What is the length of the arc of a circle of radius 30 cm subtends an angle 5 by 6 at the Centre?

Unless it is specifically requested, the $\pi$ value obtained by accessing the appropriate function in a scientific or graphic calculator should be used in any calculation involving ${\pi}$.

Índice Show

What is Sector of a circle?
Area of a sector
Length of the Arc of Sector Formula
Video Lessons on Circles
Perimeter of a Sector
Perimeter of a Sector Formula
Practice Questions

ALL angles in this unit will be specified in degrees.

Measuring arc length

In any circle, the length of an arc ($l$) is proportional to the angle ($\theta$) it creates (subtends) at the centre.

For a circle of radius $r$, the circumference will be $2\pi$ units.

Thus,

$$\frac{\text{the length of the arc}}{\text{circumference}}=\frac{\theta}{360^{\circ}},$$

so $\begin{aligned}[t] &\frac{l}{2\pi r } = \frac{\theta}{360^{\circ}}.\\ \therefore l &= \frac{\theta}{360^{\circ}} \times {2\pi r}\\ &= \frac{\theta}{180^{\circ}} \times {\pi r}\\ &= {r} \times \frac{\pi}{180^{\circ}} \times {\theta} \text{ (Alternate formulations)} \end{aligned}$

A circle has a radius of 30 cm. Find the length of an arc subtending an angle of ${75^{\circ}}$ at the centre, correct to one decimal place.

Solution

$\begin{aligned}[t] \mbox{Length of arc}&=\frac{r\pi}{180^{\circ}}\times\theta\\ &=\frac{30\pi}{180^{\circ}}\times 75^{\circ}\\ &=39.2699\ldots\\ &\approx 39.3\text{ cm} \end{aligned}$

An arc of a circle has a length of 30 cm. If the radius of the circle is 25 cm, what angle, correct to the nearest whole degree, does the arc subtend at the centre?

Solution

$\begin{aligned}[t] 30&=\frac{r\pi}{180^{\circ}}\times {\theta}\\ &=\frac{25\pi}{180^{\circ}}\times {\theta}\\ \theta &=\frac{30 \times 180^{\circ}}{25 \times \pi}\\ &=68.754\ldots\approx {69^{\circ}} \end{aligned}$

Sector

The area of a sector depends not only on the radius of the circle involved, but also the angle between the two straight edges.

We can see this in the following table :

Angle	Fraction of circle area	Area rule
$180^{\circ}$	$\dfrac{180^{\circ}}{360^{\circ}}=\dfrac{1}{2}$	${A}=\dfrac{1}{2}\times {\pi r^2}$
$90^{\circ}$	$\dfrac{90^{\circ}}{360^{\circ}}=\dfrac{1}{4}$	${A}=\dfrac{1}{4}\times {\pi r^2}$
$45^{\circ}$	$\dfrac{45^{\circ}}{360^{\circ}}=\dfrac{1}{8}$	${A}=\dfrac{1}{8}\times {\pi r^2}$
$30^{\circ}$	$\dfrac{30^{\circ}}{360^{\circ}}=\dfrac{1}{12}$	${A}=\dfrac{1}{12}\times {\pi r^2}$
${\theta}$	$\dfrac{\theta}{360^{\circ}}$	${A}=\dfrac{\theta}{360}\times {\pi r^2}$

Find the area of a sector with a radius of 40 mm, that contains an angle of ${60^{\circ}}$, correct to the nearest square millimetre.

Solution

$$A=\frac{\theta}{360^{\circ}}\times {\pi r^2}=\frac{60^{\circ}}{360^{\circ}}\times {\pi 40^2}=923.628\ldots \approx 924\text{ mm}^2$$

Areas of segments

Where a chord subtends an angle ${\theta}$ at the centre of a circle of radius $r$, the area of the minor segment is given by:

\[\text{ Area} = \frac{1}{2} \times {r^2} \left(\frac{2 \pi \theta}{360^{\circ}} - {\sin \theta}\right)\]

The area of the sector that includes the segment is given by $A=\dfrac{\theta}{360^{\circ}}\times {\pi r^2}$

or rewritten as

\[A = \dfrac{1}{2}\times {r^2} \times \dfrac{2 \pi \theta}{360^{\circ}}.\]

The area of the triangle whose boundaries are the two radii and the chord is given by

$\dfrac{1}{2}\times {r^2} \times {\sin \theta} \qquad \left(\text{from the Sine Area Rule} \quad \dfrac{1}{2}\times {ab} \sin {C}\right).$

Hence the area of the segment (minor) can be calculated by subtracting the area of the triangle from the area of the sector.

The area of the major segment can be calculated by taking the area of the minor segment from the total area of the circle.

Find the area, correct to two decimal places, of the minor segment in a circle of radius 10 cm where the angle subtended at the centre of the circle by the chord is ${85^{\circ}}$.

Solution

$\begin{aligned}[t] &\mbox{ Area}\\ &= \frac{1}{2} \times {r^2} \left(\frac{2 \pi \theta}{360^{\circ}} - {\sin \theta}\right)\\ & = \frac{1}{2} \times {10^2} \left(\frac{2 \times \pi \times 85^{\circ}}{360^{\circ}} - {\sin {85}^{\circ}}\right) \\ &= 24.366\ldots \\ &\approx 24.37\text{cm}^2 \end{aligned}$

Find the area, correct to one decimal place, of the minor segment in a circle, of radius 15 cm where the chord length is also 15 cm.

Solution

If the chord length is the same as the radius of the circle, then the triangle that is formed will be equilateral (all three sides are 15 cm!), and the angle subtended at the centre by the chord will be ${60^{\circ}}$ (the angle inside an equilateral triangle).

$$ \text{Area} = \frac{1}{2} \times {r^2} \left(\frac{2 \pi \theta}{360^{\circ}} - {\sin \theta}\right) = \frac{1}{2} \times {15^2} \left(\frac{2 \times \pi \times 60^{\circ}}{360^{\circ}} - {\sin 60^{\circ}}\right)= 20.381\ldots \approx 20.4\text{ cm}^2$$

Find the area, correct to two decimal places, of the minor segment in a circle, of radius 13 cm where the chord length is 21 cm.

Solution

To find the angle subtended at the centre by the chord, we will have to use the Cosine Rule

\begin{align*} A& = \cos^{-1} \left(\frac{b^2 + c^2 - a^2}{2bc}\right)\\ \mbox{ where} b& = c = \mbox{radius}\\ a& = \mbox{chord length}, \\ A&= \mbox{angle subtended at centre of circle}.\\ \text{So A}& = \cos^{-1} \left(\frac{13^2 + 13^2 - 21^2}{2 \times 13 \times 13}\right)\\ & = 107.74^{\circ} \end{align*} \begin{align*} \text{ Area}&= \frac{1}{2} \times {r^2} \left(\frac{2 \pi \theta}{360^{\circ}} - {\sin \theta}\right)\\ &= \frac{1}{2} \times {13^2} \left(\frac{2 \times \pi \times 107.74^{\circ}}{360^{\circ}} - {\sin 107.74^{\circ}}\right)\\ &= 78.409\ldots\\ &\approx 78.41\text{ cm}^2 \end{align*}

Next page - Content - The Earth as a sphere

A circle has always been an important shape among all geometrical figures. There are various concepts and formulas related to a circle. The sectors and segments are perhaps the most useful of them. In this article, we shall focus on the concept of a sector of a circle along with area and perimeter of a sector.

A sector is said to be a part of a circle made of the arc of the circle along with its two radii. It is a portion of the circle formed by a portion of the circumference (arc) and radii of the circle at both endpoints of the arc. The shape of a sector of a circle can be compared with a slice of pizza or a pie.

Before we start learning more about the sector, first let us learn some basics of the circle.

What is a Circle?

A circle is a locus of points equidistant from a given point located at the centre of the circle. The common distance from the centre of the circle to its point is called the radius. Thus, the circle is defined by its centre (o) and radius (r). A circle is also defined by two of its properties, such as area and perimeter. The formulas for both the measures of the circle are given by;

- Area of a circle = πr2
- The perimeter of a circle = 2πr

What is Sector of a circle?

The sector is basically a portion of a circle which could be defined based on these three points mentioned below:

A circular sector is the portion of a disk enclosed by two radii and an arc.
A sector divides the circle into two regions, namely Major and Minor Sector.
The smaller area is known as the Minor Sector, whereas the region having a greater area is known as Major Sector.

Area of a sector

In a circle with radius r and centre at O, let ∠POQ = θ (in degrees) be the angle of the sector. Then, the area of a sector of circle formula is calculated using the unitary method.

For the given angle the area of a sector is represented by:

The angle of the sector is 360°, area of the sector, i.e. the Whole circle = πr2

When the Angle is 1°, area of sector = πr2/360°

So, when the angle is θ, area of sector, OPAQ, is defined as;

Let the angle be 45 °. Therefore the circle will be divided into 8 parts, as per the given in the below figure;

Now the area of the sector for the above figure can be calculated as (1/8) (3.14×r×r).

Thus the Area of a sector is calculated as:

Length of the Arc of Sector Formula

Similarly, the length of the arc (PQ) of the sector with angle θ, is given by;

l = (θ/360) × 2πr (or) l = (θπr) /180

Area of Sector with respect to Length of the Arc

If the length of the arc of the sector is given instead of the angle of the sector, there is a different way to calculate the area of the sector. Let the length of the arc be l. For the radius of a circle equal to r units, an arc of length r units will subtend 1 radian at the centre. Hence, it can be concluded that an arc of length l will subtend l/r, the angle at the centre. So, if l is the length of the arc, r is the radius of the circle and θ is the angle subtended at the centre, then;

θ = l/r, where θ is in radians.

When the angle of the sector is 2π, then the area of the sector (whole sector) is πr2

When the angle is 1, the area of the sector = πr2/2π = r2/2

So, when the angle is θ, area of the sector = θ × r2/2

A = (l/r) × (r2/2)

Video Lessons on Circles

Some examples for better understanding are discussed here.

Examples

Example 1: If the angle of the sector with radius 4 units is 45°, then find the length of the sector.

Solution: Area = (θ/360°) × πr2

= (45°/360°) × (22/7) × 4 × 4

= 44/7 square units

The length of the same sector = (θ/360°)× 2πr

l = (45°/360°) × 2 × (22/7) × 4

l = 22/7

Example 2: Find the area of the sector when the radius of the circle is 16 units, and the length of the arc is 5 units.

Solution: If the length of the arc of a circle with radius 16 units is 5 units, the area of the sector corresponding to that arc is;

A = (lr)/2 = (5 × 16)/2 = 40 square units.

Perimeter of a Sector

The perimeter of the sector of a circle is the length of two radii along with the arc that makes the sector. In the following diagram, a sector is shown in yellow colour.

The perimeter should be calculated by doubling the radius and then adding it to the length of the arc.

Perimeter of a Sector Formula

The formula for the perimeter of the sector of a circle is given below :

Perimeter of sector = radius + radius + arc length

Perimeter of sector = 2 radius + arc length

Arc length is calculated using the relation :

Arc length = l = (θ/360) × 2πr

Therefore,

Perimeter of a Sector = 2 Radius + ((θ/360) × 2πr )

Example

A circular arc whose radius is 12 cm, makes an angle of 30° at the centre. Find the perimeter of the sector formed. Use π = 3.14.

Solution :

Given that r = 12 cm,

θ = 30° = 30° × (π/180°) = π/6

Perimeter of sector is given by the formula;

P = 2 r + r θ

P = 2 (12) + 12 ( π/6)

P = 24 + 2 π

P = 24 + 6.28 = 30.28

Hence, Perimeter of sector is 30.28 cm

Practice Questions

A sector is cut from a circle of radius 21 cm. The angle of the sector is 150o. Find the length of the arc, perimeter and area of the sector.
A pizza with 21 cm radius is divided into 6 equal slices (slices are in the shape of a sector). Find the area of each slice.
The minute hand of a clock is 7 cm long. Find the area swept by the minute hand in 35 minutes.

The sector of a circle is the region bounded by two radii and an arc of a circle.

Let PQ is an arc of a circle of radius r and centre at O if PQ subtends angle 𝜃 at the centre of the circle. Then, the arc length of PQ = 𝜃/360o (2𝜋r), where 𝜃 is measured in degrees.

The formula for the area of the sector of a circle is 𝜃/360o (𝜋r2) where r is the radius of the circle and 𝜃 is the angle of the sector.

The formula for the perimeter of the sector of a circle is [2r + 𝜃/360o (2𝜋r)] where r is the radius of the circle and 𝜃 is the angle of the sector.