Arc Length vs. Chord Length: Understanding the Difference

Question

Hey everyone! 👋 Ever get confused between arc length and chord length? 🤔 They sound similar, but they're totally different! Let's break it down so it's super easy to understand! 🤓

smith.vicki45 · Accepted Answer

📚 Understanding Arc Length vs. Chord Length
In geometry, especially when dealing with circles, it's easy to mix up arc length and chord length. Let's clarify the difference:

📏 Definition of Arc Length
Arc length is the distance along the curved line making up the arc. Think of it as walking along the curved edge of a slice of pizza 🍕. It's a portion of the circumference of the circle.

➖ Definition of Chord Length
Chord length is the straight-line distance between the two endpoints of an arc. Imagine cutting straight across that slice of pizza 🍕 from one side to the other. That straight cut is the chord.

🆚 Arc Length vs. Chord Length: A Detailed Comparison

Feature
  Arc Length
  Chord Length

Definition
  📏 The distance along the curved line forming the arc.
  ➖ The straight-line distance between the endpoints of the arc.

Path
   🛤️ Curved
    직선 Straight

Measurement
   📐 Measured in units of length (e.g., cm, inches) and is part of the circumference.
  📏 Measured in units of length (e.g., cm, inches) and is always shorter than the corresponding arc length.

Formula
   ➗ $Arc Length = r 	heta$, where $r$ is the radius and $	heta$ is the central angle in radians.
  📐 $Chord Length = 2r \sin(\frac{	heta}{2})$, where $r$ is the radius and $	heta$ is the central angle in radians.

Relationship
   🔄 Directly proportional to the radius and central angle.
   📉 Depends on the sine of half the central angle.

Occurrence
   🍕 Found along the edge of circular shapes.
   ✂️ Connects two points on a circle.

Example
   🧭 The distance you travel along a curved road.
   🌉 A bridge spanning between two points on a circular lake.

🔑 Key Takeaways

📏 Definition: Arc length measures the distance along a curve, while chord length measures the straight-line distance between the arc's endpoints.
 📐 Formulas: Arc length is calculated using $r	heta$, and chord length uses $2r \sin(\frac{	heta}{2})$.
 💡 Visual: Imagine a slice of pizza; the crust is the arc length, and a cut straight across is the chord length.

Arc Length vs. Chord Length: Understanding the Difference

1 Answers

📚 Understanding Arc Length vs. Chord Length

📏 Definition of Arc Length

➖ Definition of Chord Length

🆚 Arc Length vs. Chord Length: A Detailed Comparison

🔑 Key Takeaways

Join the discussion

Feature	Arc Length	Chord Length
Definition	📏 The distance along the curved line forming the arc.	➖ The straight-line distance between the endpoints of the arc.
Path	🛤️ Curved	직선 Straight
Measurement	📐 Measured in units of length (e.g., cm, inches) and is part of the circumference.	📏 Measured in units of length (e.g., cm, inches) and is always shorter than the corresponding arc length.
Formula	➗ $Arc Length = r \theta$, where $r$ is the radius and $\theta$ is the central angle in radians.	📐 $Chord Length = 2r \sin(\frac{\theta}{2})$, where $r$ is the radius and $\theta$ is the central angle in radians.
Relationship	🔄 Directly proportional to the radius and central angle.	📉 Depends on the sine of half the central angle.
Occurrence	🍕 Found along the edge of circular shapes.	✂️ Connects two points on a circle.
Example	🧭 The distance you travel along a curved road.	🌉 A bridge spanning between two points on a circular lake.