of an angle in its standard position which extends away from the x-axis.

**Vertex**—The point at which the two sides of an angle meet.

**Vertical angles**—Angles created by the intersection of two lines which are directly across from each other and share a common vertex.

# angle

an·gle^{1 } / ˈanggəl/ •
n. 1. the space (usually measured in degrees) between two intersecting lines or surfaces at or close to the point where they meet. ∎ a corner, esp. an external projection or an internal recess of a part of a building or other structure: *a skylight in the angle of the roof.* ∎ slope; a measure of the inclination of two lines or surfaces with respect to each other, equal to the amount that one would have to be turned in order to point in the same direction as the other: *sloping at an angle of 33° to the horizontal.* ∎ a position from which something is viewed or along which it travels or acts, often as measured by its inclination from an implicit horizontal or vertical baseline: *camera angles.*2. a particular way of approaching or considering an issue or problem: *discussing the problems from every conceivable angle.* ∎ one part of a larger subject, event, or problem: *a black prosecutor who downplayed the racial angle.* ∎ a bias or point of view: *Zimmer saw the world from an angle that few could understand.*3. Astrol. each of the four mundane houses (the first, fourth, seventh, and tenth of the twelve divisions of the heavens) that extend counterclockwise from the cardinal points of the compass.•
v. [tr.] direct or incline at an angle: *Anna angled her camera toward the tree.* ∎ [intr.] move or be inclined at an angle: *the cab angled across two lanes and skidded to a stop.* ∎ [tr.] present (information) to reflect a particular view or have a particular focus.PHRASES: at an angle in a direction or at an inclination markedly different from parallel, vertical, or horizontal with respect to an implicit baseline: *she wore her beret at an angle.*from all angles from every direction or point of view: *looking at the problem from all angles.*

^{2 }• v. [intr.] fish with rod and line:

*there are no big fish left to*∎ seek something desired by indirectly prompting someone to offer it:

**angle for**.*Ralph had begun to*• n. archaic a fishhook.

**angle for**an invitation.# Angle

**Angle** a member of a Germanic people, originally inhabitants of what is now Schleswig-Holstein, who came to England in the 5th century ad. The Angles founded kingdoms in Mercia, Northumbria, and East Anglia and gave their name to England and the English.

The name comes from Latin *Angli* ‘the people of Angul’, a district of Schleswig (now in northern Germany), so named because of its angular shape.

not Angles but Angels comment attributed to Gregory the Great (ad *c.*540–604), on seeing fair-haired English slaves in Rome; the story is oral tradition, based on Bede's Historia Ecclesiastica.

# Angle

# Angle

An·gle / ˈanggəl/ • n. a member of a Germanic people, originally inhabitants of what is now Schleswig-Holstein, who migrated to England in the 5th century ad.

# angle

# angle

**angle** Measure of the inclination of two straight lines or planes to each other. One complete revolution is divided into 360 degrees or 2π radians. One degree may be subdivided into 60 minutes, and one minute into 60 seconds.

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**angle**