Area of Sector and Length of Arc Revision Notes for Leaving Cert Mathematics

Area of Sector and Length of Arc

What are sectors and arcs?

When working with circles, you'll often need to calculate measurements for parts of circles rather than the whole circle. This is where sectors and arcs become essential.

A sector is a slice or wedge-shaped portion of a circle, similar to a slice of pizza. It's formed when two radii extend from the centre to the circumference.

An arc is the curved boundary that forms the edge of a sector. Think of it as the curved part of the pizza crust.

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The pizza analogy is particularly helpful for visualising sectors - just imagine cutting a circular pizza into slices. Each slice is a sector, and the curved crust of each slice is the arc.

The minor arc refers to the shorter arc between two points on a circle's circumference, whilst the central angle is the angle formed at the centre of the circle between the two radii.

Essential formulas

Before diving into sector calculations, remember these fundamental circle formulas:

Circumference of a circle = $2\pi r$
Area of a circle = $\pi r^2$

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Critical Formulas for Sectors:

The sector formulas build directly on the full circle formulas:

Area of sector = $\frac{\theta}{360°} \times \pi r^2$
Length of arc = $\frac{\theta}{360°} \times 2\pi r$

Where:

$\theta$ = central angle in degrees
$r$ = radius of the circle

These formulas are essential for all sector calculations!

Understanding the formulas

These sector formulas work because they calculate what fraction of the full circle your sector represents. The fraction $\frac{\theta}{360°}$ tells you what portion of the complete 360° circle you're working with.

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Understanding the Fraction:

Think of $\frac{\theta}{360°}$ as asking: "What fraction of a complete rotation is my angle?"

For example:

A 90° sector is $\frac{90°}{360°} = \frac{1}{4}$ of the full circle
A 180° sector is $\frac{180°}{360°} = \frac{1}{2}$ of the full circle

This fraction then multiplies the full circle area or full circle circumference to give you the sector measurement.

Worked example 1: Finding arc length and sector area

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Worked Example: Basic Sector Calculations

Question: A circle has radius 9cm and central angle 120°. Using $\pi = \frac{22}{7}$ , find: (i) the length of the minor arc AB (ii) the area of the shaded sector AOB

Solution:

(i) Length of arc AB:

Length of arc = $\frac{\theta}{360°} \times 2\pi r$

Length of arc AB = $\frac{120°}{360°} \times 2\pi r$

= $\frac{120}{360} \times 2 \times \frac{22}{7} \times 9$

= $\frac{1}{3} \times 2 \times \frac{22}{7} \times 9$

= 18.86 cm

(ii) Area of sector AOB:

Area of sector = $\frac{\theta}{360°} \times \pi r^2$

Area of sector AOB = $\frac{120°}{360°} \times \pi r^2$

= $\frac{120}{360} \times \frac{22}{7} \times 9^2$

= $\frac{1}{3} \times \frac{22}{7} \times 81$

= 84.86 cm²

Worked example 2: Finding the radius

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Worked Example: Finding Radius from Sector Area

Question: The area of sector AOB is 205 cm². The central angle is 120°. Find the length of the radius.

Solution:

Let $r$ be the radius.

Area of sector AOB = $\frac{120°}{360°} \times \pi r^2 = 205$ cm²

$\frac{\pi r^2}{3} = 205$

$\pi r^2 = 205 \times 3$ (multiply both sides by 3)

$r^2 = \frac{205 \times 3}{\pi}$

$r^2 = 195.76$ (using π key on calculator)

$r = \sqrt{195.76} = 13.99$

Therefore, radius = 14 cm (correct to the nearest cm)

Worked example 3: Different angle and area

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Worked Example: Finding Radius with 45° Sector

Question: Find the radius of a circle if a sector with central angle 45° has area 77 cm².

Solution:

Area of sector = $\frac{\theta}{360°} \times \pi r^2$

$77 = \frac{45°}{360°} \times \pi r^2$

$77 = \frac{1}{8} \times \pi r^2$

$\pi r^2 = 77 \times 8 = 616$

$r^2 = \frac{616}{\pi}$

$r^2 = 196.13$ (using π key on calculator)

$r = \sqrt{196.13} = 14.0$

Therefore, radius = 14.0 cm

Worked example 4: Larger sector

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Worked Example: Finding Radius with 140° Sector

Question: A sector has central angle 140° and area 276 cm². Find the radius.

Solution:

Area of sector = $\frac{\theta}{360°} \times \pi r^2$

$276 = \frac{140°}{360°} \times \pi r^2$

$276 = \frac{140}{360} \times \pi r^2$

$\pi r^2 = 276 \times \frac{360}{140}$

$\pi r^2 = 276 \times \frac{18}{7} = 712.3$

$r^2 = \frac{712.3}{\pi} = 226.7$

$r = \sqrt{226.7} = 15.1$

Therefore, radius = 15.1 cm

Exam tips and common mistakes

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Key Exam Strategies:

Always check whether the angle is given in degrees or radians
Round your final answer to the precision requested in the question
Use the π key on your calculator for accuracy, unless told to use a specific value
Show all working clearly - marks are awarded for method even if the final answer is incorrect

Common Mistakes to Avoid:

Forgetting to convert the angle to degrees if given in another unit
Using the wrong formula (arc length vs sector area)
Not simplifying fractions before calculating
Rounding too early in multi-step calculations

Memory Tip: Remember that both formulas start with the same fraction $\frac{\theta}{360°}$ - this represents what portion of the full circle you're calculating.

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Key Points to Remember:

A sector is a slice of a circle formed by two radii, while an arc is the curved boundary of that sector
Area of sector = $\frac{\theta}{360°} \times \pi r^2$ and Length of arc = $\frac{\theta}{360°} \times 2\pi r$
Both formulas use the fraction $\frac{\theta}{360°}$ to find what portion of the complete circle you're working with
When finding an unknown radius, rearrange the sector area formula and solve for $r^2$ , then take the square root
Always show your working step-by-step and round to the precision requested in the question

Area of Sector and Length of Arc (Leaving Cert Mathematics): Revision Notes