I’m Up and Down, and Round and Round · Lesson 1 of 8
Introduction to Circles
“A circle is one curved line with an endless supply of geometry.”
• Understand how circles arise from a distance condition. • Define circle, locus, centre, radius, chord and diameter. • Understand why every diameter is a line of reflection symmetry. • Understand complete rotational symmetry of a circle. • Connect circular shapes in nature with their mathematical definition.
Circle
Circles are one of the most familiar shapes around us. We see circular forms in ripples spreading across water, the cross-sections of stems and fruits, the apparent shapes of the Sun and the full Moon, wheels, rings, clocks, and many decorative designs. Although these objects may look different, they all share the same basic idea of roundness.
Mathematics describes this idea very precisely. A circle is not simply any round shape; it is the collection of all points in a plane that are at the same fixed distance from one particular point. This fixed point is called the centre, and the fixed distance is called the radius. Once the centre and radius are known, the entire circle is determined. This simple definition becomes the foundation for understanding chords, arcs, angles, and many other important properties of circles.
The Defining Property of a Circle
Pick one fixed point on a plane. Now imagine marking every point that is exactly the same distance from that fixed point. The complete collection of all such points forms a circle.
A circle is the set of all points in a plane that are at the same fixed distance from a given fixed point.
The set of all points that satisfy a given condition is called the locus of points satisfying that condition.
The fixed point from which every point of the circle is equally distant is called the centre of the circle.
The fixed distance from the centre to any point on the circle is called the radius.
Chord and Diameter
A line segment joining any two points on a circle is called a chord.
A chord that passes through the centre of the circle is called a diameter.
Every diameter is a chord, but every chord is not a diameter. A diameter has length twice the radius because it consists of two radii placed end to end.
Why the Diameter Is the Longest Chord
Among all chords, the diameter passes through the centre and stretches from one side of the circle to the opposite side. Later we will prove a stronger result: the closer a chord is to the centre, the longer it is. Since the diameter has distance 0 from the centre, it is the longest possible chord.
Problem
A circle has radius 5 cm. What is the length of its longest chord?
- 1.The longest chord is the diameter.
- 2.Diameter = 2 × radius.
- 3.Diameter = 2 × 5 = 10 cm.
- 4.So the longest chord is 10 cm.
Reflection Symmetry
If a circular paper is folded exactly along any diameter, one half of the circle falls perfectly on the other half. This means every diameter acts as a line of reflection symmetry.
A figure has reflection symmetry about a line if reflecting the figure across that line leaves the figure unchanged.
Because there are infinitely many diameters through the centre, a circle has infinitely many lines of reflection symmetry.
Rotational Symmetry
A circle also has complete rotational symmetry. If a circle is rotated about its centre through any angle—10°, 37°, 90°, 180° or any other amount—it looks exactly the same.
A figure has rotational symmetry if rotating it about a fixed point through a certain angle leaves its appearance unchanged.
A regular polygon matches itself only for certain rotations. A circle matches itself after every possible rotation about its centre.
The Locus Idea Beyond Circles
The idea of locus is useful beyond circles. For example, the locus of points equidistant from two fixed points A and B is the perpendicular bisector of AB. This fact becomes essential when we ask how many circles can pass through two or three fixed points.
Practice Problems
- List five natural or everyday objects that resemble circles and explain what feature makes them circular.
- A circle has radius 7 cm. Find its diameter and the length of its longest chord.
- Explain why every diameter is a chord but every chord is not a diameter.
- Explain why a circle has infinitely many lines of reflection symmetry.
- What is the locus of points at a fixed distance 4 cm from a fixed point O?
- What is the locus of points equidistant from two fixed points A and B?
Key Takeaways
• A circle is defined by equal distance from a fixed centre. • A locus is a set of points satisfying a condition. • Radius joins the centre to the circle; chord joins two points on the circle. • A diameter is a chord through the centre and has length 2r. • Every diameter is a line of reflection symmetry. • A circle has complete rotational symmetry about its centre.
Next, we ask a construction question: how many circles can pass through two points or three points?
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How Many Circles?