Degree — Angle Unit, Definition & Examples

Degree

A unit of angle measure equal to $The fraction 1/360$ of a complete revolution. There are 360 degrees in a circle. Degrees are indicated by the ° symbol, so 35° means 35 degrees.

See also

Minute, second

Key Formula

1\text{ full rotation} = 360°

Where:

$°$ = The degree symbol, placed after a number to indicate degrees

Worked Example

Problem: Two angles in a triangle measure 50° and 70°. Find the third angle.

Step 1: Recall that the interior angles of any triangle sum to 180°.

\angle A + \angle B + \angle C = 180°

Step 2: Substitute the two known angle measures.

50° + 70° + \angle C = 180°

Step 3: Combine the known angles.

120° + \angle C = 180°

Step 4: Subtract 120° from both sides to isolate the unknown angle.

\angle C = 180° - 120° = 60°

Answer: The third angle measures 60°.

Another Example

Problem: Convert 45° to radians.

Step 1: Use the conversion factor: 180° equals π radians.

180° = \pi \text{ radians}

Step 2: Set up the conversion by multiplying 45° by the ratio π/180°.

45° \times \frac{\pi}{180°}

Step 3: Simplify the fraction.

\frac{45\pi}{180} = \frac{\pi}{4}

Answer: 45° equals π/4 radians.

Frequently Asked Questions

Why are there 360 degrees in a circle?

The 360° convention dates back to ancient Babylonian mathematics, which used a base-60 (sexagesimal) number system. The number 360 is convenient because it has many factors (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360), making it easy to divide a circle into equal parts without fractions.

What is the difference between degrees and radians?

Both are units for measuring angles. Degrees divide a full circle into 360 equal parts, while radians relate the angle to the radius of a circle — a full circle is 2π radians. To convert, use the fact that 180° = π radians. Degrees are more common in everyday use and geometry classes, while radians are standard in higher mathematics and physics.

Degree vs. Radian

A degree is 1/360 of a full rotation, designed for practical divisibility. A radian is the angle formed when the arc length equals the radius, making a full rotation equal to 2π radians. Degrees are intuitive for everyday measurement, while radians simplify formulas in calculus and trigonometry (for example, the derivative of sin(x) is cos(x) only when x is in radians).

Why It Matters

Degrees are the most widely used angle unit in everyday life, from reading a compass bearing to measuring roof pitch to describing turns in sports and navigation. In geometry, nearly every problem involving triangles, polygons, or circles requires working with degree measures. Understanding degrees also lays the groundwork for learning radians, which become essential in trigonometry and calculus.

Common Mistakes

Mistake: Forgetting the degree symbol and mixing up degrees with radians on a calculator.

Correction: Always check whether your calculator is set to degree mode (DEG) or radian mode (RAD). Entering 90 in radian mode gives completely different results than entering 90°. For example, sin(90) in radian mode ≈ 0.894, but sin(90°) = 1.

Mistake: Assuming all angle sums are 360°.

Correction: The sum depends on the shape. Triangle angles sum to 180°. Quadrilateral angles sum to 360°. The general formula for an n-sided polygon is (n − 2) × 180°. Only the angles around a single point or the exterior angles of any convex polygon sum to 360°.

Related Terms

Measure of an Angle — How angles are quantified using degrees
Circle — A full circle spans 360 degrees
Minute — Subdivision: 1 degree = 60 minutes
Second — Subdivision: 1 minute = 60 seconds
Radian — Alternative angle unit used in advanced math
Right Angle — An angle measuring exactly 90 degrees
Protractor — Tool used to measure angles in degrees