How to Find the Length of a Chord
Finding the length of a chord in a circle is a fundamental skill in geometry, trigonometry, and many practical applications such as engineering, architecture, and even music theory. And this article explains how to find the length of a chord using basic geometric principles and simple calculations. By the end, you will be able to determine chord length confidently, whether you are working with a textbook problem or a real‑world design.
Understanding the Basics
Before diving into calculations, it helps to grasp the key concepts that define a chord. That said, the radius (denoted r) is the distance from the center of the circle to any point on its edge. A chord is a straight line segment whose endpoints both lie on the circumference of a circle. The diameter is twice the radius and passes through the center, connecting two opposite points on the circle Turns out it matters..
The relationship between a chord and the radius is governed by the central angle θ (theta) subtended by the chord at the circle’s center. But this angle measures how far the chord “spans” around the circle. When θ is known, the chord length can be derived directly.
- Key terms:
- radius (r) – distance from center to circle edge.
- diameter – 2 × radius.
- central angle (θ) – angle formed at the center by the two radii that meet the chord’s endpoints.
Formula and Geometry
The fundamental formula for chord length (c) uses the sine of half the central angle:
[ c = 2r \sin\left(\frac{\theta}{2}\right) ]
This equation arises from splitting the isosceles triangle formed by the two radii and the chord into two right‑angled triangles. In each right triangle, the hypotenuse is the radius r, the opposite side to angle θ/2 is half the chord (c/2), and the sine function relates opposite side to hypotenuse.
Not the most exciting part, but easily the most useful.
If the central angle is not given, you can still find the chord length using the perpendicular distance from the circle’s center to the chord (often called the sagitta). The relationship is:
[ c = 2\sqrt{r^{2} - d^{2}} ]
where d is the distance from the center to the chord. This version is handy when you have the radius and the sagitta but not the angle Simple, but easy to overlook..
Step‑by‑Step Guide
Below is a clear, sequential method to calculate chord length when the radius and central angle are known.
-
Identify the radius (r) of the circle Still holds up..
- Measure the distance from the center to any point on the circumference, or use the given value.
-
Determine the central angle (θ) subtended by the chord.
- This may be provided in the problem, measured with a protractor, or calculated from other geometric information.
-
Divide the angle by two to get θ/2.
-
Compute the sine of θ/2 (ensure your calculator is set to the correct mode: degrees or radians).
-
Multiply the result by 2 × r Worth knowing..
- The final product is the chord length c.
Example Calculation
Suppose a circle has a radius of 5 cm and the chord subtends a central angle of 60°.
- θ/2 = 60° / 2 = 30°.
- sin 30° = 0.5.
- c = 2 × 5 cm × 0.5 = 5 cm.
Thus, the chord length is 5 cm.
Using Trigonometry When the Angle Is Unknown
In many situations, the central angle is not directly given. Instead, you might know the perpendicular distance (d) from the center to the chord. Here’s how to adapt the method:
-
Start with the Pythagorean relationship:
[ r^{2} = d^{2} + \left(\frac{c}{2}\right)^{2} ] -
Solve for c:
[ \frac{c}{2} = \sqrt{r^{2} - d^{2}} \quad\Rightarrow\quad c = 2\sqrt{r^{2} - d^{2}} ] -
Plug in the known radius and distance, then compute the square root and multiply by 2.
Example with Sagitta
If the radius is 10 cm and the sagitta (distance from chord to circle edge) is 4 cm, then d = 10 cm − 4 cm = 6 cm.
- c = 2 √(10² − 6²) = 2 √(100 − 36) = 2 √64 = 2 × 8 = 16 cm.
The chord length is 16 cm.
Practical Examples
Example 1: Using the Central Angle
A circle has a radius of 12 cm. The chord subtends an angle of 120°.
- θ/2 = 60°.
- sin 60° ≈ 0.866.
- c = 2 × 12 cm × 0.866 ≈ 20.8 cm.
Result: The chord measures about 20.8 cm.
Example 2: Using the Sagitta
A chord is 8 cm away from the center of a circle with radius 13 cm.
- d = 8 cm.
- c = 2 √(13² − 8²) = 2 √(169 − 64) = 2 √105 ≈ 2 × 10.25 = 20.5 cm.
Result: The chord length is approximately 20.5 cm Practical, not theoretical..
Common Mistakes and Tips
- Mixing up degrees and radians: Ensure your calculator uses the same angular mode as the angle you are working with.
- Forgetting to halve the angle: The formula uses θ/2; using the full angle will give an incorrect result.
- Rounding too early: Keep extra decimal places during intermediate steps to avoid cumulative errors.
- Misidentifying d: The sagitta is the distance from the chord to the circle’s edge, not the distance from the center to the chord.
Tip: Draw a simple diagram. Visualizing the radius, chord, and perpendicular distance often clarifies which formula to apply Worth knowing..
Frequently Asked Questions
Q1: Can I find chord length without knowing the radius?
A: Yes, if you have the sagitta and the chord’s distance from the center, you can use c = 2√(r² − d²). Even so, you still need the radius or another piece of information that relates the chord to the circle’s size.
Q2: What if the chord is a diameter?
A: A diameter is a special case where the central angle θ = 180°. Substituting into the formula gives c = 2r sin(90°) = 2r × 1 = 2r, which matches the definition of a diameter Worth knowing..
Q3: Does the formula work for any circle size?
A: Absolutely. The relationship is universal because it stems from the properties of circles and right‑angled triangles, independent of the circle’s actual size Still holds up..
Q4: How accurate is the sine‑based method?
A: Very accurate, provided the angle measurement is precise. Small errors in angle measurement lead to proportionally small errors in the sine value, and thus in the chord length.
Conclusion
Understanding how to find the length of a chord involves recognizing the relationship between the circle’s radius, the central angle, and the chord itself. That said, by mastering the formula c = 2r sin(θ/2) or the alternative c = 2√(r² − d²), you can solve a wide range of geometric problems with confidence. Remember to keep your angle units consistent, double‑check each step, and use diagrams to clarify the geometry. With practice, calculating chord lengths becomes a straightforward and reliable tool in both academic and real‑world contexts Simple, but easy to overlook..
