Arc

Following the Circumference Between Two Points

A circle arc is the part of a circle's circumference between two points. You can picture it as a piece of string laid exactly along the circle's edge. Any two points on a circle determine two arcs between them.

Each arc has two endpoints on the circle. On a fixed circle, its length depends on the measure of the central angle that subtends it: a larger central angle gives a longer arc.

Types of Arcs Based on Size

Based on the measure of the central angle that subtends them, circle arcs can be distinguished into three types:

1. Minor Arc is an arc whose central angle is less than $180^\circ$. This arc is the shorter part of the two possible arcs connecting two points on the circle.

2. Major Arc is an arc whose central angle is more than $180^\circ$. This arc is the longer part of the two possible arcs connecting two points on the circle.

3. Semicircular Arc is an arc whose central angle is exactly $180^\circ$. Its endpoints divide the circumference into two arcs of equal length.

Arc Length Formula

Arc length can be calculated using the ratio between the central angle and the full angle of the circle. Since the full circumference of a circle is $2\pi r$, the arc length can be expressed as:

The variables in this formula are:

• $\alpha$ is the central angle in degrees
• $r$ is the radius of the circle

A radian measures arc length divided by radius, so an angle in radians gives the formula directly:

In the second formula, $\alpha$ is the central angle in radians.

Relationship Between Arc and Central Angle

On the same circle, arc length is proportional to the central angle that subtends it:

This ratio applies to arcs on the same circle. This means that if the central angle of one arc is twice the central angle of another arc, then the length of that arc will also be twice as long.

This ratio compares arc lengths even when the radius is not given directly.

Practice Problems

Each problem asks for the length of an arc, so identify the radius and the central angle in radians before you multiply.

1. A circle arc has a radius of $14 \text{ cm}$ and a central angle of $90^\circ$. Determine the length of the arc.

2. Given that the length of arc $AB$ is $22 \text{ cm}$ and the angle $AOB$ is $120^\circ$, where O is the center of the circle. What is the radius of the circle?

3. In a circle with radius $21 \text{ cm}$, there are two arcs. The first arc has a central angle of $60^\circ$ and the second arc has a central angle of $150^\circ$. Determine the ratio of the lengths of the two arcs.

4. A circle arc has a length of $15.7 \text{ cm}$. If the radius of the circle is $10 \text{ cm}$, determine the central angle of the arc in degrees.

5. Given that the circumference of a circle is $88 \text{ cm}$. If an arc on the circle has a central angle of $45^\circ$, determine the length of the arc.

Worked Solutions

1. Answer: $7\pi \text{ cm} \approx 21.99 \text{ cm}$

   Given: $r = 14 \text{ cm}$, $\alpha = 90^\circ$

   Using the arc length formula:

   $$\text{Arc length} = \frac{\alpha}{360^\circ} \times 2\pi r$$

   $$= \frac{90^\circ}{360^\circ} \times 2\pi \times 14$$

   $$= \frac{1}{4} \times 28\pi$$

   $$= 7\pi \text{ cm} \approx 21.99 \text{ cm}$$

2. Answer: $\frac{33}{\pi} \text{ cm} \approx 10.50 \text{ cm}$

   Given: Arc length $AB = 22 \text{ cm}$, $\alpha = 120^\circ$

   Using the arc length formula:

   $$22 = \frac{120^\circ}{360^\circ} \times 2\pi r$$

   $$22 = \frac{2\pi r}{3}$$

   $$66 = 2\pi r$$

   $$r = \frac{33}{\pi} \text{ cm} \approx 10.50 \text{ cm}$$

3. Answer: $2:5$

   Given: $r = 21 \text{ cm}$, $\alpha_1 = 60^\circ$, $\alpha_2 = 150^\circ$

   Since on the same circle, the ratio of arc lengths equals the ratio of their central angles:

   $$\frac{\text{Arc length}_1}{\text{Arc length}_2} = \frac{\alpha_1}{\alpha_2} = \frac{60^\circ}{150^\circ} = \frac{2}{5}$$

   Therefore, the ratio of the lengths of the two arcs is $2:5$.

4. Answer: $\alpha \approx 90^\circ$

   Given: arc length = $15.7 \text{ cm}$, $r = 10 \text{ cm}$

   Using the arc length formula:

   $$15.7 = \frac{\alpha}{360^\circ} \times 2\pi \times 10$$

   $$15.7 = \frac{\alpha\pi}{18}$$

   $$\alpha = \frac{15.7 \times 18^\circ}{\pi}$$

   $$\alpha = \frac{282.6^\circ}{\pi} \approx 89.95^\circ$$

5. Answer: $11 \text{ cm}$

   Given: Circumference = $88 \text{ cm}$, $\alpha = 45^\circ$

   Using the ratio concept:

   $$\begin{aligned} &\text{Arc length} \\ &= \frac{\alpha}{360^\circ} \times \text{Circumference} \end{aligned}$$

   $$= \frac{45^\circ}{360^\circ} \times 88$$

   $$= \frac{1}{8} \times 88 = 11 \text{ cm}$$