A 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°.
A degree is one of 360 equal slices of a full turn. Counting slices gives the angle’s measure.
Formal definition and notation
A one-degree angle is the angle that turns through 1360 of a circle centered at its vertex. An angle that turns through n one-degree angles has a measure of n degrees. The degree is accepted for use with the SI, with 1° = π180radians; it is divided into 60 minutes (′), and each minute into 60 seconds (″).
Symbol
Read it as
Meaning
35°
35 degrees
an angle measure; the small raised circle is the degree sign
35° 15′ 20″
35 degrees, 15 minutes, 20 seconds
degrees split into minutes and seconds, used in maps and navigation
m∠ A
the measure of angle A
the size of the angle, usually in degrees
Key facts
Angle measure does not depend on sidelength: a 30° angle drawn with short or long rays is still 30°.
Benchmarks: right angle90°, straight angle 180°, full turn 360°.
Angle measures add: two angles of 25° and 40° placed side by side make 65°.
The number 360 traces back to the base-60 counting of ancient Babylon, which is also why a degree splits into 60 minutes.
Temperature also uses degrees and the ° sign, but that is a different unit.
Degree conversions
Degree conversions
360° = 1 turn, 1° = 60′, 1° = π180 rad
1 turn
one full rotation around a point
60′
sixty minutes of arc
rad
radians, the angle unit used in calculus
Degrees to radians: multiply by π180. Radians to degrees: multiply by 180π.
Try it: measure in degrees
Interactive
Drag the ray and count the degrees. Use Shift with the arrow keys to jump in bigger steps, and find 90, 180 and 270.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Degree examples, step by step
Example 1: Fraction of a turn
Warm-up
How many degrees are in 18 of a full turn?
A full turn is 360°.
18 × 360° = 45°.
Answer:18 of a turn is 45°.
Example 2: Clock hands
Core
How many degrees does the minute hand of a clock turn in 20 minutes?
The degree of a polynomial is its highest exponent: x3 + 2x has degree 3. It has nothing to do with angles.
Degreevs.Radian
A radian is another angle unit, based on the radius: a full turn is 2π radians instead of 360°. 180° = π radians.
Quick check: 3 questions
Question 1 of 3
How many degrees are in three quarters of a full turn?
Show answer and explanation
Answer: B.34 × 360° = 270°. 135° uses 180° as the full turn, 75° treats a turn as 100, and 300° is a slip.
Question 2 of 3
An angle turns through 48 one-degree angles. What is its measure?
Show answer and explanation
Answer: A.n one-degree angles make n degrees, so 48°. 312° is 360 − 48, 132° is 180 − 48, and 42° is 90 − 48.
Question 3 of 3
What is 15° 45′ as a decimal number of degrees?
Show answer and explanation
Answer: D.45′ = 4560° = 0.75°, so 15.75°. 15.45° treats minutes as hundredths, 15.25° divides 15 by 60, and 16.5° is a slip.
Degree: frequently asked questions
Why are there 360 degrees in a circle?
The choice comes from ancient Babylonian astronomers, who counted in base 60, and 360 is a convenient number close to the days in a year. It also divides evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12 and more, so halves, thirds, quarters and sixths of a turn are all whole numbers of degrees.
What is the difference between degrees and radians?
Both measure angles. Degrees split a full turn into 360 parts. Radians measure the arc length divided by the radius, so a full turn is 2π ≈ 6.28 radians. Degrees are common in school geometry and everyday life; radians are standard in trigonometry and calculus. To convert degrees to radians, multiply by π180.
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