Relationship Between Arc Length and Sector Area

Basic Concepts of Arc and Sector Relationships

In circle geometry, there is a very close relationship between arc length and sector area. Imagine a bicycle wheel spinning, the larger the rotation angle of the wheel, the longer the path traced by a point on the edge of the wheel and the larger the area swept by the wheel's radius.

This relationship can be expressed in the form of a consistent proportion. When the central angle of a circle changes, the arc length and sector area will change proportionally with the same ratio to the circumference and total area of the circle.

Fundamental Proportion

The fundamental proportion that connects arc length and sector area to the whole circle can be expressed as follows:

This formula shows that the ratio of arc length to circle circumference equals the ratio of sector area to circle area, both of which equal the ratio of central angle to the full angle of the circle.

Mathematical Formulas

Based on this proportional relationship, we can derive formulas for calculating arc length and sector area:

From these two formulas, we can find a direct relationship between arc length and sector area:

Visualization of Proportional Relationships

Let's see how this proportional relationship applies to various central angles. Each central angle produces a consistent ratio between arc length and sector area relative to the entire circle.

From the visualization above, we can see a consistent pattern. For every central angle $\alpha$, the following mathematical relationship applies:

Or in complete formula form:

It is important to understand that these two formulas are interconnected. Two arcs are said to be congruent on the same circle if their corresponding central angles are equal. Furthermore, the arc length created by two adjacent arcs with coincident endpoints will equal the sum of the lengths of both arcs.

Application in Earth Measurement

One of the most fascinating applications of the relationship between arc length and sector area is the measurement of Earth's circumference by Eratosthenes around 276-195 BC. By observing that sunlight fell perpendicularly at Syene while forming a 7.2° angle at Alexandria, which was 500 miles away, he was able to calculate Earth's circumference.

Using the proportional relationship:

Relationship Between Arc and Sector

It is important to understand that in one circle with the same central angle, the ratio between arc length and sector area has a special relationship:

This relationship shows that the ratio of sector area to arc length is always equal to half the radius of the circle, regardless of the size of the central angle.

Exercises

1. A circle has a radius of 14 cm. If the arc length of a sector is 22 cm, determine the area of that sector.

2. Given a sector area of 154 cm² and an arc length of 22 cm. Determine the radius of the circle.

3. Two cities are located on the same latitude with a distance of 1,000 km. If the angle formed at Earth's center is 9°, determine the estimated circumference of Earth.

Answer Key

1. Solution Steps:

   Given: $r = 14 \text{ cm}$, arc length = 22 cm

   Using the relationship: $\text{Sector Area} = \frac{1}{2} \times \text{Arc Length} \times r$

   $$\text{Sector Area} = \frac{1}{2} \times 22 \times 14$$

   $$= \frac{22 \times 14}{2}$$

   $$= \frac{308}{2}$$

   $$= 154 \text{ cm}^2$$

2. Solution Steps:

   Given: Sector area = 154 cm², arc length = 22 cm

   Using the relationship: $\frac{\text{Sector Area}}{\text{Arc Length}} = \frac{r}{2}$

   $$\frac{154}{22} = \frac{r}{2}$$

   $$7 = \frac{r}{2}$$

   $$r = 14 \text{ cm}$$

3. Solution Steps:

   Given: Distance = 1,000 km, angle = 9°

   Using the proportion: $\frac{\text{Distance}}{\text{Earth Circumference}} = \frac{9°}{360°}$

   $$\frac{1,000}{\text{Earth Circumference}} = \frac{9°}{360°} = \frac{1}{40}$$

   $$\text{Earth Circumference} = 1,000 \times 40$$

   $$= 40,000 \text{ km}$$