Circles

A ban­gle, a coin, the rim of a cycle wheel and the face of a clock are all cir­cles. A cir­cle is dif­fer­ent from the shapes in the last two lessons: it has no straight sides and no cor­ners. In this les­son you will learn exactly what a cir­cle is, the names of its parts, and how to tell whether a point is inside it, on it or out­side it. Find­ing the length round a cir­cle and its area comes in the men­su­ra­tion chap­ter.

What a cir­cle is#

Push a pin into a sheet of paper, tie a loop of thread round it, put a pen­cil in the other end of the loop and, keep­ing the thread tight, draw all the way round. The curve you draw is a cir­cle. A com­pass does the same job.

Cir­cle. A cir­cle is the set of all points in a plane that are at the same dis­tance from a fixed point. The fixed point is called the cen­tre of the cir­cle.

A cir­cle is usu­ally named by its cen­tre. A cir­cle with cen­tre $O$ is called "cir­cle $O$".

The parts of a cir­cle#

Radius. A line seg­ment from the cen­tre to any point on the cir­cle is a radius. All the radii (the plural of radius) of a cir­cle are equal. Their com­mon length is also called the radius, $r$. A cir­cle has as many radii as you like.

Chord. A line seg­ment join­ing any two points on the cir­cle is a chord.

Diam­e­ter. A chord that passes through the cen­tre is a diam­e­ter. It is made of two radii in a straight line, so

$$\text{diameter} = 2 \times \text{radius} \qquad d = 2r$$

The diam­e­ter is the longest chord of a cir­cle. Every other chord is shorter.

Cir­cum­fer­ence. The curve of the cir­cle itself, the bound­ary all the way round, is called its cir­cum­fer­ence. The same word is used for its length.

Arc. A part of the cir­cle between two points on it is called an arc. Two points on a cir­cle split it into two arcs. The shorter one is the minor arc and the longer one is the major arc.

Semi­cir­cle. A diam­e­ter cuts a cir­cle into two equal arcs. Each is a semi­cir­cle, which means half a cir­cle.

Regions of a cir­cle#

A cir­cle divides the plane into three parts:

• the inte­rior, the region inside the cir­cle,
• the cir­cle itself,
• the exte­rior, the region out­side the cir­cle.

The cir­cle together with its inte­rior is called the cir­cu­lar region.

Sec­tor. The region between two radii and the arc join­ing their ends is a sec­tor. A slice of a round pizza is a sec­tor. A semi­cir­cu­lar region is a sec­tor whose angle at the cen­tre is $180^\circ$, and a quar­ter cir­cle is a sec­tor whose angle at the cen­tre is $90^\circ$.

Seg­ment. The region between a chord and the arc it cuts off is a seg­ment. A chord cuts the cir­cu­lar region into two seg­ments, the minor seg­ment and the major seg­ment.

Be care­ful not to mix up the words: a sec­tor is bounded by two radii and an arc; a seg­ment is bounded by a chord and an arc.

Inside, on or out­side?#

Let the radius of a cir­cle be $r$, and let a point $P$ be at a dis­tance $d$ from the cen­tre.

Con­cen­tric cir­cles. Cir­cles with the same cen­tre but dif­fer­ent radii are called con­cen­tric cir­cles. The rings of a dart­board are con­cen­tric cir­cles.

Exam­ple 1#

The radius of a cir­cle is 7 cm. Find its diam­e­ter.

$$d = 2r = 2 \times 7 = 14 \text{ cm}$$

Exam­ple 2#

The diam­e­ter of a cir­cu­lar plate is 9 cm. Find its radius.

$$\displaystyle r = \frac{d}{2} = \frac{9}{2} = 4.5 \text{ cm}$$

Exam­ple 3#

A cir­cle has radius 6 cm. Points $P$, $Q$ and $R$ are 5 cm, 6 cm and 8 cm from its cen­tre. Where is each point?

$P$ is 5 cm away, less than 6 cm, so it is inside. $Q$ is exactly 6 cm away, so it is on the cir­cle. $R$ is 8 cm away, more than 6 cm, so it is out­side.

Exam­ple 4#

Can a cir­cle of radius 5 cm have a chord 12 cm long?

The longest chord is the diam­e­ter, $2 \times 5 = 10$ cm. A chord of 12 cm would be longer than the diam­e­ter. So no, it can­not.

Exam­ple 5#

Two con­cen­tric cir­cles have radii 3 cm and 8 cm. How wide is the ring between them?

Along any line from the cen­tre, the inner cir­cle is 3 cm away and the outer cir­cle is 8 cm away. The ring is $8 - 3 = 5$ cm wide.

Angles at the cen­tre#

A com­plete turn round the cen­tre is $360^\circ$. So:

• a semi­cir­cle makes an angle of $180^\circ$ at the cen­tre,
• a quar­ter cir­cle makes an angle of $90^\circ$ at the cen­tre.

If a cir­cle is cut into equal sec­tors, divide $360^\circ$ by the num­ber of sec­tors to get each angle at the cen­tre. A round cake cut into 8 equal pieces gives pieces with an angle of $360^\circ \div 8 = 45^\circ$ at the cen­tre.

Prac­tice#

1. The radius of a cir­cle is 12 cm. Find its diam­e­ter.
2. The diam­e­ter of a wheel is 70 cm. Find its radius.
3. How many radii can be drawn in a cir­cle? How many diam­e­ters?
4. Which is the longest chord of a cir­cle?
5. A cir­cle has radius 4 cm. Point $A$ is 3.5 cm from the cen­tre and point $B$ is 4.5 cm from the cen­tre. Where is each point?
6. Can a cir­cle of diam­e­ter 10 cm have a chord 8 cm long? A chord 11 cm long?
7. Two con­cen­tric cir­cles have radii 5 cm and 9 cm. How wide is the ring between them?
8. A round pizza is cut into 6 equal slices. What is the angle at the cen­tre of each slice?
9. What is the dif­fer­ence between a sec­tor and a seg­ment?

Answers#