Formula For Chord Of A Circle

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. Knowing how to calculate the length of a chord is essential in many fields—from geometry and engineering to architecture and even art. 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.

Introduction

When you see a circle, the most obvious feature is its radius or diameter. Even so, the chord—an internal line connecting two points on the circumference—plays a important role in constructing arcs, sectors, and many geometric proofs. Still, 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.

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 ).	—
2	Measure or obtain the central angle ( \theta ) in radians.	—
3	Plug values into: ( c = 2r \sin(\theta/2) ).	—
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 ). On top of that,	—
2	Measure or obtain the sagitta ( d ).	—
3	Compute the inner term: ( r - d ). On the flip side,	—
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.

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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.

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. In practice, let ( A ) and ( B ) be the chord’s endpoints, and let ( M ) be the midpoint of ( AB ). Plus, 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} ). From the Pythagorean theorem, ( (\frac{c}{2})^2 = r^2 - (r - d)^2 ), leading to the sagitta formula. Solving for ( c ) gives the central‑angle 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.

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.

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 manage any circular challenge with confidence.