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:

These two formulas are connected. Two arcs are congruent on the same circle if their corresponding central angles are equal. The arc length created by two adjacent arcs with the same endpoint equals the sum of the two arc lengths.

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 $276\text{-}195 \text{ BC}$. By observing that sunlight fell perpendicularly at Syene while forming a $7.2^\circ$ angle at Alexandria, which was $500 \text{ miles}$ away, he was able to calculate Earth's circumference.

Using the proportional relationship:

Relationship Between Arc and Sector

In one circle, when the central angle is the same, arc length and sector area have this ratio:

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 $14 \text{ cm}$. If the arc length of a sector is $22 \text{ cm}$, determine the area of that sector.

2. Given a sector area of $154 \text{ cm}^2$ and an arc length of $22 \text{ cm}$. Determine the radius of the circle.

3. Two cities are located on the same latitude with a distance of $1{,}000 \text{ km}$. If the angle formed at Earth's center is $9^\circ$, determine the estimated circumference of Earth.

Answer Key

1. Solution Steps:

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

   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 = $154 \text{ cm}^2$, arc length = $22 \text{ cm}$

   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 = $1{,}000 \text{ km}$, angle = $9^\circ$

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

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

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

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