Angle

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.

The angle symbol.

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 ἀγκύλος (angulοs), meaning "crooked, curved"; both are connected with the PIE root *ank-, meaning "to bend".

Units of measure for angles

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:

$\theta ={\frac {ks}{r}}$ [[[im cool]]]

Since the circumference of a circle is proportional to its radius, and an arc in an angle is a fixed fraction of the circumference, the measure of the angle is independent of the size of the circle.

Angles are considered dimensionless, since they are defined as the ratio of lengths. There are, however, several units used to measure angles, depending on the choice of the constant k. Decimal subunits are standard, but both binary and sexagesimal subunits are still in use.

• The full circle or full turn or cycle or rotation or revolution uses k = 1/2π, making the angle of 1 full circle = 2π rad = 4 right angles = 400 gon = 360°. The rotation and revolution are abbreviated rot and rev, respectively, but just r in rpm (revolutions per minute).

• The angle of the equilateral triangle = 1/6 full circle = π/3 rad = 60° ≈ 1.047197551 rad is the unit used by the babylonians. This angle is easily constructed. The degree, minute of arc, second of arc are sexagesimal subunits of the babylonian unit.

• The right angle = 1/4 full circle = 3/2 babylonian unit = π/2 rad = 100 gon = 90° is the unit used in Euclid's Elements.

• The radian unit 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. Because of the simple relationship to arc length, radians are a special unit. Sines and cosines whose argument is in radians are derivatives of each other (modulo ±). The radian is abbreviated rad. 1 rad ≈ 0.954929659 babylonian units.

• The mil is approximately equal to a milliradian. There are several definitions.

• The degree unit of 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 advantage of this old sexagesimal subunit is that many simple angles are measured as a whole number of degrees. The problem of having all interesting angles measured as whole numbers is of course insolvable.
  • Fractions of a degree may be written as a decimal degree, e.g. 1 rad = 57.29577951°, but they are also 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. 1 rad = 57°17′45″. An mixed format with decimal minutes is also used, e.g. 1 rad = 57° 17.75′.
  • 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 a decimal subunit of the right angle. A kilometer was historically defined as a centi-gon of arc along a great circle of the Earth, so the kilometer is the decimal analog to the sexagesimal nautical mile. The gon 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 used in navigation uses k = 32/2π. 1 point = 1/32 full circle = 1/8 right angle = 11.25° = 12.5 gon. It is a binary subunit of the full circle. Naming all 32 points on a compass rose is boxing the compass.

• The astronomical hour angle is measured in hours. 1 hour = 1/24 full circle = π/12 rad = 1/6 right angle = 1/4 babylonian unit = 15° = 16.667 gon. The sexagesimal subunits were called minute of time and second of time, even if they are units of angle.

Approximations

• 1° is approximately the width of a pinky finger nail at arm's length
• 10° is approximately the width of a closed fist at arms length.
• 20° is approximately the width of a handspan at arm's length.

Conventions on measurement

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

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:

Right angle

Acute, obtuse, and straight angles (a, b, c). Here, a and b are supplementary angles.

• 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 a straight angle is termed the supplement of the angle.

Some facts

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°.

If the sum of two angles equals 90 degrees, they form a complementary angle. If the sum of two angles equals 180 degrees, they form a supplementary angle.

A formal definition

Using trigonometric functions

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

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

and

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

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

To the ratio ${\frac {y}{x}}$ there correspond two angles in the geometric range $0<\theta <2\pi$, since

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

Using rotations

Suppose we have two unit vectors ${\vec {u}}$ and ${\vec {v}}$ in the euclidean plane $\mathbb {R} ^{2}$. Then there exists one positive isometry (a rotation), and one only, from $\mathbb {R} ^{2}$ to $\mathbb {R} ^{2}$ that maps $u$ onto $v$. Let r be such a rotation. Then the relation ${\vec {a}}{\mathcal {R}}{\vec {b}}$ defined by ${\vec {b}}=r({\vec {a}})$ is an equivalence relation and we call angle of the rotation r the equivalence class $\mathbb {T} /{\mathcal {R}}$, where $\mathbb {T}$ denotes the unit circle of $\mathbb {R} ^{2}$. 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 $\mathbb {T}$ at distance $\theta$ from $(1,0)$ (on the circle), let ${\vec {u}}={\overrightarrow {OM}}$. If we call $r_{\theta }$ the rotation that transforms $(1,0)$ into ${\vec {u}}$, then $\left[r_{\theta }\right]\mapsto \theta$ is a bijection, which means we can identify any angle with a number between 0 and $2\pi$.

Angles in different contexts

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

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

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

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,

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

Angles in astronomy

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

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

External links

• 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
• Angle Estimation -- for basic astronomy.
• Angle definition pages with interactive applets that are also useful in a classroom setting. Math Open Reference