# How to use the RADIANS function

**What is the RADIANS function?**

The RADIANS function converts degrees to radians.

#### Table of Contents

## 1. Introduction

**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.

**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.

**Why is the PI function useful?**

The trigonometric functions in Excel accept radians as the angular measurement. The basic trigonometry ratios like sine, cosine and tangent come from the ratios of sides in a right triangle. Sine, cosine and tangent ratios relate the lengths of sides to angles in a right triangle. If you know one side and angle, you can use these ratios to find the other sides. The ratios link geometry and angles together.

**What is the relationship between the number pi and radians?**

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.

**What is an arc in a circle?**

The arc in a circle is a segment of the circle's circumference. Arcs are an angle measured in degrees or radians.

**What is a segment in a circle?**

A segment (also called chord) connects two points on a circle's edge by a straight line, and can be used to analyze geometric aspects of the circle like angles, area, and tangents.

**What is a sector?**

A sector of a circle is the section enclosed by two radii and an arc.

**What is radii?**

The plural form of the word "radius".

**What is the connection between a circle and trigonometry?**

The geometry of the circle underlies the theory, definitions, models, identities, and applications of trigonometry. The circle and trigonometry are deeply mathematically connected.

**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)

This is what the RADIANS function does:

Radians = Degrees * PI() / 180

## 2. Syntax

RADIANS(*angle*)

angle |
Required. The degree you want to convert. |

## 3. Example 1

**Convert the following values in degrees [0, 45, 90, 135, 180, 225, 270, 315, 360] to radians?**

The degrees are specified in cell range B16:B24.

Formula in cell D16:

The formula returns the following radians in cell range D16:D24 in decimal form, they are [0, π/4, π/2, 3π/4, π, 5/4π, 3/2π, 7/4π, 2π] in fractions of pi.

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:

- DEGREES: This column lists various angle values in degrees, including 0, 45, 90, 135, 180, 225, 270, 315, 360
- Radians: This column shows the corresponding radian values for the degree values in the previous column, calculated using the RADIANS function in Excel. The RADIANS function converts an angle value from radians to degrees.
- Radians: This column lists various angle values in radians based on fractions of π (pi), including 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, 7π/4, and 2π.

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

*In a right-angled triangle, find the length of the opposite side if A = 30 degrees and the hypotenuse is 5 units?*

What we know:

- Right-angled triangle meaning one angle is 90 degrees or π/2
- c = 5 units (hypotenuse)
- A = 30 degrees

Steps to build the Excel formula:

- Convert degrees to radians.
- Calculate the ratio between a and c (a/c) using the SIN function and the provided angle converted to radians.
- Multiply result by 5 to get the length of the opposite side.

Formula in cell C20:

The fomrula in cell C20 returns 2.5 which represents the length of the opposite side in units.

The RADIANS function allows us to convert 30 degrees to radians which is π/4. The SIN function calculates the ratio between opposite side and the hypotenuse using an angle as the input value.

SIN A = a/5

a = 5 * SIN A

a = 5 * SIN π/4

a = 5 * 0.5

a = 2.5

The length of the opposite side is 2.5 units. The image above shows a chart displaying a right triangle in blue color, angle A is 30 degrees and the hypotenuse is 5. Cell C18 contains the number of degrees which represents the angle A.

### 'RADIANS' function examples

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### Functions in 'Math and trigonometry' category

The RADIANS function function is one of 61 functions in the 'Math and trigonometry' category.

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