The Formula for a Chord of a Circle: A Complete Guide
A chord is a straight line segment whose endpoints both lie on a circle. And knowing how to calculate the length of a chord is essential in many fields—from geometry and engineering to architecture and even art. But this article walks you through the chord formula step by step, explains why it works, and shows practical examples and common pitfalls. By the end, you’ll feel confident applying the chord formula in any circular context But it adds up..
Introduction
When you see a circle, the most obvious feature is its radius or diameter. Still, the chord—an internal line connecting two points on the circumference—makes a difference in constructing arcs, sectors, and many geometric proofs. The chord formula lets you find the chord’s length when you know either the circle’s radius and the chord’s central angle, or the radius and the distance from the circle’s center to the chord (the sagitta). Understanding this relationship unlocks a deeper appreciation of circular geometry and provides a practical tool for real‑world measurements.
1. The Basic Chord Formula
1.1 When the Central Angle Is Known
If you know the radius ( r ) of the circle and the central angle ( \theta ) (measured in radians) subtended by the chord, the chord length ( c ) is:
[ c ;=; 2r \sin!\left(\frac{\theta}{2}\right) ]
Why it works: The chord, the radius, and the center form an isosceles triangle. Splitting it into two right triangles gives a half‑chord of length ( \frac{c}{2} ) opposite the angle ( \frac{\theta}{2} ). By definition of sine in a right triangle, ( \sin(\frac{\theta}{2}) = \frac{c/2}{r} ), leading directly to the formula above Still holds up..
1.2 When the Sagitta Is Known
The sagitta (also called the versine or height) is the perpendicular distance from the circle’s center to the chord. If you know the radius ( r ) and the sagitta ( d ), the chord length is:
[ c ;=; 2 \sqrt{,r^2 ;-; (r - d)^2,} ]
Derivation: Drop a perpendicular from the center to the chord; this creates a right triangle with hypotenuse ( r ), one leg ( r - d ), and the other leg ( \frac{c}{2} ). Applying the Pythagorean theorem yields ( \left(\frac{c}{2}\right)^2 = r^2 - (r - d)^2 ), which simplifies to the formula above.
2. Step‑by‑Step Application
Below are clear steps for each scenario. Pick the one that matches the data you have.
2.1 Using the Central Angle
| Step | Action | Formula |
|---|---|---|
| 1 | Identify the radius ( r ). Which means | — |
| 3 | Plug values into: ( c = 2r \sin(\theta/2) ). In practice, | — |
| 2 | Measure or obtain the central angle ( \theta ) in radians. | — |
| 4 | Compute the sine and multiply. |
Tip: If the angle is given in degrees, convert it first: ( \theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180} ).
2.2 Using the Sagitta
| Step | Action | Formula |
|---|---|---|
| 1 | Identify the radius ( r ). | — |
| 2 | Measure or obtain the sagitta ( d ). Now, | — |
| 3 | Compute the inner term: ( r - d ). Now, | — |
| 4 | Apply: ( c = 2 \sqrt{r^2 - (r - d)^2} ). | — |
| 5 | Simplify if possible: ( c = 2 \sqrt{2rd - d^2} ). |
3. Practical Examples
Example 1: Central Angle Known
Problem: A circle has radius ( 10 , \text{cm} ). A chord subtends a central angle of ( 60^\circ ). Find the chord length.
Solution:
- Convert angle: ( 60^\circ = \frac{\pi}{3} ) radians.
- Apply formula: ( c = 2 \times 10 \times \sin!\left(\frac{\pi}{6}\right) ).
- Since ( \sin(\pi/6) = 0.5 ), ( c = 20 \times 0.5 = 10 , \text{cm} ).
Answer: The chord is ( 10 , \text{cm} ) long.
Example 2: Sagitta Known
Problem: A circle’s radius is ( 15 , \text{cm} ). The sagitta (distance from center to chord) is ( 5 , \text{cm} ). Find the chord length.
Solution:
- Compute ( r - d = 15 - 5 = 10 ).
- Plug into formula: ( c = 2 \sqrt{15^2 - 10^2} = 2 \sqrt{225 - 100} = 2 \sqrt{125} ).
- ( \sqrt{125} \approx 11.18 ), so ( c \approx 22.36 , \text{cm} ).
Answer: The chord is approximately ( 22.4 , \text{cm} ) long Most people skip this — try not to..
4. Common Mistakes to Avoid
| Mistake | Why It Happens | Fix |
|---|---|---|
| Using degrees directly in the sine function | The standard chord formula assumes radians | Convert degrees to radians first |
| Forgetting the factor of 2 | The formula calculates half the chord | Multiply the result by 2 |
| Mixing up sagitta and chord | Confusing the distance from center to chord with the chord itself | Remember sagitta is the perpendicular distance |
| Ignoring the radius sign | Using negative radius values leads to nonsensical results | Always use a positive radius |
5. Scientific Explanation: Why the Formula Works
The chord formula derives from simple trigonometry and the Pythagorean theorem. Let ( A ) and ( B ) be the chord’s endpoints, and let ( M ) be the midpoint of ( AB ). Consider the circle centered at ( O ) with radius ( r ). Dropping a perpendicular from ( O ) to ( M ) splits ( \triangle OAB ) into two congruent right triangles.
- Hypotenuse ( r ) (the radius),
- Opposite side ( \frac{c}{2} ) (half the chord),
- Adjacent side ( r - d ) (if sagitta is used).
From right‑triangle trigonometry, ( \sin(\theta/2) = \frac{c/2}{r} ). Solving for ( c ) gives the central‑angle formula. But from the Pythagorean theorem, ( (\frac{c}{2})^2 = r^2 - (r - d)^2 ), leading to the sagitta formula. These derivations show that the chord length is inherently tied to the geometry of the circle’s radius and the angle or distance involved Less friction, more output..
6. FAQ
Q1: Can I use the chord formula for any circle?
A: Yes, as long as you know either the central angle or the sagitta and the radius, the formula applies to any circle, regardless of size.
Q2: What if I only know the chord length and the radius? Can I find the central angle?
A: Absolutely. Rearranging the formula:
[
\theta = 2 \arcsin!\left(\frac{c}{2r}\right)
]
Compute the arcsine of ( \frac{c}{2r} ) and multiply by 2.
Q3: How does the chord length change as the central angle increases?
A: The chord length increases monotonically with the central angle, reaching a maximum of ( 2r ) when the central angle is ( \pi ) (180°), i.e., when the chord becomes a diameter Most people skip this — try not to..
Q4: Is the chord always shorter than the radius?
A: No. A chord can be longer than the radius, especially when the central angle exceeds ( 60^\circ ). The maximum chord length equals the diameter ( 2r ).
Q5: What if the chord is a diameter?
A: When the chord is a diameter, the central angle is ( \pi ) radians, and the formula simplifies to ( c = 2r ), which matches the definition of a diameter.
7. Conclusion
The chord formula is a concise, powerful tool that links the geometry of a circle to simple trigonometric relationships. Whether you’re calculating the length of a bridge span, designing a circular garden bed, or solving a geometry problem, understanding how to apply
[ c = 2r \sin!\left(\frac{\theta}{2}\right) \quad\text{or}\quad c = 2 \sqrt{r^2 - (r - d)^2} ]
will save time and eliminate guesswork. Remember to work in radians for angles, double-check your units, and practice with both central‑angle and sagitta scenarios. With these skills, you’ll deal with any circular challenge with confidence.
