Radians

What you'll learn

How radians measure angles, and how to convert between degrees and radians.
How to use arc length and sector area formulae.
How to handle chords, sectors, and circular segments.
How to use small angle approximations safely in algebra and numerical estimates.

1. What is a radian?

You already know degrees. Radians are another way to measure angles, but they are built directly from circles, which is why they make A-Level formulae much neater.

One radian is the central angle subtended by an arc whose length equals the radius.

Definition

Radian and central angle

A central angle is an angle whose vertex is at the centre of a circle.
One radian is the central angle made when the arc length is equal to the radius.
If the arc length is $s$ , the radius is $r$ , and the angle is $θ\theta$ radians, then $θ=sr\theta=\frac{s}{r}$ .

A full turn has circumference $2πr2\pi r$ , so the angle in a full turn is:

2πrr=2π\frac{2\pi r}{r}=2\pi

So:

radians360^\circ=2\pi \text{ radians}, \qquad 180^\circ=\pi \text{ radians}

To convert:

degrees to radians: multiply by $π180\frac{\pi}{180}$
radians to degrees: multiply by $180π\frac{180}{\pi}$

Example

Converting an angle before using it

Suppose an angle is 40°. Convert it to radians by multiplying by $π180\frac{\pi}{180}$ :
$40π180=2π9\frac{40\pi}{180}=\frac{2\pi}{9}$
So the radian angle is $2π9\frac{2\pi}{9}$ .
If a formula needs radians, use $θ=2π9\theta=\frac{2\pi}{9}$ , not 40.

Common Mistake

Leaving degrees in a radian formula

Formulae like $s=rθs=r\theta$ and $A=12r2θA=\frac12 r^2\theta$ only work directly when $θ\theta$ is in radians. If the question gives degrees, convert first.

2. Arc length and sector area

A circle sector is like a slice of pizza: it is bounded by two radii and an arc.

A sector consists of two radii and the arc between them, with perimeter made from two radii plus the arc length.

Definition

Sector and arc

A sector is the region enclosed by two radii and the arc between them.
An arc is part of the circumference of a circle.

For a sector with radius $r$ and angle $θ\theta$ radians:

s=rθs=r\theta

where $s$ is the arc length.

The area of the sector is:

A=12r2θA=\frac12 r^2\theta

The perimeter of a sector is:

2 r + s

because it includes two radii and one arc.

Key Idea

Why radians are useful

When angles are in radians, arc length and sector area become simple: $s=rθs=r\theta$ and $A=12r2θA=\frac12r^2\theta$ .

Example

Arc length and sector area

A sector has radius 9 cm and angle 1.2 radians. Use $s=rθs=r\theta$ for the arc length:
$s=9⋅1.2=10.8s=9\cdot 1.2=10.8$
The arc length is 10.8 cm.
Use $A=12r2θA=\frac12r^2\theta$ for the sector area:
$A=12(9)2(1.2)=48.6A=\frac12(9)^2(1.2)=48.6$
The sector area is 48.6 cm².

Example

Sector perimeter when the angle is hidden

A sector has centre $O$ , and triangle $A O B$ is equilateral. Therefore the central angle is 60°, so:

An equilateral triangle inside the sector shows why the central angle is 60° before converting to radians.

$$
\theta=\frac{\pi}{3}
$$

2. If the sector area is $24π24\pi$ cm², use $A=12r2θA=\frac12r^2\theta$ :

$$
24\pi=\frac12r^2\cdot\frac{\pi}{3}
$$

3. Solve for $r$ :

$$
24\pi=\frac{\pi r^2}{6}\Rightarrow r^2=144\Rightarrow r=12
$$

4. Find the arc length:

$$
s=r\theta=12\cdot\frac{\pi}{3}=4\pi
$$

5. Add two radii and the arc to get the perimeter:

$$
12+12+4\pi=24+4\pi
$$

3. Chords and circular segments

Definition

Chord and minor segment

A chord is a straight line joining two points on the circumference of a circle.
A minor segment is the smaller region between a chord and the arc it cuts off.

For segment problems, split the diagram into:

A minor segment can be found by subtracting the isosceles triangle from the sector.

a sector
an isosceles triangle formed by the two radii and the chord

The area of the minor segment is:

area\text{sector area}-\text{triangle area}

If the central angle is $θ\theta$ , then the triangle area is:

12r2sin⁡θ\frac12r^2\sin\theta

So the minor segment area is:

12r2(θ−sin⁡θ)\frac12r^2(\theta-\sin\theta)

Example

Finding an angle from a chord, then a segment area

A circle has radius 6 cm and chord length 8 cm. Split the isosceles triangle in half. Half the chord is 4 cm, so:

Bisecting the chord creates a right triangle used to find half the central angle.

$$
\sin\left(\frac{\theta}{2}\right)=\frac{4}{6}=\frac23
$$

2. Find the central angle:

$$
\theta=2\arcsin\left(\frac23\right)=1.459\text{ to 3 d.p.}
$$

3. Find the sector area using the unrounded angle:

$$
A_{\text{sector}}=\frac12(6)^2\theta=26.3
$$

4. Find the minor segment area using $Asegment=12r2(θ−sin⁡θ)A_{\text{segment}}=\frac12r^2(\theta-\sin\theta)$ :

$$
A_{\text{segment}}=\frac12(6)^2(\theta-\sin\theta)=8.38
$$

5. The shaded minor segment area is about 8.38 cm².

Tip

Coordinate geometry link

If a circle is given by an equation, complete the square first to find its centre and radius. Then use distances, angles, and sector formulae.

4. Small angle approximations

For very small angles, trig functions can be approximated by simpler expressions.

Near x = 0, the curves for sin x and tan x are close to y = x, while cos x is close to 1 − x²/2.

Definition

Small angle approximations

For an angle $x$ close to 0, measured in radians:

sin⁡x≈x,tan⁡x≈x,cos⁡x≈1−x22\sin x\approx x,\qquad \tan x\approx x,\qquad \cos x\approx 1-\frac{x^2}{2}

These are especially useful when you are asked to “show that” a complicated expression can be approximated by a simpler algebraic expression.

Common Mistake

Radians only

Small angle approximations only work when the angle is measured in radians. For example, convert 3° to $π60\frac{\pi}{60}$ radians before using them.

Example

Approximating a trig fraction and finding percentage error

Use $cos⁡θ≈1−θ22\cos\theta\approx1-\frac{\theta^2}{2}$ and $sin⁡θ≈θ\sin\theta\approx\theta$ :
$cos⁡θsin⁡θ≈1−θ22θ\frac{\cos\theta}{\sin\theta}\approx\frac{1-\frac{\theta^2}{2}}{\theta}$
Simplify the expression:
$1−θ22θ=2−θ22θ\frac{1-\frac{\theta^2}{2}}{\theta}=\frac{2-\theta^2}{2\theta}$
For $θ=π24\theta=\frac{\pi}{24}$ , the approximation is:
$2−(π24)22(π24)=7.57399\frac{2-\left(\frac{\pi}{24}\right)^2}{2\left(\frac{\pi}{24}\right)}=7.57399$
Using a calculator, the true value is about 7.59575, so the percentage error is:
$∣7.57399−7.59575∣7.59575×100=0.287%\frac{\left|7.57399-7.59575\right|}{7.59575}\times100=0.287\%$

5. Multiplying and solving with approximations

When you substitute small angle approximations, keep your algebra tidy. Expand brackets carefully, especially if powers like $x^2$ or $x^3$ appear.

Example

Approximating a product

For small $x$ , approximate each factor:
$tan⁡(3x)≈3x,cos⁡(2x)≈1−(2x)22=1−2x2\tan(3x)\approx3x,\qquad \cos(2x)\approx1-\frac{(2x)^2}{2}=1-2x^2$
Multiply the approximations:
$tan⁡(3x)cos⁡(2x)≈3x(1−2x2)=3x−6x3\tan(3x)\cos(2x)\approx3x(1-2x^2)=3x-6x^3$
To approximate $tan⁡(0.3)cos⁡(0.2)\tan(0.3)\cos(0.2)$ , notice that $3 x = 0.3$ , so $x = 0.1$ :
$3(0.1)-6(0.1)^3=0.294$

Example

Solving an approximate equation

Approximate the left-hand side; the $θ2\theta^2$ terms cancel:
$2cos⁡θ+(sin⁡θ+1)2≈2(1−θ22)+(θ+1)2=2−θ2+θ2+2θ+1=3+2θ\begin{aligned} 2\cos\theta+(\sin\theta+1)^2 &\approx 2\left(1-\frac{\theta^2}{2}\right)+(\theta+1)^2\\ &=2-\theta^2+\theta^2+2\theta+1\\ &=3+2\theta \end{aligned}$
Approximate the right-hand side:
$20tan⁡θ≈20θ20\tan\theta\approx20\theta$
Solve the resulting linear equation:
$3+2θ=20θ⇒18θ=3⇒θ=16=0.1673+2\theta=20\theta\Rightarrow 18\theta=3\Rightarrow \theta=\frac16=0.167$

Key Idea

When it becomes a straight line

If a small angle approximation leaves only a constant term and an $x$ term, the graph is approximately a straight line of the form $y = m x + c$ .

Exam technique

In the exam

Check whether the angle is in degrees or radians before using any formula.
For sectors and segments, label the radius, angle, arc, chord, and triangle clearly.
In small angle questions, substitute the approximations first, then simplify algebraically.
For percentage error, use $value∣×100\frac{|\text{approximation}-\text{true value}|}{|\text{true value}|}\times100$ .

Self review

Check yourself

Why must $θ\theta$ be in radians when using $s=rθs=r\theta$ ?
How do you find the area of a minor segment from a sector and a triangle?
Which small angle approximation would you use for $cos⁡(2x)\cos(2x)$ ?

Radians

What you'll learn

How radians measure angles, and how to convert between degrees and radians.
How to use arc length and sector area formulae.
How to handle chords, sectors, and circular segments.
How to use small angle approximations safely in algebra and numerical estimates.

1. What is a radian?

You already know degrees. Radians are another way to measure angles, but they are built directly from circles, which is why they make A-Level formulae much neater.

One radian is the central angle subtended by an arc whose length equals the radius.

Definition

Radian and central angle

A central angle is an angle whose vertex is at the centre of a circle.
One radian is the central angle made when the arc length is equal to the radius.
If the arc length is $s$ , the radius is $r$ , and the angle is $θ\theta$ radians, then $θ=sr\theta=\frac{s}{r}$ .

A full turn has circumference $2πr2\pi r$ , so the angle in a full turn is:

2πrr=2π\frac{2\pi r}{r}=2\pi

So:

radians360^\circ=2\pi \text{ radians}, \qquad 180^\circ=\pi \text{ radians}

To convert:

degrees to radians: multiply by $π180\frac{\pi}{180}$
radians to degrees: multiply by $180π\frac{180}{\pi}$

Example

Converting an angle before using it

Suppose an angle is 40°. Convert it to radians by multiplying by $π180\frac{\pi}{180}$ :
$40π180=2π9\frac{40\pi}{180}=\frac{2\pi}{9}$
So the radian angle is $2π9\frac{2\pi}{9}$ .
If a formula needs radians, use $θ=2π9\theta=\frac{2\pi}{9}$ , not 40.

Common Mistake

Leaving degrees in a radian formula

Formulae like $s=rθs=r\theta$ and $A=12r2θA=\frac12 r^2\theta$ only work directly when $θ\theta$ is in radians. If the question gives degrees, convert first.

2. Arc length and sector area

A circle sector is like a slice of pizza: it is bounded by two radii and an arc.

A sector consists of two radii and the arc between them, with perimeter made from two radii plus the arc length.

Definition

Sector and arc

A sector is the region enclosed by two radii and the arc between them.
An arc is part of the circumference of a circle.

For a sector with radius $r$ and angle $θ\theta$ radians:

s=rθs=r\theta

where $s$ is the arc length.

The area of the sector is:

A=12r2θA=\frac12 r^2\theta

The perimeter of a sector is:

2 r + s

because it includes two radii and one arc.

Key Idea

Why radians are useful

When angles are in radians, arc length and sector area become simple: $s=rθs=r\theta$ and $A=12r2θA=\frac12r^2\theta$ .

Example

Arc length and sector area

A sector has radius 9 cm and angle 1.2 radians. Use $s=rθs=r\theta$ for the arc length:
$s=9⋅1.2=10.8s=9\cdot 1.2=10.8$
The arc length is 10.8 cm.
Use $A=12r2θA=\frac12r^2\theta$ for the sector area:
$A=12(9)2(1.2)=48.6A=\frac12(9)^2(1.2)=48.6$
The sector area is 48.6 cm².

Example

Sector perimeter when the angle is hidden

A sector has centre $O$ , and triangle $A O B$ is equilateral. Therefore the central angle is 60°, so:

An equilateral triangle inside the sector shows why the central angle is 60° before converting to radians.

$$
\theta=\frac{\pi}{3}
$$

2. If the sector area is $24π24\pi$ cm², use $A=12r2θA=\frac12r^2\theta$ :

$$
24\pi=\frac12r^2\cdot\frac{\pi}{3}
$$

3. Solve for $r$ :

$$
24\pi=\frac{\pi r^2}{6}\Rightarrow r^2=144\Rightarrow r=12
$$

4. Find the arc length:

$$
s=r\theta=12\cdot\frac{\pi}{3}=4\pi
$$

5. Add two radii and the arc to get the perimeter:

$$
12+12+4\pi=24+4\pi
$$

3. Chords and circular segments

Definition

Chord and minor segment

A chord is a straight line joining two points on the circumference of a circle.
A minor segment is the smaller region between a chord and the arc it cuts off.

For segment problems, split the diagram into:

A minor segment can be found by subtracting the isosceles triangle from the sector.

a sector
an isosceles triangle formed by the two radii and the chord

The area of the minor segment is:

area\text{sector area}-\text{triangle area}

If the central angle is $θ\theta$ , then the triangle area is:

12r2sin⁡θ\frac12r^2\sin\theta

So the minor segment area is:

12r2(θ−sin⁡θ)\frac12r^2(\theta-\sin\theta)

Example

Finding an angle from a chord, then a segment area

A circle has radius 6 cm and chord length 8 cm. Split the isosceles triangle in half. Half the chord is 4 cm, so:

Bisecting the chord creates a right triangle used to find half the central angle.

$$
\sin\left(\frac{\theta}{2}\right)=\frac{4}{6}=\frac23
$$

2. Find the central angle:

$$
\theta=2\arcsin\left(\frac23\right)=1.459\text{ to 3 d.p.}
$$

3. Find the sector area using the unrounded angle:

$$
A_{\text{sector}}=\frac12(6)^2\theta=26.3
$$

4. Find the minor segment area using $Asegment=12r2(θ−sin⁡θ)A_{\text{segment}}=\frac12r^2(\theta-\sin\theta)$ :

$$
A_{\text{segment}}=\frac12(6)^2(\theta-\sin\theta)=8.38
$$

5. The shaded minor segment area is about 8.38 cm².

Tip

Coordinate geometry link

If a circle is given by an equation, complete the square first to find its centre and radius. Then use distances, angles, and sector formulae.

4. Small angle approximations

For very small angles, trig functions can be approximated by simpler expressions.

Near x = 0, the curves for sin x and tan x are close to y = x, while cos x is close to 1 − x²/2.

Definition

Small angle approximations

For an angle $x$ close to 0, measured in radians:

sin⁡x≈x,tan⁡x≈x,cos⁡x≈1−x22\sin x\approx x,\qquad \tan x\approx x,\qquad \cos x\approx 1-\frac{x^2}{2}

These are especially useful when you are asked to “show that” a complicated expression can be approximated by a simpler algebraic expression.

Common Mistake

Radians only

Small angle approximations only work when the angle is measured in radians. For example, convert 3° to $π60\frac{\pi}{60}$ radians before using them.

Example

Approximating a trig fraction and finding percentage error

Use $cos⁡θ≈1−θ22\cos\theta\approx1-\frac{\theta^2}{2}$ and $sin⁡θ≈θ\sin\theta\approx\theta$ :
$cos⁡θsin⁡θ≈1−θ22θ\frac{\cos\theta}{\sin\theta}\approx\frac{1-\frac{\theta^2}{2}}{\theta}$
Simplify the expression:
$1−θ22θ=2−θ22θ\frac{1-\frac{\theta^2}{2}}{\theta}=\frac{2-\theta^2}{2\theta}$
For $θ=π24\theta=\frac{\pi}{24}$ , the approximation is:
$2−(π24)22(π24)=7.57399\frac{2-\left(\frac{\pi}{24}\right)^2}{2\left(\frac{\pi}{24}\right)}=7.57399$
Using a calculator, the true value is about 7.59575, so the percentage error is:
$∣7.57399−7.59575∣7.59575×100=0.287%\frac{\left|7.57399-7.59575\right|}{7.59575}\times100=0.287\%$

5. Multiplying and solving with approximations

When you substitute small angle approximations, keep your algebra tidy. Expand brackets carefully, especially if powers like $x^2$ or $x^3$ appear.

Example

Approximating a product

For small $x$ , approximate each factor:
$tan⁡(3x)≈3x,cos⁡(2x)≈1−(2x)22=1−2x2\tan(3x)\approx3x,\qquad \cos(2x)\approx1-\frac{(2x)^2}{2}=1-2x^2$
Multiply the approximations:
$tan⁡(3x)cos⁡(2x)≈3x(1−2x2)=3x−6x3\tan(3x)\cos(2x)\approx3x(1-2x^2)=3x-6x^3$
To approximate $tan⁡(0.3)cos⁡(0.2)\tan(0.3)\cos(0.2)$ , notice that $3 x = 0.3$ , so $x = 0.1$ :
$3(0.1)-6(0.1)^3=0.294$

Example

Solving an approximate equation

Approximate the left-hand side; the $θ2\theta^2$ terms cancel:
$2cos⁡θ+(sin⁡θ+1)2≈2(1−θ22)+(θ+1)2=2−θ2+θ2+2θ+1=3+2θ\begin{aligned} 2\cos\theta+(\sin\theta+1)^2 &\approx 2\left(1-\frac{\theta^2}{2}\right)+(\theta+1)^2\\ &=2-\theta^2+\theta^2+2\theta+1\\ &=3+2\theta \end{aligned}$
Approximate the right-hand side:
$20tan⁡θ≈20θ20\tan\theta\approx20\theta$
Solve the resulting linear equation:
$3+2θ=20θ⇒18θ=3⇒θ=16=0.1673+2\theta=20\theta\Rightarrow 18\theta=3\Rightarrow \theta=\frac16=0.167$

Key Idea

When it becomes a straight line

If a small angle approximation leaves only a constant term and an $x$ term, the graph is approximately a straight line of the form $y = m x + c$ .

Exam technique

In the exam

Check whether the angle is in degrees or radians before using any formula.
For sectors and segments, label the radius, angle, arc, chord, and triangle clearly.
In small angle questions, substitute the approximations first, then simplify algebraically.
For percentage error, use $value∣×100\frac{|\text{approximation}-\text{true value}|}{|\text{true value}|}\times100$ .

Self review

Check yourself

Why must $θ\theta$ be in radians when using $s=rθs=r\theta$ ?
How do you find the area of a minor segment from a sector and a triangle?
Which small angle approximation would you use for $cos⁡(2x)\cos(2x)$ ?

Radians

What you'll learn

1. What is a radian?

2. Arc length and sector area

3. Chords and circular segments

4. Small angle approximations

5. Multiplying and solving with approximations

Recap questions

You've reached the end

Radians

What you'll learn

1. What is a radian?

2. Arc length and sector area

3. Chords and circular segments

4. Small angle approximations

5. Multiplying and solving with approximations

Recap questions

You've reached the end

Radians

What you'll learn

1. What is a radian?

2. Arc length and sector area

3. Chords and circular segments

4. Small angle approximations

5. Multiplying and solving with approximations

Recap questions

You've reached the end

Radians Revision Guide

Radians

What you'll learn

1. What is a radian?

2. Arc length and sector area

3. Chords and circular segments

4. Small angle approximations

5. Multiplying and solving with approximations

Recap questions

You've reached the end

Radians Revision Guide