Degrees to Radians A Level Maths: How to Convert

This calculator converts an angle given in degrees to radians, the standard unit of angular measure used throughout A level maths and physics. You enter the angle in degrees, and the calculator returns the exact radian value, along with the equivalent in terms of pi where applicable. The conversion is a direct multiplication by the constant π/180, so the result is always exact when expressed symbolically.

Formula

The relationship between degrees and radians is defined by the fact that a full circle is 360° and also 2π radians. Therefore:

radians = degrees × π / 180

Where:

• degrees is the angle in degrees (°)
• radians is the angle in radians (rad, dimensionless)
• π (pi) ≈ 3.141592653589793

To go the other way, from radians to degrees, use: degrees = radians × 180 / π.

Worked Example

Using the calculator default of 180 degrees:

1. Start with the formula: radians = degrees × π / 180
2. Substitute 180 for degrees: radians = 180 × π / 180
3. The 180s cancel: radians = π
4. So 180° = π radians ≈ 3.1416 rad

The calculator shows 3.1416 rad for the default input, matching the exact value of π. The second output, ‘Radians as a multiple of π’, gives 1.0000, meaning the angle is exactly 1π.

Why A Level Maths Uses Radians

Radians are not just an alternative unit; they are the natural unit for calculus and many formulas. When you differentiate or integrate trigonometric functions, the formulas only work cleanly if angles are measured in radians. For example, the derivative of sin(x) is cos(x) only when x is in radians. If you used degrees, you would have to multiply every derivative by π/180. Similarly, the arc length formula s = rθ and the area of a sector A = ½r²θ are valid only with θ in radians. In A level maths, you will encounter radians in topics such as:

• Circular measure (arc length, sector area)
• Trigonometric equations and identities
• Calculus of trigonometric functions
• Small angle approximations (sin θ ≈ θ, tan θ ≈ θ, cos θ ≈ 1 − θ²/2)

Common Angles and Their Radian Equivalents

Memorising these common conversions will save time in exams:

• 0° = 0 rad
• 30° = π/6 rad
• 45° = π/4 rad
• 60° = π/3 rad
• 90° = π/2 rad
• 120° = 2π/3 rad
• 135° = 3π/4 rad
• 150° = 5π/6 rad
• 180° = π rad
• 270° = 3π/2 rad
• 360° = 2π rad

Notice that the radian value is often left as a multiple of π. In A level maths, you are expected to give exact answers in terms of π unless the question asks for a decimal.

How to Convert: Step-by-Step Method

To convert any angle from degrees to radians by hand:

1. Write down the angle in degrees.
2. Multiply it by π/180. Do not use a decimal approximation for π yet.
3. Simplify the fraction if possible. For example, 45° × π/180 = 45π/180 = π/4.
4. If the question asks for a decimal, multiply by 3.14159 and round to the required number of decimal places.

For example, convert 210° to radians:

210 × π/180 = 210π/180 = 7π/6 ≈ 3.6652 rad.

Common Mistakes and Misconceptions

One frequent error is forgetting to simplify the fraction. Leaving 45° as 45π/180 instead of π/4 is technically correct but will lose marks in an exam because answers are expected in simplest form. Another mistake is using a calculator in degree mode when working with radians. Always check your calculator’s mode: if you are evaluating sin(π/6), the calculator must be in radian mode, otherwise you will get sin(3.1416°) which is nearly zero, not 0.5.

A related misconception is that radians are just a different way to write degrees, like converting miles to kilometres. In reality, radians are a ratio: the angle subtended at the centre of a circle by an arc equal in length to the radius. That is why the unit is dimensionless and why formulas like arc length are so simple. Another common issue is mixing up the conversion factor: some students multiply by 180/π when they should multiply by π/180. A simple check: since 180° = π rad, a radian is roughly 57.3°, so a number in radians should be much smaller than the same angle in degrees (except for very small angles).

When to Use Degrees vs Radians

In A level maths, you will encounter both units. Degrees are used in geometry, trigonometry of triangles, and many word problems. Radians are used whenever calculus is involved, and in most physics formulas such as angular velocity (ω = θ/t) and simple harmonic motion. The exam syllabus will usually specify which unit to use. If a question does not specify, look for clues: if the angle is given as a multiple of π, use radians; if it is a whole number with a degree symbol, use degrees. In calculus, always switch to radians even if the problem is stated in degrees.

When using a calculator, always set the mode to match the unit of the angle you are working with. Many A level calculators have a degree/radian toggle. If you are solving a trigonometric equation and the solutions are expected in radians, set the calculator to radian mode before using inverse trig functions.

Limitations of the Conversion

The conversion itself is exact and has no limitations. However, when you round a radian value to a decimal, you lose exactness. In pure maths, it is almost always better to leave the answer in terms of π. Also, be aware that the radian is a dimensionless unit; it is defined as a ratio of two lengths (arc length over radius). This means that in some contexts, you may see angles written without any unit, and they are understood to be in radians. For example, in the series expansion of sin(x), x is in radians. The conversion calculator is useful for checking your work, but you should also be comfortable doing the conversion by hand.

Frequently Asked Questions

Why do we multiply by π/180 and not 180/π?
Because a full circle is 360° and also 2π radians, so 1° = 2π/360 = π/180 radians. Multiplying the number of degrees by π/180 gives the equivalent in radians. The reverse conversion multiplies by 180/π.

What is 1 radian in degrees?
1 radian = 180/π degrees ≈ 57.2958°. This is the angle subtended at the centre of a circle by an arc equal in length to the radius.

Should I always give my answer as a multiple of π?
In A level maths, yes, unless the question specifically asks for a decimal. Exact answers in terms of π are preferred because they are not rounded. For example, 60° is exactly π/3, not 1.0472.

How do I convert radians to degrees?
Multiply the radian value by 180/π. For example, 2π/3 radians × 180/π = 120°. This is the inverse of the degrees-to-radians conversion.

Why do calculus formulas only work in radians?
Because the derivative of sin(x) is cos(x) only when x is measured in radians. In degrees, the derivative would be (π/180)cos(x). Radians make the formulas simpler and are the natural unit for angular measure in calculus.

Can I use this calculator for negative angles?
Yes, the calculator accepts any non-negative input. For negative angles, you can enter a positive value and then apply the negative sign to the result, since the conversion is linear. For example, −45° = −π/4 rad.

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in d | Angle in degrees | ° | 180 | 0
out rad | Angle in radians | rad | d * pi / 180 | 4
out rad_pi | Radians as a multiple of π | π | d / 180 | 4