A radian is a unit for measuring angles: one radian is the central angle that cuts off an arc exactly as long as the circle’s radius. A full turn is 2π radians, so π radians = 180° and 1 radian is about 57.3°. To change degrees to radians, multiply by π180.
Bend the radius onto the circle: the angle it spans is 1 radian. About 6.28 radius-lengths fit around the circle, which is why a full turn is 2π radians.
Formal definition and notation
The radian measure of a central angleθ is the ratio of the lengths of the arc it cuts off to the radius r: θ = sr. Because a whole circumference is 2π r, a full rotation measures 2π radians, which equals 360°. On the unit circle (r = 1), the radian measure equals the arc length itself. Radians are a ratio of two lengths, so they have no physical unit, and angles in calculus and trigonometric functions of real numbers are measured in radians.
Symbol
Read it as
Meaning
rad
radians
the unit; often left off, so an angle written as 2 means 2 radians
The radius is a length, such as 5 cm. A radian is an angle, the one whose arc equals the radius.
Quick check: 3 questions
Question 1 of 3
How many radians are in 180°?
Show answer and explanation
Answer: B. A half turn is π radians. 2π is a full turn, π2 is 90°, and 180π multiplies instead of converting.
Question 2 of 3
About how many degrees is 1 radian?
Show answer and explanation
Answer: D.1 × 180π ≈ 57.3°. 3.14° confuses a radian with π, and a radian is much bigger than 1° but smaller than 90°.
Question 3 of 3
What is π3 radians in degrees?
Show answer and explanation
Answer: A.π3 × 180π = 1803 = 60°. 30° is π6, 120° is 2π3, and 540 multiplies by 3.
Radian: frequently asked questions
Why do we use radians instead of degrees?
Radians come straight from the circle: the angle is arc length divided by radius, so formulas get simpler. Arc length is s = rθ and sector area is 12r2θ with no extra fractions. In calculus, the derivative of sinx is cosx only when x is in radians, so higher math uses radians by default.
How do I convert degrees to radians quickly?
Multiply the degrees by π and divide by 180, then simplify the fraction. For 45°: 45π180 = π4. It helps to memorize that 30°, 45°, 60° and 90° are π6, π4, π3 and π2; other angles are multiples of these.
RadiusThe radius is a line segment from the center of a circle to any point on the circle, or the length of that segment. The radius is half the diameter: if the diameter is 10, the radius is 5. All radii in the same circle have equal length, and r is the radius in C = 2π r.Geometry · Grade 6, Grade 7, Grade 8AngleAn angle is formed by two rays that share the same endpoint, called the vertex. It is measured in degrees by how far one ray is turned from the other. A right angle measures 90 degrees, a straight angle measures 180 degrees, and one full turn is 360 degrees.Geometry · Grade 4, Grade 5, Grade 6RatioA ratio compares two quantities by division. Ratios can be written with words, a colon, or a fraction: 3 boys to 5 girls is 3 to 5, 3 : 5, or 35. Order matters, and a ratio can compare part to part or part to whole. 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Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1DegreeA degree is the standard unit for measuring an angle, written with the symbol °. One degree is 1360 of a full turn, so a full turn is 360°, a straight angle is 180° and a right angle is 90°. An angle that turns through 25 one-degree angles measures 25°.Geometry · Grade 4, Grade 5, Grade 7SimplifySimplify means to rewrite a number or expression in a shorter, easier form without changing its value. To simplify a fraction, divide the numerator and denominator by a common factor until none is left: 1824 = 34. To simplify an expression, combine like terms and do the operations you can: 3x + 2x = 5x.Number System · Grade 4, Grade 5, Grade 6CircleA circle is the set of all points in a flat plane that are the same distance from one fixed point, the center. That distance is the radius. A segment through the center from edge to edge is the diameter, twice the radius. The distance around is the circumference, 2π r, and the area inside is π r2.Geometry · Kindergarten, Grade 1, Grade 7Central AngleA central angle is an angle whose vertex is the center of a circle and whose sides are two radii. It cuts off an arc, and the arc’s degree measure equals the central angle. A 90° central angle marks off a quarter of the circle, and the central angles around the center add to 360 degrees.Geometry · Grade 7, GeometryNumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2SineSine is the trigonometric ratio that compares the side opposite an angle with the hypotenuse of a right triangle: sin θ = oppositehypotenuse. On the unit circle, sin θ is the y-coordinate of the point at angle θ, so sine values always stay between −1 and 1.Trigonometry · Geometry, Algebra 2, PrecalculusPeriodThe period of a repeating function is the horizontal length of one complete cycle: the smallest distance after which the graph repeats exactly. y = sinx and y = cosx have period 2π, or 360°, and y = tanx has period π. For y = sin(Bx) or y = cos(Bx), the period is 2π|B|.Trigonometry · Algebra 2, PrecalculusPiPi (π) is the number you get when you divide any circle’s circumference by its diameter. It is the same for every circle, about 3.14159. Pi is an irrational number: its decimal never ends or repeats, so we round it to 3.14 or use 227.Geometry · Grade 7, Grade 8, Geometry