Degrees to Radians
Introduction
Degrees and radians are the two most commonly used units to measure angles in geometry. Converting degrees to radians is a way to change the unit of measurement for angles in geometry. It's important to know how to convert degrees to radians because radians are often preferred for certain calculations.
Radian measure is commonly used in various applications such as finding the area of a sector of a circle, determining arc length, and calculating angular velocity. When dealing with objects moving in circular paths or parts of a circular path, it's recommended to use radians. Let's explore the formula for converting degrees to radians and solve some examples to understand it better.
What Are Degrees?
Degrees, denoted by the symbol `(°)`, are units used to measure angles. They represent the amount of rotation from one side of an angle to the other. A full rotation, going counterclockwise, is equal to `360°`. Each degree is a fraction of this complete rotation, with `180°` equaling half a rotation and `360°` representing a full rotation. Degrees are commonly measured using a protractor.
What Are Radians?
Radians are another unit used to measure angles. One radian is defined as the angle formed at the center of a circle by an arc length equal to the radius of the circle. A full counterclockwise rotation measures `2π` radians, where `π` `(\text{pi})` is approximately `3.14159`. Radians are often denoted by the symbol "rad" and are used in mathematical calculations involving circular motion, trigonometry, and calculus.
What Is Degrees to Radians Conversion?
The conversion from degrees to radians allows us to change the measurement of angles from one unit to another. We can easily convert angles from degrees to radians through simple calculations.
A full circle rotation is represented by `2\pi` radians or `360` degrees. Thus the degree to radian relation can be represented as:
`2\pi \text{radian} = 360°` or more precisely as
`2\pi \text{radian} = 180°`
We can use this relation to write the degrees to radians conversion formula.
Degrees to Radians Conversion Formula
The formula to convert an angle measured in degrees `(θ)` to radians (rad) is:
`\text{Radians} = {\text{Degrees} \times \pi}/{180}`
where:
- radians represent the angle in radians.
- degrees represent the angle in degrees.
- `π` `(\text{pi})` is the mathematical constant approximately equal to `3.14159`.
Derivation of Degrees to Radians Formula
To derive the degree into radian formula, we use the fact that one complete counterclockwise revolution corresponds to `360` degrees in the degree measure and `2π` radians in the radian measure.
This equation forms the basis for converting between degrees and radians.
So, to convert an angle from degrees to radians, we use the degree to radian formula:
The value of `π` is approximately `3.14`, which can be used for calculations involving degrees to radians conversion.
How to Convert Degrees to Radians?
Converting degrees to radians involves using the conversion formula:
`\text{Radians} = \text{Degrees} × (π/{180°})`.
Follow these steps to convert an angle given in degrees to radians:
`1`. Note the angle in degrees: Write down the angle given in degrees.
`2`. Apply the conversion formula: Multiply the angle in degrees by `π/(180°)` to convert it to radians.
`3`. Simplify the expression: Cancel out any common factors to simplify the expression.
`4`. Express in radians: The simplified result represents the angle in radians.
Example: Convert `60` degrees to radians.
Solution:
Given `60°`.
Angle in Radians `= 60° × (π/180°) = π/3`.
So, `60` degrees is equal to `π/3` radians.
Negative Degrees to Radians
Converting negative degrees to radians follows the same process as converting positive degrees. Simply multiply the given angle in degrees by `π/180`.
For instance, if we need to convert `-180` degrees to radians:
`\text{Radian} = \frac{\pi}{180} \times \text{(degrees)}`
`\text{Radian} = \frac{\pi}{180} \times (-180^\circ)`
\( \text{Angle in radian} = -\pi \)
So, converting `-180` degrees to radians yields an angle of `-π` radians.
Degrees to Radians Conversion Table
Below are two tables displaying the conversion of angles from degrees to radians for some of the most commonly used degree measures.
Table `1`: Degrees to Radians (`0°` to `90°`)
Table `2`: Degrees to Radians (`120°` to `360°`)
Degrees to Radians Chart
Solved Examples
Example 1: Convert `200°` into radians.
Solution:
To convert `200°` to radians, we use the formula:
` \text{Angle in radians} = \text{Angle in degrees} \times \frac{\pi}{180}`
So,
`200° \times \frac{\pi}{180} = \frac{200\pi}{180} = \frac{10\pi}{9}`
Thus, `200°` is equal to ` \frac{10\pi}{9} ` radians.
Example `2`: Triangle `ABC` is an equilateral triangle. Find the measure of each angle in radians.
Solution:
In an equilateral triangle, all the three angles measure the same. As per the Triangle Sum Theorem, the measure of each angle ` = (180°)/ 3 = 60°`.
To convert `60°` to radians, we use the formula:
`\text{Angle in radians} = \text{Angle in degrees} \times \frac{\pi}{180}`
So,
`60° \times \frac{\pi}{180} = \frac{60\pi}{180} = \frac{\pi}{3}`
Thus, `60°` is equal to ` \frac{\pi}{3} ` radians.
Example 3: Convert `3.25°` degrees into radians.
Solution:
To convert `3.25°` degrees to radians, we apply the formula:
`\text{Angle in radians} = \text{Angle in degrees} \times \frac{\pi}{180}`
Therefore,
`3.25° \times \frac{\pi}{180} = \frac{13\pi}{720}`
So, `3.25°` is equivalent to `\frac{13\pi}{720}` radians.
Example `4`: In the circle with center `O`, points `P` and `Q` are `2` points on the circle subtending an arc of `45°`. Convert the measure of angle `POQ` from degrees to radians.
Solution:
To convert `45°` degrees to radians, we apply the formula:
`\text{Angle in radians} = \text{Angle in degrees} \times \frac{\pi}{180}`
Therefore,
`45° \times \frac{\pi}{180} = \frac{\pi}{4}`
So, `45°` is equivalent to `\frac{\pi}{4}` radians.
Example `5`: Katie is learning to read a clock. She observes that the minute hand is pointing at `12` and the hour hand is pointing `4`. What is the central angle that the minute and hour hands make at the center of the clock. Write the angle measure in radians.
Solution:
The center of the clock makes a `360°` full circle.
For every `5` minutes of arc, the angle subtended at the ceter is `360/(12°) = 30°`.
There are `4` intervals of `5` minutes between the hour and minute hand making an agle of `4 \times 30° = 120°` at the center.
To convert `120°` degrees to radians, we apply the formula:
`\text{Angle in radians} = \text{Angle in degrees} \times \frac{\pi}{180}`
Therefore,
`120° \times \frac{\pi}{180} = \frac{2\pi}{3}`
So, `120°` is equivalent to `\frac{2\pi}{4} ` radians.
Practice Problems
Q`1`. How many radians does a full circle make?
- `360` radians
- `180` radians
- `2π` radians
- `π` radians
Answer: c
Q`2`. Convert `150°` degrees into radians.
- `150° = frac{5π}{6}` radians
- `150° = frac{4π}{5}` radians
- `150° = frac{3π}{4}` radians
- `150° = frac{2π}{3}` radians
Answer: a
Q`3`. Convert `40°` degrees into radians.
- `40° = frac{5π}{8}` radians
- `40° = frac{4π}{5}` radians
- `40° = frac{3π}{7}` radians
- `40° = frac{2π}{9}` radians
Answer: d
Q`4`. `"_______"` is an SI unit used to measure angles.
- Degrees
- Radians
- Both a and b
- None of the above
Answer: b
Frequently Asked Questions
Q`1`. What is the relationship between degrees and radians?
Answer: Degrees and radians are units of measurement for angles. One complete revolution in degrees equals `360°`, while one complete revolution in radians equals `2\pi` radians.
Q`2`. How to find the radian equivalent for a measure given in degrees?
Answer: To convert degrees to radians, you use the formula: Angle in radians `=` Angle in degrees `×` `\frac{\pi}{180}`. Simply multiply the angle in degrees by `\frac{\pi}{180}` to get the equivalent angle in radians.
Q`3`. What are some common angle measurements in radians?
Answer: Sure! Some common angle measurements in radians are:
- `30^\circ = \frac{\pi}{6}` radians
- `45^\circ = \frac{\pi}{4}` radians
- `60^\circ = \frac{\pi}{3}` radians
- `90^\circ = \frac{\pi}{2}` radians
- `180^\circ = \pi` radians
Q`4`. How do I convert radians to degrees?
Answer: For conversion of radian into degree, you use the formula: Angle in degrees `=` Angle in radians `×` `\frac{180}{\pi}`. Multiply the angle in radians by `\frac{180}{\pi}` to obtain the equivalent angle in degrees.
Q`5`. What is the value of `\pi` in radians?
Answer: The value of `\pi` in radians is the ratio of the circumference of a circle to its diameter, which is approximately `3.14159`. In radians, `\pi` represents half of a complete circle, or `180^\circ`.