Circles, arcs, and curve patterns

A circle is the set of all points the same distance from a centre. It is one of the most perfect and most useful shapes in mathematics , appearing in the moon, in raindrops on a pond, in the wheels of every vehicle, in plates and bangles and rotis.

Concept

The circle and its parts

To draw a circle, you fix a centre and choose a radius , a distance. The circle is then every point exactly that distance from the centre.

Parts of a circle:

• Centre , the fixed point.
• Radius , the distance from centre to any point on the circle (also the segment from centre to circle). All radii of one circle are equal.
• Diameter , a straight line passing through the centre, touching the circle on both sides. The diameter is twice the radius: $d = 2r$.
• Chord , a straight segment joining two points on the circle. The diameter is the longest chord.
• Arc , a piece of the circle (a curved portion).
• Sector , a "pizza slice" of the circle, bounded by two radii and an arc.

A semicircle is half a circle, formed by a diameter.

Arcs and their uses

When you only need a piece of a circle, draw an arc instead of a full circle , it's cleaner and easier to read. You will use arcs heavily in constructions, especially when finding intersection points between two arcs.

Patterns from circles

By drawing several circles together, you can make beautiful patterns:

• Flower of life. Start with one circle. Then draw six more circles, each centred on a point of the first circle, with the same radius. You get a six-petalled flower. Add more circles around , the pattern fills the plane.
• Concentric circles. All sharing the same centre, with different radii. Think of a target board or ripples in a pond.
• Vesica Piscis. Two circles of equal radius, each passing through the centre of the other. The almond-shaped overlap is sacred in many traditions.
• Spirals. A sequence of arcs of growing radius, each picking up where the last left off.

A property worth remembering

The same circle can be drawn from many different centres at many different radii? No , the centre and radius uniquely determine a circle. But three points (not on the same straight line) also determine a unique circle through them. This will be very useful later when constructing circles through given points.

Worked examples

Example 1. Draw a circle of radius $5$ cm. What is its diameter?

• Diameter $= 2 \times 5 = 10$ cm.

Example 2. A circle has diameter $14$ cm. Find its radius.

• $r = d / 2 = 7$ cm.

Example 3. Draw a circle of radius $4$ cm. Mark a chord that is $3$ cm long. Mark a chord that is $6$ cm long. Mark the diameter.

• Use a ruler to draw chords on the circle. The longest chord is the diameter ($8$ cm).

Example 4. Construction: Draw a "Vesica Piscis".

• Mark two points $A$ and $B$, $5$ cm apart.
• Draw a circle of radius $5$ cm centred at $A$ (passing through $B$).
• Draw a circle of radius $5$ cm centred at $B$ (passing through $A$).
• The two circles intersect at two points; the almond-shaped region between them is the Vesica Piscis.

Example 5. Draw three concentric circles of radii $1, 2, 3$ cm.

• Mark one centre.
• Set compass to $1$ cm, draw circle.
• Reset to $2$ cm, draw.
• Reset to $3$ cm, draw.

Try it yourself

1. Draw a circle of radius $3$ cm. Sketch any three chords. Compare their lengths to the diameter.
2. Draw a sector that is exactly half the circle. What is the angle of the sector?
3. Draw two circles with the same centre but different radii. What is the region between them called? (Hint: it has a special name , an annulus.)
4. Construct the "flower of life" first step: a circle and $6$ circles of the same radius around it.
5. A clock's minute hand traces a circle every hour. If the hand is $10$ cm long, what is the diameter of the circle it traces?
6. Draw two intersecting circles of the same radius. Connect their two intersection points with a line. What property does this line have? (It passes through the centres? Hint: try this.)
7. Sketch a spiral made of $4$ arcs, each a quarter-circle, with radii $1, 2, 3, 4$ cm.
8. Investigate: if you draw three circles, each passing through the centres of the other two (all same radius), how many intersection points do they have?

Activity

Flower of life. Mark a point in the middle of a fresh sheet of paper. Draw a circle around it (radius about $3$ cm). Without changing the compass, pick any point on the circle and centre your compass there. Draw another circle. Then move to a new point on the original circle and repeat. Six circles around the first will form the "flower". Continue outward to fill the page.