Arc length and sector area

IB Mathematics: Applications and Interpretation SL· 15 min read

1. 1. Arcs, sectors and the fraction idea★☆☆☆☆⏱ 8 min

📘 Definition

Arc and sector

An arc is a portion of the circumference of a circle between two points. A sector is the pie-slice region enclosed by two radii and the arc between them. The angle at the centre between the two radii is called the central angle $θ$ , measured in degrees.

Example:

A pizza cut into 8 equal slices: each slice is a sector with central angle $36 0^{\circ} \div 8 = 4 5^{\circ}$ , and the curved crust of one slice is an arc.

The key idea is that a sector is simply a fraction of the whole circle. A full circle turns through $36 0^{\circ}$ , so a sector with central angle $θ$ represents the fraction $\frac{θ}{360}$ of the circle. To find the arc length, take that same fraction of the full circumference; to find the sector area, take that same fraction of the full area.

2. 2. Length of an arc★★☆☆☆⏱ 12 min

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📘 Definition

Arc length (degrees)

$l = \frac{θ}{360} \times 2 π r$

The length $l$ of an arc is the fraction $\frac{θ}{360}$ of the full circumference $2 π r$ , where $θ$ is the central angle in degrees and $r$ is the radius.

l = \frac{\theta}{360} \times 2\pi r

📐 Worked Example

A circle has radius 9 cm. Find the length of an arc that subtends a central angle of $8 0^{\circ}$ , correct to 2 decimal places.

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Step 1: Write down the arc length formula with $θ = 8 0^{\circ}$ and $r = 9$ .
2
$l = \frac{80}{360} \times 2\pi(9)$
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Step 2: The fraction of the circle is $\frac{80}{360} = \frac{2}{9}$ , and the full circumference is $2 π (9) = 18 π$ .
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$l = \frac{2}{9} \times 18\pi = 4\pi$
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Step 3: Evaluate on your GDC: $l = 4 π \approx 12.57$ cm.

Exam tip:

The arc length and sector area formulas are both in the IB formula booklet, so you do not need to memorise them, but you must know how to apply them and enter them correctly on your GDC.

3. 3. Area of a sector★★☆☆☆⏱ 12 min

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📘 Definition

Sector area (degrees)

$A = \frac{θ}{360} \times π r^{2}$

The area $A$ of a sector is the fraction $\frac{θ}{360}$ of the full circle area $π r^{2}$ , where $θ$ is the central angle in degrees and $r$ is the radius.

A = \frac{\theta}{360} \times \pi r^2

📐 Worked Example

A circular pizza has radius 15 cm and is cut into slices. One slice is a sector with a central angle of $4 5^{\circ}$ . Find the area of one slice, correct to 1 decimal place.

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Step 1: Write the sector area formula with $θ = 4 5^{\circ}$ and $r = 15$ .
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$A = \frac{45}{360} \times \pi (15)^2$
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Step 2: The fraction is $\frac{45}{360} = \frac{1}{8}$ , and the full area is $π (15)^{2} = 225 π$ .
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$A = \frac{1}{8} \times 225\pi = 28.125\pi$
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Step 3: Evaluate on your GDC: $A \approx 88.4 cm^{2}$ .

Exam tip:

Always include the correct units: arc length is a length (cm, m), while sector area is an area (cm squared, m squared). Dropping or squaring units incorrectly is an easy way to lose marks.

4. 4. Working backwards and combined problems★★★☆☆⏱ 15 min

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Exam questions often give you the arc length or sector area and ask you to find the radius or the central angle. Substitute the known values into the formula and rearrange, or enter the equation on your GDC and solve for the unknown. Many applied problems also ask for the perimeter of a sector, which is the arc length plus the two radii.

📐 Worked Example

A radar at an airport sweeps a sector-shaped region of radius 40 km. In one sweep it covers an area of $1400 km^{2}$ . Find the angle, in degrees, through which the radar sweeps, correct to the nearest degree.

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Step 1: Substitute $A = 1400$ and $r = 40$ into the sector area formula.
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$1400 = \frac{\theta}{360} \times \pi (40)^2$
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Step 2: Simplify $π (40)^{2} = 1600 π$ , so the equation becomes:
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$1400 = \frac{\theta}{360} \times 1600\pi$
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Step 3: Rearrange to make $θ$ the subject.
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$\theta = \frac{1400 \times 360}{1600\pi}$
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Step 4: Evaluate on your GDC: $θ \approx 10 0^{\circ}$ .

Exam tip:

When solving for an unknown angle or radius, you can enter the whole equation into your GDC's equation solver rather than rearranging by hand, which reduces the chance of algebra errors.

5. Common Pitfalls

Wrong move:

Using radians or the radian formulas $l = r θ$ and $A = \frac{1}{2} r^{2} θ$

Why:

IB AI SL works with the degree formulas only; the radian versions belong to the AI HL course and give wrong answers if the angle is in degrees

Correct move:

Always use $l = \frac{θ}{360} \times 2 π r$ and $A = \frac{θ}{360} \times π r^{2}$ with $θ$ in degrees

Wrong move:

Confusing the arc length formula with the sector area formula

Why:

The two formulas look similar but arc length uses the circumference $2 π r$ while sector area uses the area $π r^{2}$

Correct move:

Ask whether the answer should be a length or an area, and pick the matching formula and units

Wrong move:

Forgetting to include the two radii when finding the perimeter of a sector

Why:

The perimeter of a sector is the arc plus the two straight edges, not just the curved arc

Correct move:

Use $P = l + 2 r$ for the full perimeter of a sector

Wrong move:

Using the diameter instead of the radius in the formula

Why:

Both formulas are defined in terms of the radius $r$ ; substituting the diameter doubles the answer

Correct move:

If the diameter is given, halve it to get the radius before substituting

6. Quick Reference Cheatsheet

Concept	Rule	Notation
Fraction of circle	Central angle over 360	$\frac{θ}{360}$
Full circumference	Distance around the whole circle	$2 π r$
Full circle area	Area of the whole circle	$π r^{2}$
Arc length	Fraction of the circumference	$l = \frac{θ}{360} \times 2 π r$
Sector area	Fraction of the circle area	$A = \frac{θ}{360} \times π r^{2}$
Perimeter of sector	Arc plus two radii	$P = l + 2 r$
Find angle	Rearrange or use GDC solver	$θ = \frac{360 l}{2 π r}$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 1
Find the perimeter of a sector given radius and angle
2023 · 2
Modelling problem using sector area for a garden bed

What's Next

Arc length and sector area build directly on the circumference and area of a full circle, and they appear in many applied modelling questions across IB AI SL. Next, you will use right-angled and non-right-angled trigonometry to analyse triangles inside and around circles, and you will meet these circle measures again when working with three-dimensional shapes such as cones and cylinders, where curved surfaces are unrolled into sectors. Being confident with the fraction-of-a-circle idea makes all of these later topics much easier.