Chord
In short: A chord is a straight segment joining two points that both sit on a circle, and the widest one through any circle is the diameter.
GeometryGeometryG-C.A.2
What is Chord?
Choose two spots on a circle and stretch a straight string between them: the string is a chord. Unlike a radius, it need not reach the center, and both of its ends rest on the circle itself.
Three facts drive most problems. The diameter is the widest one. A radius drawn perpendicular to it splits it into two equal halves. Equal lengths sit at equal distances from the center. With the radius, the half-length and the center distance \(d\), a right triangle appears: \(r^2 = d^2 + \left(\frac{\text{length}}{2}\right)^2\).
How to spot it
- A figure shows a straight segment with both endpoints on a circle, not passing through the center.
- A line from the center meets a segment inside the circle at a right angle.
- The wording gives the distance from the center to a straight cut and asks how long the cut is.
- A bridge, arch, window or pizza slice is modeled by a circle with a straight edge across it.
The method
- Draw the radius to one endpoint and the perpendicular from the center to the segment.
- Spot the right triangle: hypotenuse \(r\), legs \(d\) and half the segment.
- Use \(\left(\frac{\text{length}}{2}\right)^2 = r^2 – d^2\) to find the half-length, or solve for \(d\) or \(r\).
- Double the half-length for the full length, and verify it does not exceed the diameter.
Chord: solved example
Problem
A circular window has a radius of 13 inches. A straight support bar is fixed across it, 5 inches from the center, with both ends on the frame. How long is the bar?
Solution
The perpendicular from the center splits the bar in half. In the right triangle, the hypotenuse is the radius 13 and one leg is the distance 5, so the other leg is \(\sqrt{13^2 – 5^2} = \sqrt{169 – 25} = \sqrt{144} = 12\).
That leg is half the bar, so the bar measures \(2 \times 12 =\) 24 inches.
Try it
Question 1
A circle has radius 10 cm and a segment of length 16 cm has both ends on it. How far is it from the center?
Show the answer
Half is 8, so \(d = \sqrt{10^2 – 8^2} = \sqrt{36} = 6\) cm.
Question 2
What is the length of the longest segment with both ends on a circle of radius 15?
Show the answer
The longest one passes through the center: the diameter, \(2 \times 15 = 30\).
Chord vs. Radius and Diameter
A radius runs from the center to the circle, so only one end is on the circle. A diameter is the special case that passes through the center and measures twice the radius. Every diameter is a chord, but most chords miss the center and are shorter.
Keep going
Lessons
Worksheets and quizzes
Related words
- Arc (Circle)
- Central Angle
- Radius and Diameter
- Tangent to a Circle
- Inscribed Angle
- Perpendicular Bisector
Whole dictionary: A–Z · Geometry · Geometry