Length of an Arc of a Sector

The length of an arc of a sector is the distance along the curved edge of the sector. A sector is a portion of a circle enclosed by two radii and the arc intercepted by those radii. Understanding how to calculate arc length is crucial in geometry and trigonometry, as it allows us to determine the distance along a curved path, which is a key concept in measuring shapes in the real world.

Formulae

The formula to calculate the length of an arc is derived directly from the concept that the arc length is a fraction of the circumference of the entire circle. The fraction is determined by the central angle ($ \theta $) of the sector (measured in degrees) divided by 360 degrees (the total degrees in a circle). The formula is as follows:

$ \text{Arc Length} = \left( \frac{\theta}{360^\circ} \right) \times 2\pi r $

Where:

• $ \theta $ is the central angle of the sector in degrees.
• $ r $ is the radius of the circle.
• $ \pi $ (pi) is a mathematical constant approximately equal to 3.14159.

Examples

Example-1: Calculate the arc length of a sector with a central angle of 60 degrees and a radius of 5 cm.

Solution:

Using the formula: $ \text{Arc Length} = \left( \frac{60^\circ}{360^\circ} \right) \times 2\pi (5\text{ cm}) $

$ \text{Arc Length} = \frac{1}{6} \times 10\pi \text{ cm} $

$ \text{Arc Length} \approx 5.24 \text{ cm} $ (rounded to two decimal places)

Example-2: A sector has a radius of 8 inches and an arc length of $4\pi$ inches. Determine the central angle of the sector.

Solution:

We know that $ \text{Arc Length} = \left( \frac{\theta}{360^\circ} \right) \times 2\pi r $

Substitute given values: $4\pi = \left( \frac{\theta}{360^\circ} \right) \times 2\pi (8)$

Simplify and solve for $ \theta $: $4\pi = \left( \frac{\theta}{360^\circ} \right) \times 16\pi $

$ \frac{4\pi}{16\pi} = \frac{\theta}{360^\circ} $

$ \frac{1}{4} = \frac{\theta}{360^\circ} $

$ \theta = \frac{1}{4} \times 360^\circ $

$ \theta = 90^\circ $

Real Life Application

Calculating the length of an arc has several real-world applications. For instance, it is essential in:

• Architecture and Construction: Designing curved structures such as arches, domes, and curved walls requires precise arc length calculations.
• Navigation and GPS: Determining distances along curved paths, such as the paths of ships or airplanes, relies on arc length calculations.
• Engineering: Designing gear systems and other rotating mechanisms, often incorporating arcs.
• Landscaping: Calculating the length of fencing or edging needed for curved garden beds.

Fun Fact

The formula for the arc length is directly related to the formula for the circumference of a circle ($2\pi r$). The arc length formula essentially calculates what fraction of the total circumference is occupied by the arc.

Recommended YouTube Videos for Deeper Understanding

Q.1 What is the length of the arc of a sector with a central angle of $60^\circ$ and a radius of 12 cm?

Check Solution

Ans: B

Using the formula, length of arc = $(\theta/360^\circ) \times 2\pi r$ = $(60^\circ/360^\circ) \times 2\pi(12)$ = $(1/6) \times 24\pi$ = $4\pi$ cm

Q.2 A sector has an arc length of $3\pi$ cm and a radius of 9 cm. What is the central angle of the sector?

Check Solution

Ans: B

Arc length = $(\theta/360^\circ) \times 2\pi r$. So, $3\pi = (\theta/360^\circ) \times 2\pi(9)$. Simplifying, $3\pi = (\theta/360^\circ) \times 18\pi$. Then, $\theta/360^\circ = 1/6$, which means $\theta = 60^\circ$.

Q.3 A circle has a radius of 5 cm. If a sector of this circle has a central angle of $120^\circ$, what is the arc length of the sector?

Check Solution

Ans: B

Using the formula, arc length = $(\theta/360^\circ) \times 2\pi r$ = $(120^\circ/360^\circ) \times 2\pi(5)$ = $(1/3) \times 10\pi$ = $\frac{10\pi}{3}$ cm.

Q.4 The arc length of a sector is $2\pi$ cm, and the central angle is $90^\circ$. What is the radius of the circle?

Check Solution

Ans: B

Arc length = $(\theta/360^\circ) \times 2\pi r$. So, $2\pi = (90^\circ/360^\circ) \times 2\pi r$. Then, $2\pi = (1/4) \times 2\pi r$, which means $r = 4$ cm.

Q.5 A sector of a circle has an arc length of $5\pi$ cm and a radius of 15 cm. What is the central angle in degrees?

Check Solution

Ans: A

Arc length = $(\theta/360^\circ) \times 2\pi r$. So, $5\pi = (\theta/360^\circ) \times 2\pi(15)$. Simplifying, $5\pi = (\theta/360^\circ) \times 30\pi$. Then, $\theta/360^\circ = 1/6$, which means $\theta = 60^\circ$.

Next Topic: Area of a Sector