Use radians, arc length, and sector area

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

Geometry and Trigonometry

IB Mathematics: Analysis and Approaches SL — Study topic SL 3.4

A central angle describes how much of a circle is covered by a turn. Degrees divide a full turn into 360 equal parts; radians describe the turn using the circle’s radius and the length of its arc. Radians are especially useful when finding arc length and sector area. This lesson develops the needed formulas, shows how to select and use them, and checks results with sensible units and calculator settings. Assume a circle has radius

r>0

and that the angle describes the sector or arc being considered.

What you will learn

Explain a radian as an angle defined by a ratio of arc length to radius.
Convert between degrees and radians, and use radians in calculations.
Calculate arc length and sector area from a radius and central angle.
Interpret results in diagrams and real contexts, including units and calculator settings.

1. Prior knowledge: angles and the radian measure

A circle’s circumference is

2\pi r

, where

r

is its radius. A full turn is therefore

360^\circ

or

2\pi

radians. The symbol “rad” is often omitted after a radian measure, but the angle is still understood to be in radians.

One radian is the angle at the centre of a circle that subtends an arc whose length equals the radius. More generally, if an arc has length

s

in a circle of radius

r

, its angle in radians is the ratio

s/r

. This definition is why radians connect angles directly to lengths.

Because

180^\circ=\pi

radians, conversion uses a scale factor. To convert degrees to radians, multiply by

\pi/180

; to convert radians to degrees, multiply by

180/\pi

. Keep the exact form when it is useful, such as

150^\circ=5\pi/6

radians.

180^\circ=\pi\text{ rad}

A full turn is $360^\circ=2\pi$ radians.
Degrees to radians: multiply by $\pi/180$ .
Radians to degrees: multiply by $180/\pi$ .

2. Arc length: from angle to distance along the circle

For a central angle

\theta

measured in radians, the arc length is

s=r\theta

. The angle must be in radians for this formula. The result has the same length unit as the radius: if

r

is in centimetres, then

s

is in centimetres.

The formula can be understood by comparing the arc with the whole circumference. An angle of

\theta

radians is a fraction

\theta/(2\pi)

of a full turn, so the corresponding arc is that same fraction of

2\pi r

. Simplifying gives

s=r\theta

. For an angle given in degrees, either convert it first or use the matching fraction of the full circumference.

A useful diagram labels the centre, radius, and central angle, and marks the curved part as the arc. Numerically, a larger radius or a larger positive angle gives a longer arc. In a context such as a rotating wheel, the distance travelled by a point on its rim is the arc length traced by that point. The wheel’s radius and the angle turned must use consistent units and radians in the formula.

s=r\theta

Use $s=r\theta$ only when $\theta$ is in radians.
Arc length is a distance, not an angle.
Use the radius, not the diameter, in the formula.

3. Sector area and connected representations

A sector is the region bounded by two radii and the arc between them. Its area is the same fraction of the circle’s total area as its central angle is of a full turn. Since the full angle is

2\pi

radians and the circle’s area is

\pi r^2

, this gives

A=\frac12r^2\theta

for an angle in radians.

Here,

A

is an area and is measured in square units. Check that the radius is squared: if it is measured in metres, the answer is in square metres. For a degree angle, the sector is the fraction

\theta/360^\circ

of the full circle, so its area can be found using that fraction or by converting the angle to radians before applying the radian formula.

The arc and sector formulas are related: the same radius and angle determine both. If the radius stays fixed and the angle doubles, both arc length and sector area double. In a sketch, the arc is a curved boundary, while the sector is the whole slice-shaped region. This distinction helps prevent reporting a length when the question asks for an area.

A graphing calculator can check numerical evaluation, especially when an angle is not a simple fraction of

\pi

. Set the calculator to radian mode before evaluating expressions such as

r\theta

or

\frac12r^2\theta

. The calculator does not decide which formula fits the situation: identify whether the requested quantity is an arc distance or a sector area, and show the formula and substitution.

A=\frac12r^2\theta

Use $A=\frac12r^2\theta$ for a sector when $\theta$ is in radians.
Sector area uses square units; arc length uses length units.
A calculator’s angle mode matters when evaluating trigonometric or angle-based calculations.

4. Choosing a method and checking a result

First identify what is known and what is required. Convert the angle if needed, then select the formula that matches the quantity: arc length for a distance along the circumference, sector area for a region. Rearrange algebraically if a radius or angle is unknown. For example, from

s=r\theta

, the radius is

s/\theta

when

\theta\ne0

.

In a diagram, the angle is located at the centre, not at the circumference. In a numerical solution, retain an exact value such as a multiple of

\pi

where possible; if a decimal is requested, use a calculator in radian mode and state the requested precision. In a context, attach units and consider whether the size is plausible: an angle smaller than a full turn should give an arc shorter than the circumference and a sector smaller than the whole circle.

For an exam-style response, make the reasoning visible: write the relevant formula, substitute values with units, calculate, and state the result with suitable units and accuracy. A calculator display alone is not a complete explanation.

\frac{s}{2\pi r}=\frac{\theta}{2\pi}

Match the formula to the quantity being asked for.
Check angle units before substituting.
Use units and a magnitude check to assess the answer.

Worked example

Convert an angle and find an arc

A circular track has radius

12

m. Find the arc length for a central angle of

75^\circ

. Give an exact answer and a decimal to three significant figures.

Convert the angle
The arc formula requires radians. Multiply the degree measure by $\pi/180$ .
$75^\circ\times\frac{\pi}{180^\circ}=\frac{5\pi}{12}$
Apply the arc formula
Substitute the radius and angle into $s=r\theta$ . The result is a length, so its unit is metres.
$s=12\times\frac{5\pi}{12}=5\pi\text{ m}$
Evaluate and round
Use a calculator to evaluate the exact result, then round to three significant figures.
$5\pi\text{ m}\approx15.7\text{ m}$

Answer:

5\pi

m, approximately

15.7

m.

Check: The angle is less than a quarter-turn, so the arc should be less than one quarter of the circumference. The circumference is

24\pi

m, and one quarter is

6\pi

m; the result

5\pi

m is plausible.

Worked example

Find a sector area from a radian angle

A fan-shaped region has radius

8

cm and central angle

1.2

radians. Find its area to one decimal place.

Select the area formula
The question asks for the area of a sector, and the angle is already in radians.
$A=\frac12r^2\theta$
Substitute the values
Square the radius before multiplying by half the angle. The area unit is square centimetres.
$A=\frac12(8)^2(1.2)=38.4\text{ cm}^2$
Report the requested accuracy
The calculation is exact at the stated input precision, and to one decimal place it remains $38.4$ .
$A=38.4\text{ cm}^2$

Answer:

38.4\text{ cm}^2

.

Check: The full circle has area

64\pi\text{ cm}^2

, about

201.1\text{ cm}^2

. Since

1.2

radians is less than a full turn of

2\pi

radians, the sector area must be less than the full-circle area; it is.

Worked example

Recover an angle from an arc

An arc is

9.6

cm long in a circle of radius

6

cm. Find its central angle in radians and degrees.

Rearrange the arc formula
Start with $s=r\theta$ and divide both sides by the non-zero radius to make the angle the subject.
$\theta=\frac{s}{r}$
Substitute the measurements
The centimetre units cancel, leaving an angle in radians.
$\theta=\frac{9.6}{6}=1.6\text{ rad}$
Convert to degrees
Multiply the radian value by $180/\pi$ . Round the degree measure to one decimal place.
$1.6\times\frac{180}{\pi}\approx91.7^\circ$

Answer:

1.6

radians, approximately

91.7^\circ

.

Check: The arc length is greater than the radius, so the angle is greater than one radian. The result of

1.6

radians is therefore reasonable.

Common mistakes and how to avoid them

Substituting a degree measure directly into

s=r\theta

or

A=\frac12r^2\theta

.

Correction: Convert degrees to radians first, or use the appropriate fraction of the full circumference or full-circle area.

Using the diameter instead of the radius.

Correction: Identify the radius from the centre to the circle’s edge. If only the diameter is given, halve it before substitution.

Giving sector area in linear units or arc length in square units.

Correction: Arc length is measured in units such as cm or m; sector area is measured in square units such as

\text{cm}^2

or

\text{m}^2

.

Reporting an unexplained calculator decimal.

Correction: Show the formula and substitution, check the calculator is in radian mode when appropriate, and state units and rounding.

Lesson summary

A full turn measures $360^\circ$ or $2\pi$ radians; convert degrees and radians using $180^\circ=\pi$ radians.
For a radian angle, arc length is $s=r\theta$ .
For a radian angle, sector area is $A=\frac12r^2\theta$ .
Check what is being measured, use the correct units, and keep exact values where useful.

Check your understanding

Question 1

What is

120^\circ

in radians?

$\frac{2\pi}{3}$
$\frac{3\pi}{2}$
$\frac{2\pi}{5}$
correctIndex של

Show answer and explanation

\frac{2\pi}{3}

Multiply by

\pi/180

:

120\times\pi/180=2\pi/3

.

Question 2

A circle has radius

5

cm and angle

0.8

radians. What is the arc length?

$4$ cm
$6.25$ cm
$10$ cm
correctIndex st

Show answer and explanation

4

cm

Use

s=r\theta

:

5(0.8)=4

cm.

Question 3

A sector has radius

3

m and angle

2

radians. What is its area?

$9\text{ m}^2$
$6\text{ m}^2$
$18\text{ m}^2$
correctIndex

Show answer and explanation

9\text{ m}^2

Use

A=\frac12r^2\theta=\frac12(9)(2)=9\text{ m}^2

.

Key terms

Radian: An angle measure defined by the ratio of the subtended arc length to the circle’s radius.
Central angle: An angle whose vertex is at the centre of a circle.
Arc: A portion of a circle’s circumference.
Sector: A region of a circle bounded by two radii and the arc between them.

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Published by DoAssignment. This reviewed lesson follows International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL, study topic SL 3.4. It is a study resource, not an official curriculum publication.

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