# Revolutions to Radians – Formulas and Examples

Radians are a way of measuring angles. Radians are mainly used when we want to perform advanced mathematical operations such as differential or integral calculus. This is because the radian has a relationship to the radius of the circle. On the other hand, revolutions are a way of considering a complete turn around a circle. This means that one revolution is equal to 2π. Therefore, to convert revolutions to radians, we multiply the number of revolutions by 2π.

Here, we will apply the process of transforming revolutions to radians by solving some practice problems.

##### TRIGONOMETRY

Relevant for

Learning to transform from revolutions to radians with examples.

See examples

##### TRIGONOMETRY

Relevant for

Learning to transform from revolutions to radians with examples.

See examples

## How to convert from revolutions to radians?

To convert from revolutions to radians, we have to multiply the number of revolutions by 2π and we will get the angle in radians that corresponds to the given number of revolutions. Therefore, we have the following formula:

where x represents the number of revolutions and y is the answer in radians.

This formula is derived considering a circle. If we go around a full circle, we have an angle of 2π radians. Also, by definition, one revolution equals one complete turn of the circle. This means that we can form the relation 1 rev = 2π.

Therefore, having any number of revolutions, we simply have to multiply by 2π to find the equivalent radians.

The following practice examples are solved using the formula for the transformation of revolutions to radians given above. Try solving the problems yourself before looking at the answer.

### EXAMPLE 1

If we have 3 revolutions, how many radians do we have?

We substitute the value given in the transformation formula, to obtain:

$latex (x\text{ rev})\times 2\pi=y \text{ rad}$

$latex (3\text{ rev})\times 2\pi=6 \pi\text{ rad}$

Therefore, 3 revolutions equal 6π radians.

### EXAMPLE 2

Using $latex x = 6$ which is the number of revolutions, we have:

$latex (x\text{ rev})\times 2\pi=y \text{ rad}$

$latex (6\text{ rev})\times 2\pi=12 \pi\text{ rad}$

Therefore, 6 revolutions equal 12π radians.

### EXAMPLE 3

How many radians is equal to 2.5 revolutions?

In this case, we have a fractional number, but the formula to use is the same. We use the value in the formula and we have:

$latex (x\text{ rev})\times 2\pi=y \text{ rad}$

$latex (2.5\text{ rev})\times 2\pi=5 \pi\text{ rad}$

Therefore, 2.5 revolutions is equal to 5π radians.

### EXAMPLE 4

If we have 3.8 revolutions, how many radians do we have?

We use the value $latex x = 3.8$ in the transformation formula and solve:

$latex (x\text{ rev})\times 2\pi=y \text{ rad}$

$latex (3.8\text{ rev})\times 2\pi=7.6 \pi\text{ rad}$

### EXAMPLE 5

How many radians is equal to 6.5 revolutions?

We can use the formula with the value $latex x = 6.5$ to obtain:

$latex (x\text{ rev})\times 2\pi=y \text{ rad}$

$latex (6.5\text{ rev})\times 2\pi=13 \pi\text{ rad}$

Therefore, 6.5 revolutions equal 13π radians.

## Revolutions to radians – Practice problems

Practice using the revolutions to radians transformation formula by solving the following problems. Select an answer and check it to see if you got the correct answer.