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Degrees and Radians



Key Terms
Degree Measurement
Radian Measurement
Converting Between Degrees and Radians
Arc Length
Sector Area
Standard Position of an Angle
Coterminal Angles
Complementary and Supplementary Angles
Summary: The Page's Key Formulas



Two Systems for Measuring Rotation

Angles quantify rotation, but rotation needs a unit before it can be expressed as a number. Two systems serve this purpose throughout mathematics: degrees and radians. Degrees divide a full rotation into 360 equal parts — a convention with roots in Babylonian base-60 arithmetic and the approximate length of a solar year. Radians take a fundamentally different approach, defining an angle through the geometry of the circle itself: one radian is the angle that subtends an arc equal in length to the radius. A full rotation is 2π2\pi radians, and the conversion between the two systems rests on the identity 180°=π180° = \pi radians.

The distinction matters far beyond notation. Arc length, sector area, and every calculus formula involving trigonometric functions assume radian input. The derivative ddxsin⁡(x)=cos⁡(x)\frac{d}{dx}\sin(x) = \cos(x) holds only when xx is in radians. For this reason, radians are the default in any mathematical context beyond basic geometry and everyday measurement. Fluency in both systems — and the ability to convert between them without hesitation — is a prerequisite for the unit circle, trigonometric functions, graphs, and everything that follows in this section.

Key Terms

Units

Degree— 1360\frac{1}{360} of a full rotation
Radian— the angle subtended by an arc equal in length to the radius

Angle Anatomy

Angle— rotation between two rays sharing a common endpoint
Initial Side— the fixed starting ray
Terminal Side— the ray after rotation
Positive Angle— counterclockwise rotation
Negative Angle— clockwise rotation
Standard Position— vertex at origin, initial side on positive xx-axis

Circle Measurements

Arc Length— distance along a circular arc: s=rθs = r\theta
Central Angle— an angle with vertex at the center of a circle
Sector— region between two radii and their arc

Angle Relationships

Coterminal Angles— angles that share the same terminal side
Complementary Angles— two angles summing to 90°90°
Supplementary Angles— two angles summing to 180°180°

Formulas Used

Measurement & Conversion

Degree-Radian Conversion— θrad=θdeg×π180\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}
Arc Length— s=rθs = r\theta
Sector Area— A=12r2θA = \frac{1}{2}r^2\theta

Supplement & Shift Identities

Sine Supplement Identity— sin⁡(π−θ)=sin⁡θ\sin(\pi - \theta) = \sin\theta
Cosine Supplement Identity— cos⁡(π−θ)=−cos⁡θ\cos(\pi - \theta) = -\cos\theta
Tangent Supplement Identity— tan⁡(π−θ)=−tan⁡θ\tan(\pi - \theta) = -\tan\theta

See All Trigonometry Definitions →


Degree Measurement

A degree is 1360\frac{1}{360} of a full rotation. The number 360 has no deep geometric significance — it persists because it is highly divisible (by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, and more), which makes it convenient for subdividing rotations into equal parts without fractions. A right angle is 90°90°, a straight angle is 180°180°, and a complete rotation returns to the starting position at 360°360°.

For precision beyond whole degrees, two conventions exist. The older system uses minutes and seconds of arc: 1°=60′1° = 60' (sixty minutes) and 1′=60′′1' = 60'' (sixty seconds). A measurement like 41°24′36′′41°24'36'' is common in navigation, cartography, and astronomy. The modern alternative is decimal degrees — for example, 41.41°41.41° — which is simpler for computation. Converting between the two is arithmetic: divide the minutes by 60 and the seconds by 3600, then add.

Degrees are intuitive and widely understood outside of mathematics. Compass bearings, latitude and longitude, architectural plans, and most geometric software use degrees as the default. Within trigonometry, degrees remain useful for visualizing angles and for problems stated in everyday terms. Their limitation appears when formulas demand a unit tied to the geometry of the circle — which is precisely what radians provide.

Radian Measurement

A radian is defined by a direct geometric relationship: it is the angle subtended at the center of a circle by an arc whose length equals the radius of that circle. If a circle has radius rr and an arc of length ss is marked along its circumference, the central angle θ\theta in radians is:

θ=sr\theta = \frac{s}{r}


Because both ss and rr are lengths, their ratio is dimensionless — radians carry no physical unit. This is not a technicality; it is the reason radians integrate seamlessly into formulas where degrees would introduce unwanted conversion factors.

A full circumference has length 2πr2\pi r, so a full rotation corresponds to 2πrr=2π\frac{2\pi r}{r} = 2\pi radians. A half rotation is π\pi radians, a quarter rotation is π2\frac{\pi}{2}, and so on. One radian is approximately 57.296°57.296° — a fact occasionally useful for quick estimation, though exact values in terms of π\pi are always preferred in mathematical work.

The radian is not an arbitrary alternative to degrees. It is the angle measure that makes the core formulas of trigonometry and calculus as clean as possible. Arc length becomes s=rθs = r\theta with no extra constants. Sector area becomes A=12r2θA = \frac{1}{2}r^2\theta. The Taylor series sin⁡(x)=x−x33!+x55!−⋯\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots holds only for xx in radians. The small-angle approximation sin⁡(θ)≈θ\sin(\theta) \approx \theta for θ\theta near zero — essential in physics and engineering — relies on radian measure. Every simplification that makes trigonometric calculus workable traces back to defining angle through the ratio of arc to radius.

Converting Between Degrees and Radians

    The two systems are linked by a single relationship: a half rotation is simultaneously 180°180° and π\pi radians. Every conversion follows from this.

    To convert degrees to radians, multiply by π180\frac{\pi}{180}:

    θrad=θdeg×π180\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}


    To convert radians to degrees, multiply by 180π\frac{180}{\pi}:

    θdeg=θrad×180π\theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180}{\pi}


    The standard angles that appear throughout trigonometry should be known in both units without computation:

  • 30°=π630° = \frac{\pi}{6}
  • 45°=π445° = \frac{\pi}{4}
  • 60°=π360° = \frac{\pi}{3}
  • 90°=π290° = \frac{\pi}{2}
  • 120°=2π3120° = \frac{2\pi}{3}
  • 135°=3π4135° = \frac{3\pi}{4}
  • 150°=5π6150° = \frac{5\pi}{6}
  • 180°=π180° = \pi
  • 210°=7π6210° = \frac{7\pi}{6}
  • 240°=4π3240° = \frac{4\pi}{3}
  • 270°=3π2270° = \frac{3\pi}{2}
  • 300°=5π3300° = \frac{5\pi}{3}
  • 315°=7π4315° = \frac{7\pi}{4}
  • 330°=11π6330° = \frac{11\pi}{6}
  • 360°=2π360° = 2\pi

    • These values all follow a pattern: they are multiples of π6\frac{\pi}{6} and π4\frac{\pi}{4}. Recognizing this pattern eliminates the need for memorization through brute force — the structure itself carries the information. These angles recur on the unit circle, in the evaluation of trigonometric functions, and throughout the solving of equations and inequalities.
Degrees Radians
0° 0
30° π ⁄ 6
45° π ⁄ 4
60° π ⁄ 3
90° π ⁄ 2
120° 2π ⁄ 3
135° 3π ⁄ 4
150° 5π ⁄ 6
180° π
210° 7π ⁄ 6
225° 5π ⁄ 4
240° 4π ⁄ 3
270° 3π ⁄ 2
300° 5π ⁄ 3
315° 7π ⁄ 4
330° 11π ⁄ 6
360° 2π

Arc Length

When an angle θ\theta (in radians) is subtended at the center of a circle with radius rr, the length of the intercepted arc is:

s=rθs = r\theta


This formula is a direct consequence of the radian definition. Since θ=sr\theta = \frac{s}{r}, multiplying both sides by rr gives s=rθs = r\theta. No conversion constant is needed — the formula works cleanly because radians are a ratio, not an imposed unit.

If the angle is given in degrees, the formula requires adjustment:

s=θ360×2πr=πrθ180s = \frac{\theta}{360} \times 2\pi r = \frac{\pi r \theta}{180}


The additional factors are the price of using a unit not inherently tied to the circle. This is one of the clearest practical demonstrations of why radians simplify computation.

Arc length problems typically involve three quantities — ss, rr, and θ\theta — and any one of them can be found given the other two. A common variation asks for the angle subtended by a known arc on a known circle: θ=sr\theta = \frac{s}{r}. Another asks for the radius of a circle given an arc length and central angle: r=sθr = \frac{s}{\theta}. In each case, the formula remains s=rθs = r\theta with θ\theta in radians.

Sector Area

A sector is the region enclosed by two radii and the arc between them — a "pie slice" of the circle. Its area is a fraction of the full circle's area, proportional to the central angle:

A=12r2θA = \frac{1}{2}r^2\theta


where θ\theta is in radians. The derivation is straightforward: the sector represents θ2π\frac{\theta}{2\pi} of the full circle, and the full circle has area πr2\pi r^2, so:

A=θ2π×πr2=12r2θA = \frac{\theta}{2\pi} \times \pi r^2 = \frac{1}{2}r^2\theta


If the angle is given in degrees:

A=θ360×πr2A = \frac{\theta}{360} \times \pi r^2


As with arc length, the radian version is simpler and is the standard in mathematical work. Problems involving sector area follow the same pattern as arc length: three quantities (AA, rr, θ\theta), any one computable from the other two. A typical application is finding the area swept by a rotating object — a windshield wiper, a radar beam, a door opening through a measured angle.

The relationship between arc length and sector area is worth noting: A=12rsA = \frac{1}{2}rs, obtained by substituting s=rθs = r\theta into the sector area formula. This parallels the triangle area formula A=12×base×heightA = \frac{1}{2} \times \text{base} \times \text{height}, with the arc length playing the role of the base and the radius playing the role of the height. For very small angles, the sector closely approximates a triangle, and this analogy becomes nearly exact.

Standard Position of an Angle

In trigonometry, angles are placed on the coordinate plane in a standardized way. An angle is in standard position when its vertex is at the origin and its initial side lies along the positive xx-axis. The terminal side is the ray that results from rotating the initial side through the given angle. This convention fixes a reference frame for all trigonometric evaluation: every angle, regardless of its measure, has a definite position on the plane and a definite intersection with the unit circle.

The direction of rotation determines the sign of the angle. Counterclockwise rotation produces a positive angle; clockwise rotation produces a negative angle. The angle 90°90° is a quarter turn counterclockwise, while −90°-90° is a quarter turn clockwise. Both terminate at the same position on the coordinate plane — along the negative or positive yy-axis respectively — but they are different angles with different trigonometric interpretations: sin⁡(90°)=1\sin(90°) = 1 while sin⁡(−90°)=−1\sin(-90°) = -1.

There is no restriction on the size of an angle. A rotation of 450°450° passes through a full rotation (360°360°) and continues another 90°90°, terminating at the same position as 90°90°. A rotation of −270°-270° goes 270°270° clockwise, also terminating at the 90°90° position. These are coterminal angles — different rotations that share the same terminal side.

Coterminal Angles

Two angles are coterminal if their terminal sides coincide when both are placed in standard position. This happens whenever the angles differ by a full rotation — that is, by 360°360° or 2π2\pi radians. More generally, angles θ\theta and θ+360°n\theta + 360°n (or θ+2πn\theta + 2\pi n) are coterminal for any integer nn.

Every angle has infinitely many coterminal partners. The angle 50°50° is coterminal with 410°410°, 770°770°, −310°-310°, −670°-670°, and so on. In radian terms, π3\frac{\pi}{3} is coterminal with π3+2π=7π3\frac{\pi}{3} + 2\pi = \frac{7\pi}{3}, with π3−2π=−5π3\frac{\pi}{3} - 2\pi = -\frac{5\pi}{3}, and with any expression of the form π3+2πn\frac{\pi}{3} + 2\pi n.

Coterminal angles produce identical trigonometric values. Since the terminal side determines the intersection point with the unit circle, and the coordinates of that point define sine and cosine, all coterminal angles yield the same sine, cosine, tangent, and so on. This is the geometric source of periodicity: sin⁡(θ+2π)=sin⁡(θ)\sin(\theta + 2\pi) = \sin(\theta) because adding 2π2\pi brings the terminal side back to where it started.

A standard task is finding the coterminal angle within [0°,360°)[0°, 360°) or [0,2π)[0, 2\pi). This amounts to dividing by 360°360° (or 2π2\pi) and taking the remainder. For example, θ=850°\theta = 850°: dividing 850850 by 360360 gives quotient 2 with remainder 130130, so the coterminal angle in [0°,360°)[0°, 360°) is 130°130°. For negative angles, add full rotations until the result falls in the desired range: θ=−200°\theta = -200° becomes −200°+360°=160°-200° + 360° = 160°.

Complementary and Supplementary Angles

Two angles are complementary if they sum to 90°90° (or π2\frac{\pi}{2} radians). Two angles are supplementary if they sum to 180°180° (or π\pi radians). These relationships appear constantly in trigonometry, geometry, and the analysis of triangles.

In a right triangle, the two acute angles are always complementary — they must sum to 90°90° because the three angles of any triangle sum to 180°180° and one angle is already 90°90°. This geometric fact is the origin of the cofunction relationships in right triangle trigonometry: sin⁡(θ)=cos⁡(90°−θ)\sin(\theta) = \cos(90° - \theta), tan⁡(θ)=cot⁡(90°−θ)\tan(\theta) = \cot(90° - \theta), and sec⁡(θ)=csc⁡(90°−θ)\sec(\theta) = \csc(90° - \theta). The complement of an angle swaps each trigonometric function with its cofunction — a pattern formalized in the cofunction identities.

Supplementary angles arise naturally in the context of the unit circle. The angles θ\theta and 180°−θ180° - \theta are supplementary, and their reference angles are equal. This produces a useful relationship: sin⁡(θ)=sin⁡(180°−θ)\sin(\theta) = \sin(180° - \theta) and cos⁡(θ)=−cos⁡(180°−θ)\cos(\theta) = -\cos(180° - \theta). In other words, supplementary angles have equal sines but opposite cosines — a fact that plays a direct role in solving trigonometric equations and in analyzing the ambiguous case of the Law of Sines.

For example, the angles 40°40° and 50°50° are complementary (40°+50°=90°40° + 50° = 90°), while 40°40° and 140°140° are supplementary (40°+140°=180°40° + 140° = 180°). The complement of π6\frac{\pi}{6} is π2−π6=π3\frac{\pi}{2} - \frac{\pi}{6} = \frac{\pi}{3}. The supplement of π6\frac{\pi}{6} is π−π6=5π6\pi - \frac{\pi}{6} = \frac{5\pi}{6}. Recognizing these pairs accelerates computation across the subject.
Relationship Condition Example pair Trigonometric consequence
Coterminal differ by 360°  (or 2π)  — same terminal side 50°, 410°, −310° identical sin, cos, tan, etc.
Complementary sum to 90°  (or π ⁄ 2) 40° & 50°;  π ⁄ 6 & π ⁄ 3 cofunction swap: sin θ = cos(90° − θ)
Supplementary sum to 180°  (or π) 40° & 140°;  π ⁄ 6 & 5π ⁄ 6 equal sines, opposite cosines: sin θ = sin(180° − θ)

Summary: The Page's Key Formulas

Five formulas carry most of the practical weight on this page — two conversions, two area/length formulas keyed by radian measure, and one alternative form linking arc length to sector area. The capstone table below collects all five with the required unit for the angle and the derivation each rests on.
Quantity Formula Required unit for θ Origin / note
Degrees → radians θrad = θdeg × π ⁄ 180 — from 180° = π
Radians → degrees θdeg = θrad × 180 ⁄ π — inverse of the above
Arc length s = r θ radians direct from θ = s ⁄ r
Sector area A = ½ r2 θ radians fraction θ ⁄ 2π of πr2
Sector area  (alt.) A = ½ r s —  (uses s, not θ) substitute s = rθ into the formula above