Radians and degrees are types of units for measuring angles. There are other types of units, but radians and degrees are the most used. Each of these units has its respective application. Radians are mainly used in differential and integral calculus. On the other hand, degrees are used in geometry and allow expressing directionality and angle size. We can convert radians to degrees by multiplying by 180° and dividing by π.
Here, we will look at some problems where we will learn the process used to convert from radians to degrees.
TRIGONOMETRY
Relevant for…
Learning to transform from radians to degrees with examples.
TRIGONOMETRY
Relevant for…
Learning to transform from radians to degrees with examples.
How to convert from radians to degrees?
To convert from radians to degrees, we need to multiply the radians by 180° and divide by π. Therefore, we have the formula:
| $latex x\cdot \frac{180^{\circ}}{\pi}=y^{\circ}$ |
where x is the angle in radians and y is the angle in degrees.
This formula is derived from the fact that we have 360° in a circle. When we express the angle in radians, we know that we have 2π radians in a circle. This means that 2π = 360°. We can divide this expression by two to obtain π = 180°.
Therefore, this is the relationship we use to convert from radians to degrees.
Transformation from radians to degrees – Examples with answers
The following examples are solved using the radians to degrees transformation formula seen above. Each example has its respective solution, but it is recommended that you try to solve the problems yourself before looking at the answer.
EXAMPLE 1
Transform $latex \frac{4 \pi}{5}$ radians to degrees.
Solution
We simply use the radians given in the conversion formula to get the degrees:
$latex x\cdot \frac{180^{\circ}}{\pi}=y^{\circ}$
$latex \frac{4\pi}{5}\cdot \frac{180^{\circ}}{\pi}=144^{\circ}$
Therefore, $latex \frac{4\pi}{5}$ radians is equivalent to 144°.
EXAMPLE 2
How many degrees is equal to $latex \frac{2 \pi}{9}$ radians?
Solution
We substitute the given value in the conversion formula. Therefore, we have:
$latex x\cdot \frac{180^{\circ}}{\pi}=y^{\circ}$
$latex \frac{2\pi}{9}\cdot \frac{180^{\circ}}{\pi}=40^{\circ}$
Thus, $latex \frac{2\pi}{9}$ radians is equivalent to 40°.
EXAMPLE 3
How many degrees is equal to $latex \frac{12 \pi}{7}$ radians?
Solution
We use the radians to degrees conversion formula along with the given value to get the degrees:
$latex x\cdot \frac{180^{\circ}}{\pi}=y^{\circ}$
$latex \frac{12\pi}{7}\cdot \frac{180^{\circ}}{\pi}=308.6^{\circ}$
Therefore, 308.6° is equal to $latex \frac{12\pi}{7}$ radians.
EXAMPLE 4
Convert 2.5 radians to degrees.
Solution
In this case, we simply have $latex x=2.5$. Therefore, we have:
$latex x\cdot \frac{180^{\circ}}{\pi}=y^{\circ}$
$latex 2.5\cdot \frac{180^{\circ}}{\pi}=143.2^{\circ}$
Thus, 2.5 radians is equivalent to 143.2°.
EXAMPLE 5
What is 3.2 radians in degrees?
Solution
We have the value $latex x=3.2$. Using this value in the formula, we have:
$latex x\cdot \frac{180^{\circ}}{\pi}=y^{\circ}$
$latex 3.2\cdot \frac{180^{\circ}}{\pi}=183.3^{\circ}$
Thus, 3.2 radians is equivalent to 183.3°.
Radians to degrees – Practice problems
Use what you have learned about transforming radians to degrees and the derived formula to solve the following practice problems. Select an answer and click “Check” to check your answer.
See also
Interested in learning more about radians and degrees? Take a look at these pages:
