This article is about angles in geometry. For other articles, see Angle (disambiguation)

An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. Angles provide a means of expressing the difference in slope between two rays meeting at a vertex without the need to explicitly define the slopes of the two rays. Angles are studied in geometry and trigonometry.

Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. According to Proclus an angle must be either a quality or a quantity, or a relationship. The first concept was used by Eudemus, who regarded an angle as a deviation from a straight line; the second by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definitions of right, acute, and obtuse angles are certainly quantitative.

The word angle comes from the Latin word angulus, meaning "a corner". The word angulus is a diminutive, of which the primitive form, angus, does not occur in Latin. Cognate are the Latin angere, meaning "to compress into a bend" or "to strangle", and the Greek (angulοs), meaning "crooked, curved"; both are connected with the Aryan or Indo-European root ank-, meaning "to bend".

Units of measure for angles

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In order to measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g. with a compass. The length of the arc s is then divided by the radius of the circle r, and possibly multiplied by a constant k:

     θ = kˑs/r

Since the circumference of a circle is always directly proportional to its radius, and an arc is a fixed fraction of the circle, the measure of the angle is independent of the size of the circle. An angle is often measured with a protractor.

Angles are dimensionless, since they are defined as the ratio of lengths. There are, however, many different units used to measure angles, depending on the choice of the constant k:

• The radian measure of the angle uses k = 1, making the angle of a full circle 2π radians. The SI system of units uses radians as the (derived) unit for angles. This is also roughly subdivided into the mil, which has several definitions in practice. Because of the simple relationship to arc length, radians are a special unit. For example, sines and cosines whose argument is in radians are derivatives of each other (modulo ±). The radian is abbreviated rad.

• The degree measure of the angle uses k = 360/2π, making the angle of a full circle 360 degrees. The symbol for degrees is a small superscript circle, as in 360°. One radian is about 57° and one degree is π/180 rad ≈ 0.0175 rad.
  • Fractions of a degree may be written as a decimal degree, e.g. 42.3753°, but for historic reasons they are also commonly expressed in sexagesimal or degree-minute-second format, using minutes of arc, which are 1/60th of a degree, and seconds of arc, which are 1/60th of a minute of arc and 1/3600th of a degree. The minute of arc is symbolized by a single prime, and the second of arc by a double prime, e.g. 42° 22′ 31″. An intermediate format with decimal minutes is also used, e.g. 42° 22.52′.
  • Minutes of arc are commonly encountered in discussions of external ballistics, as a minute of arc covers almost exactly 1 inch at 100 yards (1 m at 1200 m). A rifle capable of shooting "1 MOA", one minute of arc, can place all shots within 1 inch at 100 yards, 2 inches at 200 yards, etc. Minutes of arc are also used in navigation: a nautical mile was historically defined as a minute of arc along a great circle of the Earth. Seconds of arc routinely show up in astronomical applications, as most objects in the sky have tiny angular diameters (e.g. Venus varies between 10″ and 60″).

• The grad, also called grade, gradian, or gon, uses k = 400/2π, making the angle of a full circle 400 grads, and a right angle 100 grads. It is used mostly in triangulation, and represents the percentage rise or grade of a slope (such as a hill, roof, path, or road).

• The point uses k = 32/2π, making the angle of a full circle 32 points. One point is exactly 11.25°. The point is used in navigation.

• The full circle or full turns or cycle or rotation or revolution uses k = 1/2π, making the angle of a full circle exactly 1; it represents the number or fraction of complete full turns. For example, π/2 rad = 90° = 1/4 full circle. The rotation and revolution are abbreviated rot and rev, respectively, but may also be abbreviated as r, as in rpm (revolutions per minute).

Conventions on measurement

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A convention universally adopted in mathematical writing is that angles given a sign are positive angles if measured counterclockwise, and negative angles if measured clockwise, from a given line. If no line is specified, it can be assumed to be the x-axis in the Cartesian plane. In navigation, bearings are measured from north, increasing clockwise, so a bearing of 45 is north-east. Negative bearings are not used in navigation, so north-west is 315.

In mathematics radians are assumed unless specified otherwise because this removes the arbitrariness of the number 360 in the degree system and because the trigonometric functions can be developed into particularly simple Taylor series if their arguments are specified in radians.

Types of angles

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An angle of π/2 radians or 90°, one-quarter of the full circle is called a right angle.

Two line segments, rays, or lines (or any combination) which form a right angle are said to be either perpendicular or orthogonal:

• Angles smaller than a right angle are called acute angles (less than 90 degrees)
• Angles larger than a right angle are called obtuse angles (more than 90 degrees, less than 180).
• Angles equal to two right angles are called straight angles (equal to 180 degrees).
• Angles larger than two right angles are called reflex angles (more than 180 degrees).

• The difference between an acute angle and a right angle is termed the complement of the angle
• The difference between an angle and two right angles is termed the supplement of the angle.

Some facts

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In Euclidean geometry, the inner angles of a triangle add up to π radians or 180°; the inner angles of a simple quadrilateral add up to 2π radians or 360°. In general, the inner angles of a simple polygon with n sides add up to (n − 2) ×   π radians or (n − 2)  ×  180°.

If two straight lines intersect, four angles are formed. Each one has an equal measure to the angle across from it; these congruent angles are called vertical angles.

If a straight transversal line intersects two parallel lines, corresponding (alternate) angles at the two points of intersection are equal; adjacent angles are supplementary, that is they add to π radians or 180°.

A formal definition

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Using trigonometric functions

A Euclidean angle is completely determined by the corresponding right triangle. In particular, if $\backslash theta$ is a Euclidean angle, it is true that

$\backslash cos\; \backslash theta\; =\; \backslash frac\{x\}\{\backslash sqrt\{x^2\; +\; y^2\}\}$

and

$\backslash sin\; \backslash theta\; =\; \backslash frac\{y\}\{\backslash sqrt\{x^2\; +\; y^2\}\}$

for two numbers $x$ and $y$. So an angle can be legitimately given by two numbers $x$ and $y$.

To the ratio $\backslash frac\{y\}\{x\}$ there correspond two angles in the geometric range , since

$\backslash frac\{\backslash sin\; \backslash theta\; \}\{\backslash cos\; \backslash theta\; \}\; =\; \backslash frac\{\backslash frac\{y\}\{\backslash sqrt\{x^2\; +\; y^2\}\}\}\{\backslash frac\{x\}\{\backslash sqrt\{x^2\; +\; y^2\}\}\}\; =\; \backslash frac\{y\}\{x\}\; =\; \backslash frac\{-y\}\{-x\}\; =\; \backslash frac\{\backslash sin\; (\backslash theta\; +\; \backslash pi)\}\{\backslash cos\; (\backslash theta\; +\; \backslash pi)\; \}$

Using rotations

Suppose we have two unit vectors $\backslash vec\{u\}$ and $\backslash vec\{v\}$ in the euclidean plane $\backslash mathbb\{R\}^2$. Then there exists one positive isometry (a rotation), and one only, from $\backslash mathbb\{R\}^2$ to $\backslash mathbb\{R\}^2$ that maps $u$ onto $v$. Let r be such a rotation. Then the relation $\backslash vec\{a\}\backslash mathcal\{R\}\backslash vec\{b\}$ defined by $\backslash vec\{b\}=r(\backslash vec\{a\})$ is an equivalence relation and we call angle of the rotation r the equivalence class $\backslash mathbb\{T\}/\backslash mathcal\{R\}$, where $\backslash mathbb\{T\}$ denotes the unit circle of $\backslash mathbb\{R\}^2$. The angle between two vectors will simply be the angle of the rotation that maps one onto the other. We have no numerical way of determining an angle yet. To do this, we choose the vector $(1,0)$, then for any point M on $\backslash mathbb\{T\}$ at distance $\backslash theta$ from $(1,0)$ (on the circle), let $\backslash vec\{u\}=\backslash overrightarrow\{OM\}$. If we call $r\_\backslash theta$ the rotation that transforms $(1,0)$ into $\backslash vec\{u\}$, then $\backslash left*\backslash mapsto\backslash theta$ is a bijection, which means we can identify any angle with a number between 0 and $2\backslash pi$.

Angles in different contexts

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In the Euclidean plane, the angle θ between two vectors u and v is related to their dot product and their lengths by the formula

$\backslash mathbf\{u\}\; \backslash cdot\; \backslash mathbf\{v\}\; =\; \backslash cos(\backslash theta)\backslash \; \backslash |\backslash mathbf\{u\}\backslash |\backslash \; \backslash |\backslash mathbf\{v\}\backslash |.$

This allows one to define angles in any real inner product space, replacing the Euclidean dot product · by the Hilbert space inner product <·,·>.

The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr. ἀμφί, on both sides, κυρτόσ, convex) or cissoidal (Gr. κισσόσ, ivy), biconvex; xystroidal or sistroidal (Gr. ξυστρίσ, a tool for scraping), concavo-convex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave.

Two intersecting planes form an angle, called their dihedral angle. It is defined as the acute angle between two lines normal to the planes.

Also a plane and an intersecting line form an angle. This angle is equal to ninety degrees minus the angle between the intersecting line and the line that goes through the point of intersection and is normal to the plane.

Angles in Riemannian geometry

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In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Where U and V are tangent vectors and g_ij are the components of the metric tensor G,

$$

\cos \theta = \frac{g_{ij}U^iV^j} {\sqrt{ \left| g_{ij}U^iU^j \right| \left| g_{ij}V^iV^j \right|}}.

Angles in astronomy

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In astronomy, one can measure the angular separation of two stars by imagining two lines through the Earth, each one intersecting one of the stars. Then the angle between those lines can be measured; this is the angular separation between the two stars.

Astronomers also measure the apparent size of objects. For example, the full moon has an angular measurement of approximately 0.5°, when viewed from Earth. One could say, "The Moon subtends an angle of half a degree." The small-angle formula can be used to convert such an angular measurement into a distance/size ratio.

See also

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• Complementary angles
• Supplementary angles
• Central angle
• Inscribed angle
• Solid angle for a concept of angle in three dimensions.
• Astrological aspect

External links

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• Angle Bisectors at cut-the-knot
• Angle Bisectors and Perpendiculars in a Quadrilateral at cut-the-knot
• Angle Bisectors in a Quadrilateral at cut-the-knot
• Constructing a triangle from its angle bisectors at cut-the-knot
• Convert angles in sexagesimal degree format to decimal degrees, and vice-versa
• All types angles and their properties defined With interactive applets.

Elementary geometry | Trigonometry | Angle

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