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Arc length and sector area are controlled by the same central angle. On a turning bicycle wheel, a larger rotation makes a point on the rim travel farther and makes the radius sweep a larger area.
As the central angle changes, both quantities change by the same fraction of the full circumference and total circle area.
Arc length and sector area take the same fraction of their full-circle values:
All three fractions describe the same share of a complete circle.
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:
The following examples show the same proportion for several central angles.
The visualization follows the same relationship for every central angle :
Or in complete formula form:
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.
Eratosthenes, who lived around , 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 . The two cities were taken to be apart.
Using the proportional relationship:
The exact length of the ancient stadion is uncertain. Using a commonly cited conversion, this estimate is roughly .
For a circle with radius and a nonzero central angle, dividing the sector area by its arc length gives :
Changing the central angle changes both quantities by the same factor, so their ratio stays .
A circle has a radius of . If the arc length of a sector is , determine the area of that sector.
Given a sector area of and an arc length of . Determine the radius of the circle.
Two cities lie on the same great circle and are apart along that circle. If the corresponding angle at Earth's center is , determine the estimated circumference of Earth.
Solution Steps:
Given: , arc length =
Using the relationship:
Solution Steps:
Given: sector area = , arc length =
Using the relationship:
Solution Steps:
Given: distance = , angle =
Using the proportion:
Published: . Updated: .
