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A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of a plane angle in which one full rotation is 360 degrees. [4] It is not an SI unit—the SI unit of angular measure is the radian—but it is mentioned in the SI brochure as an accepted unit. [5]
One could say, "The Moon's diameter subtends an angle of half a degree." The small-angle formula can convert such an angular measurement into a distance/size ratio. Other astronomical approximations include: 0.5° is the approximate diameter of the Sun and of the Moon as viewed from Earth. 1° is the approximate width of the little finger at ...
One radian is defined as the angle at the center of a circle in a plane that subtends an arc whose length equals the radius of the circle. [6] More generally, the magnitude in radians of a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, =, where θ is the magnitude in radians of the subtended angle, s is arc length, and r is radius.
Degrees are traditionally used in navigation, surveying, and many applied disciplines, while radians are more common in mathematics and mathematical physics. [ 9 ] The angle φ is defined to start at 0° from a reference direction , and to increase for rotations in either clockwise (cw) or counterclockwise (ccw) orientation.
The standard "physics convention" 3-tuple set (,,) conflicts with the usual notation for two-dimensional polar coordinates and three-dimensional cylindrical coordinates, where θ is often used for the azimuth. [3] Angles are typically measured in degrees (°) or in radians (rad), where 360° = 2 π rad. The use of degrees is most common in ...
A solid angle of one steradian subtends a cone aperture of approximately 1.144 radians or 65.54 degrees. In the SI, solid angle is considered to be a dimensionless quantity, the ratio of the area projected onto a surrounding sphere and the square of the sphere's radius. This is the number of square radians in the solid angle.
The solid angle of a sphere measured from any point in its interior is 4 π sr. The solid angle subtended at the center of a cube by one of its faces is one-sixth of that, or 2 π /3 sr. The solid angle subtended at the corner of a cube (an octant) or spanned by a spherical octant is π /2 sr, one-eight of the solid angle of a sphere. [1]
The angle adds the third degree of freedom to this rotation representation. One may wish to express rotation as a rotation vector , or Euler vector , an un-normalized three-dimensional vector the direction of which specifies the axis, and the length of which is θ , r = θ e ^ . {\displaystyle \mathbf {r} =\theta {\hat {\mathbf {e} }}\,.}