Relationship Between Arc Length and Sector Area

Arc Length and Sector Area Proportion

Arc length and sector area take the same fraction of their full-circle values, so both change by the same fraction as the central angle changes:

\begin{aligned} &\frac{\text{Arc Length}}{\text{Circle Circumference}} \\ &=\frac{\text{Sector Area}}{\text{Circle Area}} \\ &=\frac{\alpha}{360^\circ} \end{aligned}

All three fractions describe the same share of a complete circle.

Formulas for Arc Length and Sector Area

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

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

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

\begin{aligned} &\text{Sector Area} = \frac{1}{2} \times \text{Arc Length} \times r \end{aligned}

Visualization of Proportional Relationships

The following examples show the same proportion for several central angles.

Angle

45^\circ

produces ratio

\frac{1}{8}

For a central angle of

45^\circ

, the arc length is

\frac{1}{8}

of the circumference and the sector area is

\frac{1}{8}

of the circle area.

Angle

90^\circ

produces ratio

\frac{1}{4}

For a central angle of

90^\circ

, the arc length is

\frac{1}{4}

of the circumference and the sector area is

\frac{1}{4}

of the circle area.

Angle

180^\circ

produces ratio

\frac{1}{2}

For a central angle of

180^\circ

, the arc length is

\frac{1}{2}

of the circumference and the sector area is

\frac{1}{2}

of the circle area.

Angle

\alpha

for all general cases

For a central angle of

\alpha

, the arc length is

\frac{\alpha}{360^\circ}

of the circumference and the sector area is

\frac{\alpha}{360^\circ}

of the circle area.

The visualization follows the same relationship for every central angle $\alpha$ :

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

Or in complete formula form:

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

On the same circle, equal central angles produce congruent arcs and sectors. For adjacent angles, their arc lengths and sector areas can also be added.

Application in Earth Measurement

Eratosthenes, who lived around $276\text{-}194 \text{ BC}$ , used the same proportional idea to estimate Earth's circumference. At the summer solstice the Sun was overhead at Syene, while a vertical object in Alexandria indicated an angular difference of $7.2^\circ$ . The two cities were taken to be $5{,}000 \text{ stadia}$ apart.

Earth Circumference Measurement by Eratosthenes

The model relates an angle of

7.2^\circ

to a surface distance of

5{,}000 \text{ stadia}

Using the proportional relationship:

Let $D$ be the distance from Alexandria to Syene and $C$ the circumference of the Earth:

\frac DC=\frac{7.2^\circ}{360^\circ}

The exact length of the ancient stadion is uncertain. Using a commonly cited conversion, this estimate is roughly $40{,}000 \text{ km}$ .

Relationship Between Arc and Sector

For a circle with radius $r$ and a nonzero central angle, dividing the sector area by its arc length gives $r/2$ :

\frac{\text{Sector Area}}{\text{Arc Length}} = \frac{r}{2}

Changing the central angle changes both quantities by the same factor, so their ratio stays $r/2$ .

Exercises

Each problem links an arc length to a sector area on the same circle, so write both quantities from the same radius and central angle.

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.
Given a sector area of $154 \text{ cm}^2$ and an arc length of $22 \text{ cm}$ . Determine the radius of the circle.
Two cities lie on the same great circle and are $1{,}000 \text{ km}$ apart along that circle. If the corresponding angle at Earth's center is $9^\circ$ , determine the estimated circumference of Earth.

Worked Solutions

Solution Steps:

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

Using the relationship: $\text{Sector Area} = \frac{1}{2} \times \text{Arc Length} \times r$
$\text{Sector Area} = \frac{1}{2} \times 22 \times 14$
Solution Steps:

Given: sector area = $154 \text{ cm}^2$ , arc length = $22 \text{ cm}$

Using the relationship: $\frac{\text{Sector Area}}{\text{Arc Length}} = \frac{r}{2}$
$\frac{154}{22} = \frac{r}{2}$
Solution Steps:

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

Using the proportion: $\frac{\text{Distance}}{\text{Earth Circumference}} = \frac{9^\circ}{360^\circ}$
$\begin{aligned} \frac{1{,}000}{\text{Earth Circumference}} &= \frac{9^\circ}{360^\circ} \\ &= \frac{1}{40} \end{aligned}$

Arc Length and Sector Area Proportion

Arc length and sector area take the same fraction of their full-circle values, so both change by the same fraction as the central angle changes:

\begin{aligned} &\frac{\text{Arc Length}}{\text{Circle Circumference}} \\ &=\frac{\text{Sector Area}}{\text{Circle Area}} \\ &=\frac{\alpha}{360^\circ} \end{aligned}

All three fractions describe the same share of a complete circle.

Formulas for Arc Length and Sector Area

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

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

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

\begin{aligned} &\text{Sector Area} = \frac{1}{2} \times \text{Arc Length} \times r \end{aligned}

Visualization of Proportional Relationships

The following examples show the same proportion for several central angles.

Angle

45^\circ

produces ratio

\frac{1}{8}

For a central angle of

45^\circ

, the arc length is

\frac{1}{8}

of the circumference and the sector area is

\frac{1}{8}

of the circle area.

Angle

90^\circ

produces ratio

\frac{1}{4}

For a central angle of

90^\circ

, the arc length is

\frac{1}{4}

of the circumference and the sector area is

\frac{1}{4}

of the circle area.

Angle

180^\circ

produces ratio

\frac{1}{2}

For a central angle of

180^\circ

, the arc length is

\frac{1}{2}

of the circumference and the sector area is

\frac{1}{2}

of the circle area.

Angle

\alpha

for all general cases

For a central angle of

\alpha

, the arc length is

\frac{\alpha}{360^\circ}

of the circumference and the sector area is

\frac{\alpha}{360^\circ}

of the circle area.

The visualization follows the same relationship for every central angle $\alpha$ :

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

Or in complete formula form:

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

On the same circle, equal central angles produce congruent arcs and sectors. For adjacent angles, their arc lengths and sector areas can also be added.

Application in Earth Measurement

Earth Circumference Measurement by Eratosthenes

The model relates an angle of

7.2^\circ

to a surface distance of

5{,}000 \text{ stadia}

Using the proportional relationship:

Let $D$ be the distance from Alexandria to Syene and $C$ the circumference of the Earth:

\frac DC=\frac{7.2^\circ}{360^\circ}

The exact length of the ancient stadion is uncertain. Using a commonly cited conversion, this estimate is roughly $40{,}000 \text{ km}$ .

Relationship Between Arc and Sector

For a circle with radius $r$ and a nonzero central angle, dividing the sector area by its arc length gives $r/2$ :

\frac{\text{Sector Area}}{\text{Arc Length}} = \frac{r}{2}

Changing the central angle changes both quantities by the same factor, so their ratio stays $r/2$ .

Exercises

Each problem links an arc length to a sector area on the same circle, so write both quantities from the same radius and central angle.

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.
Given a sector area of $154 \text{ cm}^2$ and an arc length of $22 \text{ cm}$ . Determine the radius of the circle.
Two cities lie on the same great circle and are $1{,}000 \text{ km}$ apart along that circle. If the corresponding angle at Earth's center is $9^\circ$ , determine the estimated circumference of Earth.

Worked Solutions

Solution Steps:

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

Using the relationship: $\text{Sector Area} = \frac{1}{2} \times \text{Arc Length} \times r$
$\text{Sector Area} = \frac{1}{2} \times 22 \times 14$
Solution Steps:

Given: sector area = $154 \text{ cm}^2$ , arc length = $22 \text{ cm}$

Using the relationship: $\frac{\text{Sector Area}}{\text{Arc Length}} = \frac{r}{2}$
$\frac{154}{22} = \frac{r}{2}$
Solution Steps:

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

Using the proportion: $\frac{\text{Distance}}{\text{Earth Circumference}} = \frac{9^\circ}{360^\circ}$
$\begin{aligned} \frac{1{,}000}{\text{Earth Circumference}} &= \frac{9^\circ}{360^\circ} \\ &= \frac{1}{40} \end{aligned}$