How to use the DEGREES function
What is the DEGREES function?
The DEGREES function calculates degrees based on radians. Degrees are denoted by °.
Table of Contents
1. Introduction
What is degree?
Degrees are a unit used to measure angles. It is based on dividing a full circle into 360 equal parts. Degrees are divided into fractional parts like minutes and seconds for more precision.
What is radian?
Radians measure angles by the length of the arc they make in a circle rather than degrees. The full circumference of any circle is 2π multiplied by the circle's radius (2πr).
Since the circumference goes all the way around a circle, that means the full circle measures 2π radians. Half a circle would be π radians (half of 2π). A quarter circle is 2π/4 = π/2 radians. An eighth of a circle is 2π/8 = π/4 radians.
Excel has a function that returns the number pi: PI function
What is the number pi?
Pi is a irrational number meaning it cannot be expressed as a ratio of two integers. In other words, it has an infinite number of decimal places with no repeating pattern.
It is calculated by the ratio of the circumference of a circle to its diameter. circumference = diameter * π or circumference = 2 * π * radius, this means that the diameter = 2 x radius.
When is the DEGREES function useful?
The trigonometric arc functions return a value in radians. If you need the value in degrees you need to convert it to degrees using the DEGREES function.
| Excel Function | Right Triangle Relationship |
|---|---|
| ASIN | sin(θ) = opposite/hypotenuse |
| ACOS | cos(θ) = adjacent/hypotenuse |
| ATAN | tan(θ) = opposite/adjacent |
| ACOT | tan(θ) = opposite/adjacent |
The trigonometric functions relate an angle expressed in radians or degrees to two out of three sides in a right triangle.
What is the angle θ?
The Greek letter theta (θ) is commonly used to represent an unknown angle in a right triangle.
What is a right triangle?
A right triangle is a type of triangle that contains one internal angle measuring 90 degrees or π/2 radians (a right angle).
What is the relationship between radians and degrees?
The circumference of a circle is 360 degrees or 2π radians.
360 degrees = 2π radians which is degrees = radians x (180 / π)
The following table shows degrees from 0 to 360 in 15 degree increments along with the corresponding radian values as fractions of pi:
| Degrees | Radians |
| 15 | 1/12π |
| 30 | 1/6 π |
| 45 | 1/4 π |
| 60 | 1/3 π |
| 75 | 5/12π |
| 90 | 1/2 π |
| 105 | 7/12π |
| 120 | 2/3 π |
| 135 | 3/4 π |
| 150 | 5/6 π |
| 165 | 11/12π |
| 180 | 1/1 π |
| 195 | 13/12π |
| 210 | 7/6 π |
| 225 | 5/4 π |
| 240 | 4/3 π |
| 255 | 17/12π |
| 270 | 3/2 π |
| 285 | 19/12π |
| 300 | 5/3 π |
| 315 | 7/4 π |
| 330 | 11/6 π |
| 345 | 23/12π |
| 360 | 2/1 π |
How is a circle related to trigonometry?
The geometry and periodicity of circles interconnect deeply with the definitions, identities, and applications of trigonometry. The unit circle bridges the geometry of the circle with the behavior of trig functions.
Right angle is 90 degrees, straight angle is 180 degrees, and a full circle is 360 degrees.
What is the unit circle?
The unit circle is a circle with a radius of 1 unit that is centered at the origin (0, 0) on the Cartesian coordinate plane.
What is the circumference of a circle?
The circumference is the linear distance enclosing the circle.
What is the diameter of a circle?
The diameter of a circle is the straight line distance that passes through the center of the circle connecting one point on the circumference to another, going all the way across the circle.
What is the radius of a circle?
The radius of a circle is the distance from the center point to any point on the circle's edge or circumference.
What is the math formula behind converting radians to degrees?
The formula to convert radians to degrees is:
degrees = (radians * 180) / π
- degrees is the angle measure in degrees
- radians is the angle measure in radians
- π (pi) is approximately equal to 3.14159...
The steps to derive this formula are:
- Define a full circle/revolution as 360 degrees or 2π radians.
- Set up a proportion: degrees/360 = radians/2π
- Cross-multiply: degrees = (radians * 360)/2π
- Simplify by dividing both sides by 2: degrees = (radians * 180)/π
To convert from radians to degrees you multiply the radian value by 180/π which is approximately 180/3.14159 = 57.2957795... This makes sense geometrically because π radians corresponds to 180 degrees (a straight angle), so 1 radian is (180/π) degrees.
The inverse formula to convert from degrees to radians is:
radians = (degrees * π) / 180
2. Syntax
DEGREES(angle)
| angle | Required. The radian you want to convert into degrees. |
3. Example 1
Convert the following values in radians [0, π/4, π/2, 3π/4, π, 5/4π, 3/2π, 7/4π, 2π] to degrees?
The radians are specified in cell range B16:B24.
Formula in cell E16:
The formula in cell E16 returns 0 degrees, this means that 0 (zero) radians is equal to 0 (zero) degrees.
The image above displays a chart that illustrates the relationship between angles expressed in radians and their corresponding values in degrees. The chart consists of two main sections: a graphical representation and a tabular data section.
The graphical representation shows a unit circle divided into quadrants, with angles represented in both radians and degrees. The angles are marked along the circumference of the circle, with their radian values indicated below and their degree values indicated above. For example, the angle π/4 radians is equivalent to 45 degrees, and the angle 3π/2 radians is equivalent to 270 degrees.
The tabular data section consists of two columns:
- Radians: This column lists various angle values in radians, including 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, 7π/4, and 2π.
- DEGREES function: This column shows the corresponding degree values for the radian values in the previous column, calculated using the DEGREES function in Excel. The DEGREES function converts an angle value from radians to degrees.
For example, the radian value π/4 corresponds to 45 degrees, π/2 corresponds to 90 degrees, and 2π corresponds to 360 degrees.
The chart serves as a visual aid and reference for understanding the conversion between radian and degree measurements of angles which is essential in various mathematical and scientific applications.
4. Example 2
The measure of angle N is 2π/3 radians. The measure of angle D is 5π/12 radians greater that the measure of angle N. What is the measure of angle D, in degrees?
| Angle | Radians | |
| N | 2.0943951023932 | 2π/3 |
| D | N + 1.30899693899575 | N + 5π/12 |
Formula in cell E16:
Here is how to calculate degrees based on 2π/3 radians:
- Divide 2π/3 by 2π to get the ratio: (2π/3)/2π = 1/3
- Multiply the ratio by 360: (1/3)* 360 = 120 degrees
Formula in cell E17:
Here is how to calculate degrees based on 5π/12 radians:
- Divide 5π/12 by 2π to get the ratio: (5π/12)/2π = 5/24
- Multiply the ratio by 360: (5/24)* 360 = 75 degrees
Formula in cell E19:
To calculate angle N we need to add the angles to get a total: 120 + 75 = 195 degrees.
The image shows a graphical representation of angles in a coordinate plane:
- The angle N 2π/3 radians (approximately 120 degrees) is shown.
- The angle 5π/12 radians (approximately 75 degrees) is shown.
- The angle 13π/12 radians (approximately 195 degrees) is shown.
'DEGREES' function examples
Table of Contents How to use the DEC2BIN function How to use the DEC2HEX function How to use the DEC2OCT […]
Functions in 'Math and trigonometry' category
The DEGREES function function is one of 61 functions in the 'Math and trigonometry' category.
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