What is the angle subtended at the centre of a circle of radius 10cm by an arc of length 5pie cm?

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 ${\pi}$.

Table of Contents Show

How to Calculate the Area of a Sector and the Length of an Arc
How to Find the Arc Length
How to Find the Sector Area

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*}

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Find the angle subtended at the centre of circle of radius 5cm by an arc of length `((5pi)/3)` cm

Radius (r) = 5 cm

𝜃 = angle subtended at centre (degrees)

Length of Arc =`theta/360^@`× 2𝜋r cm

But arc length =`5pi/3`𝑐𝑚

`theta/360^@× 2pi × 5 =(5pi)/3`

`theta=(360^@×pi)/(3×2pi)= 60^@`

∴ Angle subtended at centre = 60°

Concept: Circumference of a Circle

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Page 2

An arc of length 20𝜋 cm subtends an angle of 144° at centre of circle. Find the radius of the circle.

Length of arc = 20𝜋cm

Let radius = ‘r’ cm

O = angle subtended at centre = 144°

Length of arc =`theta/360^@`× 2𝜋𝑟

=`144/360× 2pir =(4pi)/5r`

=`(4pi)/5r = 20pi`

`r =(20pi×5)/(4pi)`= 25 cms

Concept: Circumference of a Circle

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Page 3

An arc of length 15 cm subtends an angle of 45° at the centre of a circle. Find in terms of 𝜋, radius of the circle.

Length of arc = 15cm

𝜃 = angle subtended at centre = 45°

Let radius = r cm

arc length = `theta/360^@`× 2𝜋𝑟

=`45^@/360^@`× 2𝜋𝑟

`45/360`× 2𝜋𝑟 = 15

𝑟 =`(15×360)/(45×2pi)=60/pi`cms

𝑅𝑎𝑑𝑖𝑢𝑠 =`60/pi`cms

Concept: Circumference of a Circle

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Go back to Calculators page

To use the arc length calculator, simply enter the central angle and the radius into the top two boxes. If we are only given the diameter and not the radius we can enter that instead, though the radius is always half the diameter so it’s not too difficult to calculate.

The calculator will then determine the length of the arc. It will also calculate the area of the sector with that same central angle.

How to Calculate the Area of a Sector and the Length of an Arc

Our calculators are very handy, but we can find the arc length and the sector area manually. It’s good practice to make sure you know how to calculate these measurements on your own.

How to Find the Arc Length

An arc length is just a fraction of the circumference of the entire circle. So we need to find the fraction of the circle made by the central angle we know, then find the circumference of the total circle made by the radius we know. Then we just multiply them together. Let’s try an example where our central angle is 72° and our radius is 3 meters.

First, let’s find the fraction of the circle’s circumference our arc length is. The whole circle is 360°. Let’s say our part is 72°. We make a fraction by placing the part over the whole and we get $\frac{72}{360}$, which reduces to $\frac{1}{5}$. So, our arc length will be one fifth of the total circumference. Now we just need to find that circumference.

The circumference can be found by the formula C = πd when we know the diameter and C = 2πr when we know the radius, as we do here. Plugging our radius of 3 into the formula, we get C = 6π meters or approximately 18.8495559 m.

Now we multiply that by $\frac{1}{5}$ (or its decimal equivalent 0.2) to find our arc length, which is 3.769911 meters. Note that our units will always be a length.

How to Find the Sector Area

Just as every arc length is a fraction of the circumference of the whole circle, the sector area is simply a fraction of the area of the circle. So to find the sector area, we need to find the fraction of the circle made by the central angle we know, then find the area of the total circle made by the radius we know. Then we just multiply them together. Let’s try an example where our central angle is 72° and our radius is 3 meters.

First, let’s find the fraction of the circle’s area our sector takes up. The whole circle is 360°. Our part is 72°. We make a fraction by placing the part over the whole and we get $\frac{72}{360}$, which reduces to $\frac{1}{5}$. So, our sector area will be one fifth of the total area of the circle. Now we just need to find that area.

The area can be found by the formula A = πr2. Plugging our radius of 3 into the formula we get A = 9π meters squared or approximately 28.27433388 m2.

Now we multiply that by $\frac{1}{5}$ (or its decimal equivalent 0.2) to find our sector area, which is 5.654867 meters squared. Note that our answer will always be an area so the units will always be squared.