Arcs & Sectors

When studying discs and circles, arcs and sectors are absolute must-knows. Here, we'll be learning how to calculate:

the length of a cicular arc, also called arc length
the area of a circular sector

both of these are illustrated here.

We start this section by stating the formula for both the arc length as well as the area of a sector. To understand these formula, make sure to watch the tutorial underneath them.

Further-on we then look at some typical quiz-type, and exam-style, questions on this topic.

What we'll Learn

To learn about the arc length and area of a circular sector, this page is organized as follows:

Video tutorial: formula and worked examples
Formula for the arc length, \(l\)
Formula for the area of a circular sector, \(A\)
Worked Examples 1 & 2
Exam Style Question 1
Exam Style Question 2

Arcs & Sectors Formula - Lesson

In the following tutorial we learn formula for the arc length and the area of a circular sector.

Formula : Arc Length

Given a circular sector whose arc length \(l\) is subtended (formed) by an angle \(\theta\), the arc length can be calculated using the following formula: \[l = \frac{\theta}{360^{\circ}}\times 2\pi r\]

Explanation

The arc length of a circular sector can be thought of as a fraction of a disc's circumference.

Reminder: Circumference of a Disc/Circle

Given a disc (or a circle) of radius \(r\), its circumference (also called perimeter) is given by the formula: \[\text{Perimeter}=2\pi r\]

Since a disc's total circumference is \(2 \pi r\) and since moving along the entire length of the circumference forms a \(360^{\circ}\) angle at the center of a disc : if an arc forms an angle \(\theta \) at the center of the disc then the arc length will be: \[\frac{\theta}{360} \times \text{Circumference}\] Which leads to the formula for the arc length: \[l = \frac{\theta}{360^{\circ}}\times 2\pi r\]

Formula : Area of a Sector

Given a circular sector of radius \(r\), which forms an angle \(\theta \) at the center of the disc it was taken from, the area of the circular sector is given by: \[A = \frac{\theta}{360^{\circ}}\pi r^2 \]

Explanation

A circular sector can be thought of as a fraction of (or a piece of) a disc; like a slice of pizza.

Reminder: Area of a Disc

Given a disc of radius \(r\), its area is given by the formula: \[\text{Area} = \pi r^2\]

Given the angle at the center of a disc is \(360^{\circ}\), if the arc formed by the sector is \(\theta \) then the area the area of the sector, \(A\), will be: \[A = \frac{\theta}{360^{\circ}}\times \text{Area of Disc}\]

Finally since the area of a disc is \(\pi r^2\) we can write the formula for the area of a circular sector: \[A = \frac{\theta}{360^{\circ}}\times \pi r^2\]

Example 1

Given the circular sector shown here, calculate:

its arc length \(l\)

its area, \(A\).

Solution

To find the arc length \(l\) we use the fomula (stated further up): \[l = \frac{\theta}{360}\times 2 \pi r\] Now, using the fact that:
- \(\theta = 48^{\circ}\)
- \(r = 7\ \text{cm}\)
and using our calculator, we obtain: \[\begin{aligned} l & = \frac{48}{360}\times 2 \pi \times 7 \\ & = 5.8643063 \\ l &= 5.86\ \text{cm} \quad \text{(rounded to 3 significant figures)} \end{aligned}\]

To find the area \(A\) we use the formula: \[A = \frac{\theta}{360}\times \pi r^2\] Once more, we use the fact that:
- \(\theta = 48^{\circ}\)
- \(r = 7\ \text{cm}\)
and using our calculator, we obtain: \[\begin{aligned} A & = \frac{48}{360}\times \pi \times 7^2 \\ & = 20.525072 \\ A &= 20.5\ \text{cm}^2 \quad \text{(rounded to 3 significant figures)} \end{aligned}\]

Example 2

Given the circular sector shown here, calculate:

Without using a calculator and writing your answers in terms of \(\pi \):
1. its arc length \(l\)
2. its area, \(A\).

Using your calculator, write each of your previous answers rounded to 3 significant figures.

Solution

1. To find the arc length \(l\) we use the fomula (stated further up): \[l = \frac{\theta}{360}\times 2 \pi r\] Now, using the fact that:
  - \(\theta = 300^{\circ}\)
  - \(r = 6\ \text{cm}\)
  and using our calculator, we obtain: \[\begin{aligned} l & = \frac{300}{360}\times 2 \pi \times 6 \\ & = \frac{300 \times 2 \times \pi \times 6}{360} \\ & = \frac{3600 \pi}{360} \\ l &= 10 \pi \ \text{cm} \end{aligned}\]
2. To find the area \(A\) we use the formula: \[A = \frac{\theta}{360}\times \pi r^2\] Once more, we use the fact that:
  - \(\theta = 300^{\circ}\)
  - \(r = 6\ \text{cm}\)
  and using our calculator, we obtain: \[\begin{aligned} A & = \frac{300}{360}\times \pi \times 7^2 \\ & = \frac{5}{6}\times \pi \times 49 \\ \quad \text{Note: } \frac{300}{360} = \frac{5}{6} & = \frac{5\times 49}{6}\times \pi \\ A & = \frac{245}{6}\pi \ \text{cm}^2 \quad \text{(rounded to 3 significant figures)} \end{aligned}\]

Plugging each of our previous results into our calculator we can find their values rounded to 3 significant figures:

Arc Length

Plugging: \[10 \pi \] into our calculator we obtain: \[\begin{aligned} l & = 10 \pi \\ & = 31.415927 \\ l & = 31.4 \ \text{cm} \quad \text{(rounded to 3 significant figures)} \end{aligned}\]

Sector's Area

Plugging: \[ \frac{245}{6}\pi \] into our calculator we obtain: \[\begin{aligned} A & = \frac{245}{6}\pi \\ & = 128.2817 \\ A & = 128 \ \text{cm}^2 \quad \text{(rounded to 3 significant figures)} \end{aligned}\]