Degrees to Radians – Formulas and Examples

Degrees and radians are the two most common types of units for measuring angles. Each of these units is suitable in different situations. Degrees are used in geometry as they allow us to measure an angle and indicate a direction. However, technically speaking, angles are not numbers and we can only perform mathematical calculations with numbers. For this reason, radians are more convenient when working on calculus or other advanced mathematical topics. To convert from degrees to radians, we have to multiply by π and divide by 180°.

Here, we will learn about the process used to convert from degrees to radians with practice problems.

TRIGONOMETRY
formula to transform from degrees to radians

Relevant for

Learning to transform from degrees to radians with examples.

See examples

TRIGONOMETRY
formula to transform from degrees to radians

Relevant for

Learning to transform from degrees to radians with examples.

See examples

How to convert from degrees to radians?

We can convert from degrees to radians by multiplying the given angle by π and dividing by 180°. This means that we have the following formula:

y^{\circ}\cdot \frac{\pi}{180^{\circ}}=x rad

Here, y represents the angle in degrees and x is the answer in radians.

relationship radians degrees

We can derive this formula by considering that we have 360° in a complete circle. Furthermore, we also know that one complete turn is equivalent to 2π radians. Therefore, we can form the relationship 2π = 360°.

Now, we can simplify this relationship by dividing both sides by two. Dividing by two, we get π = 180°. This is the relationship used to convert from degrees to radians.


Transformation from degrees to radians – Examples with answers

The formula for the transformation from degrees to radians is used to solve the following examples. Try to solve the problems yourself before looking at the answer.

EXAMPLE 1

How many radians is equal to 240°?

We use the transformation formula with the given value. Therefore, we have:

y^{\circ}\cdot \frac{\pi}{180^{\circ}}=x rad

240^{\circ}\cdot \frac{\pi}{180^{\circ}}= \frac{4\pi}{3} rad

Thus, 240° is equivalent to \frac{4\pi}{3} radians.

EXAMPLE 2

Convert 200° to radians.

We use the angle given in the formula seen above to obtain:

y^{\circ}\cdot \frac{\pi}{180^{\circ}}=x rad

200^{\circ}\cdot \frac{\pi}{180^{\circ}}= \frac{10\pi}{9} rad

Therefore, 200° is equal to \frac{10\pi}{9} radians.

EXAMPLE 3

How many radians is equal to 86°?

We substitute the given angle in the transformation formula to obtain:

y^{\circ}\cdot \frac{\pi}{180^{\circ}}=x rad

86^{\circ}\cdot \frac{\pi}{180^{\circ}}= \frac{43\pi}{90} rad

Then, the angle 86° is equivalent to\frac{43\pi}{90} radians or 1.5 radians.

EXAMPLE 4

Convert 110° to radians.

We substitute the given value in the transformation formula and we have:

y^{\circ}\cdot \frac{\pi}{180^{\circ}}=x rad

110^{\circ}\cdot \frac{\pi}{180^{\circ}}= \frac{11\pi}{18} rad

Therefore, 110° is equivalent to \frac{11\pi}{18} radians or 1.92 radians.

EXAMPLE 5

What is 68° in radians?

We use the transformation formula with the given value. Therefore, we have:

y^{\circ}\cdot \frac{\pi}{180^{\circ}}=x rad

68^{\circ}\cdot \frac{\pi}{180^{\circ}}= \frac{17\pi}{45} rad

Thus, 68° is equivalent to \frac{17\pi}{45} radians or 1.19 radians.


Degrees to radians – Practice problems

Solve the following practice problems using the degrees to radians transformation formula. If you need help with this, you can look at the solved examples above.

If we have 150°, what is its equivalent in radians?

Choose an answer






If we have 280°, which of the following is equivalent?

Choose an answer






If we have 76°, what is its equivalent in radians?

Choose an answer






How many radians is equal to 270°?

Choose an answer







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