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 are studied in geometry and trigonometry.

Measuring angles

In order to measure an angle, a circle centered at the vertex is drawn. The radian measure of the angle is the length of the arc cut out by the angle, divided by the circle's radius. The degree measure of the angle is the length of the arc, divided by the circumference of the circle, and multiplied by 360. The symbol for degrees is a small superscript circle, as in 360°. The grad, also called grade or gon, is an angular measure where the arc is divided by the circumference, and multiplied by 400. It is used mostly in triangulation. The point is used in navigation, and is defined as one thirty-second of a circle, or exactly 11.25°.

2π radians is equal to 360° (a full circle), so one radian is about 57° and one degree is π/180 radians.

Mathematicians generally prefer angle measurements in radians 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. The SI system of units uses radians as the (derived) unit for angles.

Note that angles are dimensionless, since they are defined as the ratio of lengths.

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 perpendicular:

Angles smaller than a right angle are called acute angles; angles larger than a right angle are called obtuse angles. Angles equal to two right angles are called straight angles. Angles larger than two right angles are called reflex angles.

See also:

• Central angle
• Complementary angles
• Inscribed angle
• Supplementary angles

Some facts

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

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 line intersects two parallel lines, corresponding angles at the two points of intersection are equal.

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

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 of two intersecting curves is defined to be the angle between the tangents at the point of intersection.

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

Also a plane and an intersecting line form an angle. This angle is equal to π/2 minus the angle between the intersecting line and the line that goes through the point of intersection and is perpendicular to the plane.

See also solid angle for a concept of angle in three dimensions.

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,

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 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.

Angles in maritime navigation

The modern format of angle used to indicate longitude or latitude is hemisphere degree minute.decimal, where there are 60 minutes in a degree, for instance N 51 23.438 or E 090 58.928.

The obsolete format of angle used to indicate longitude or latitude is hemisphere degree minute' second", where there are 60 minutes in a degree and 60 seconds in a minute, for instance N 51 23′26″ or E 090 58′57″