1. What is the Diameter?
The diameter of a circle is a line segment that passes through the centre of the circle and has its endpoints on the circle.
If the centre is \(O\) and the endpoints are \(A\) and \(B\), then:
\(AB\text{ is a diameter of Circle } O.\)
The diameter divides the circle into two equal semicircles.
2. Relation Between Diameter and Radius
The diameter is exactly twice the radius. If the radius measures how far the boundary is from the centre, the diameter measures the full distance across the circle.
2.1. Formula
\(d = 2r\)
or equivalently:
\(r = \dfrac{d}{2}\)
2.2. Example
If radius \(r = 9\text{ cm}\), then:
\(d = 2r = 2 \times 9 = 18\text{ cm}\)
3. Diameter as the Longest Chord
The diameter is a special type of chord. A chord is any line segment with endpoints on the circle, but:
the diameter is the longest possible chord because it passes through the centre.
3.1. Why It Is the Longest Chord
A chord gets longer as it moves closer to the centre. When a chord passes exactly through the centre, it becomes the diameter, the maximum possible length.
4. Using Diameter in Formulas
The diameter appears in important circle formulas:
4.1. Circumference
\(C = \pi d\)
4.2. Area
\(A = \pi r^2 = \pi \left(\dfrac{d}{2}\right)^2\)
5. Real-Life Understanding
The diameter helps measure how wide a circular object is. Examples include:
- The width of a circular plate.
- The size of a bicycle wheel.
- The width of a round clock or table.
When someone says a pipe is "10 cm wide", they usually mean the diameter.