Arc

Understanding Circle Arcs

A circle arc is a part of the circumference of a circle that is bounded by two points on the circle. Imagine it like a piece of string that curves following the shape of the circle. When we have a circle and mark two points on it, the part of the circumference that connects these two points is called an arc.

Each arc has two endpoints located on the circle, and the length of the arc depends on the size of the central angle that faces the arc. The larger the central angle, the longer the arc.

Types of Arcs Based on Size

Based on the size of the central angle that faces them, circle arcs can be distinguished into several 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 $180^\circ$. This arc divides the circle into two equal parts.

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:

where:

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

If the central angle is expressed in radians, the formula becomes simpler:

where $\alpha$ is the central angle in radians.

Relationship Between Arc and Central Angle

There is a very close relationship between arc length and the size of the central angle that faces it. This relationship can be expressed in the form of a ratio:

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 concept is very useful in solving various problems involving circle arcs, especially when we need to find arc length without knowing the radius of the circle directly.

Practice Problems

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.

Answer Key

1. Answer: $22 \text{ cm}$

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

   Using the arc length formula:

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

   $$= \frac{90^\circ}{360^\circ} \times 2 \times \frac{22}{7} \times 14$$

   $$= \frac{1}{4} \times 2 \times \frac{22}{7} \times 14$$

   $$= \frac{1}{4} \times 88 = 22 \text{ cm}$$

2. Answer: $10.5 \text{ cm}$

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

   Using the arc length formula:

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

   $$22 = \frac{1}{3} \times 2 \times \frac{22}{7} \times r$$

   $$22 = \frac{44r}{21}$$

   $$r = \frac{22 \times 21}{44} = \frac{462}{44} = 10.5 \text{ cm}$$

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: $90^\circ$

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

   Using the arc length formula:

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

   $$15.7 = \frac{\alpha}{360^\circ} \times 2 \times 3.14 \times 10$$

   $$15.7 = \frac{\alpha}{360^\circ} \times 62.8$$

   $$\alpha = \frac{15.7 \times 360^\circ}{62.8} = \frac{5652^\circ}{62.8} = 90^\circ$$

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

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

   Using the ratio concept:

   $$\text{Arc length} = \frac{\alpha}{360^\circ} \times \text{Circumference}$$

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

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