The length of a major arc is found by applying the standard arc length formula using the central angle of the major arc, which is always greater than 180 degrees (or π radians), or by subtracting the length of the corresponding minor arc from the total circumference of the circle.
Understanding Major Arcs
In a circle, an arc is a continuous segment of the circle's circumference. An arc is classified based on the measure of its central angle:
- Minor Arc: An arc whose central angle is less than 180 degrees (or π radians).
- Major Arc: An arc whose central angle is greater than 180 degrees (or π radians).
- Semicircle: An arc whose central angle is exactly 180 degrees (or π radians), dividing the circle into two equal halves.
The formula for arc length is a proportion of the circle's circumference, determined by the central angle.
Formulas for Major Arc Length
Calculating the length of a major arc involves using the general arc length formulas, but specifically for an angle that represents more than half the circle.
1. Using the Central Angle of the Major Arc
The most direct way to calculate the length of a major arc is to use the standard arc length formula, where the angle ($\theta_{\text{major}}$) is the central angle of the major arc itself.
| Arc Length Formula (if $\theta_{\text{major}}$ is in degrees) | $s = 2 \pi r \left( \frac{\theta_{\text{major}}}{360^\circ} \right)$ |
|---|---|
| Arc Length Formula (if $\theta_{\text{major}}$ is in radians) | $s = \theta_{\text{major}} \times r$ |
Where:
- $s$ is the length of the major arc.
- $r$ is the radius of the circle.
- $\pi$ (pi) is approximately 3.14159.
- $\theta_{\text{major}}$ is the central angle of the major arc, measured in degrees or radians. This angle will always be $> 180^\circ$ or $> \pi$ radians.
2. Using the Minor Arc Length and Circumference
An alternative method is to calculate the length of the corresponding minor arc and subtract it from the total circumference of the circle.
The length of the major arc ($s_{\text{major}}$) can be found using the formula:
$s{\text{major}} = \text{Circumference} - s{\text{minor}}$
Where:
- $\text{Circumference} = 2 \pi r$
- $s_{\text{minor}}$ is the length of the minor arc that completes the circle with the major arc.
The central angle of the minor arc ($\theta{\text{minor}}$) is related to the central angle of the major arc ($\theta{\text{major}}$) by:
- $\theta{\text{minor}} = 360^\circ - \theta{\text{major}}$ (if in degrees)
- $\theta{\text{minor}} = 2\pi - \theta{\text{major}}$ (if in radians)
Then, $s_{\text{minor}}$ can be calculated using the same general arc length formulas:
- If $\theta{\text{minor}}$ is in degrees: $s{\text{minor}} = 2 \pi r \left( \frac{\theta_{\text{minor}}}{360^\circ} \right)$
- If $\theta{\text{minor}}$ is in radians: $s{\text{minor}} = \theta_{\text{minor}} \times r$
Therefore, the major arc length can be expressed as:
- Degrees: $s{\text{major}} = 2 \pi r - 2 \pi r \left( \frac{\theta{\text{minor}}}{360^\circ} \right)$
- Radians: $s{\text{major}} = 2 \pi r - (\theta{\text{minor}} \times r)$
Key Components of the Formula
To calculate the length of a major arc, you need two fundamental pieces of information:
- Radius (r): The distance from the center of the circle to any point on its circumference.
- Central Angle ($\theta{\text{major}}$ or $\theta{\text{minor}}$): The angle formed by two radii connecting the endpoints of the arc to the center of the circle. This angle must be in the correct unit (degrees or radians) for the chosen formula.
Practical Examples
Let's illustrate with a couple of examples.
Example 1: Major Arc Length (Degrees)
Question: A circle has a radius of 7 cm. A minor arc has a central angle of $120^\circ$. What is the length of the corresponding major arc?
Solution:
- Find the central angle of the major arc ($\theta_{\text{major}}$):
$\theta{\text{major}} = 360^\circ - \theta{\text{minor}} = 360^\circ - 120^\circ = 240^\circ$ - Apply the major arc length formula (in degrees):
$s{\text{major}} = 2 \pi r \left( \frac{\theta{\text{major}}}{360^\circ} \right)$
$s{\text{major}} = 2 \times \frac{22}{7} \times 7 \times \left( \frac{240^\circ}{360^\circ} \right)$
$s{\text{major}} = 2 \times 22 \times \frac{2}{3}$
$s_{\text{major}} = \frac{88}{3} \approx 29.33 \text{ cm}$
Example 2: Major Arc Length (Radians)
Question: A circle has a radius of 10 meters. A major arc subtends a central angle of $4\pi/3$ radians. What is the length of this major arc?
Solution:
- Identify the central angle of the major arc ($\theta_{\text{major}}$):
$\theta_{\text{major}} = \frac{4\pi}{3}$ radians (This is greater than $\pi$ radians, confirming it's a major arc.) - Apply the major arc length formula (in radians):
$s{\text{major}} = \theta{\text{major}} \times r$
$s{\text{major}} = \frac{4\pi}{3} \times 10$
$s{\text{major}} = \frac{40\pi}{3} \approx 41.89 \text{ meters}$
Why Understanding Arc Length is Important
Understanding arc length, including major arcs, is crucial in various fields:
- Engineering: Designing curved structures, roads, or machine parts.
- Architecture: Calculating materials for curved walls, domes, or decorative elements.
- Navigation: Determining distances along curved paths, such as flight routes or nautical tracks on Earth's surface.
- Cartography: Accurately representing curved features on maps.
For further exploration of arc length concepts and formulas, you can refer to resources like BYJU'S on Arc Length Formula.
