Arcs, sectors and segments

You already know the cir­cum­fer­ence and area of a whole cir­cle. Many real shapes use only part of a cir­cle: the path swept by a clock's minute hand, a slice of a round cake, the region a wind­screen wiper cleans, a cir­cu­lar run­ning track. This les­son finds the length of an arc and the areas of sec­tors, seg­ments and rings, all by tak­ing the right frac­tion of the whole cir­cle.

The whole cir­cle#

For a cir­cle of radius $r$:

$$\text{Circumference} = 2\pi r, \qquad \text{Area} = \pi r^2$$

Unless a ques­tion says oth­er­wise, take $\displaystyle \pi = \frac{22}{7}$. It makes the arith­metic easy when the radius is a mul­ti­ple of $7$.

Exam­ple 1#

A cycle wheel has radius $35$ cm. How far does the cycle travel when the wheel turns $100$ times?

In one turn the cycle moves one cir­cum­fer­ence: $\displaystyle 2 \times \frac{22}{7} \times 35 = 220$ cm. In $100$ turns it moves $22000$ cm, that is $220$ m.

Arcs and sec­tors#

An arc whose angle at the cen­tre is $\theta$ (in degrees) is the frac­tion $\displaystyle \frac{\theta}{360}$ of the whole cir­cle. So is the sec­tor it cuts off. There­fore

$$\displaystyle \text{Length of arc} = \frac{\theta}{360} \times 2\pi r, \qquad \text{Area of sector} = \frac{\theta}{360} \times \pi r^2$$

The perime­ter of a sec­tor is the arc plus the two radii: $\displaystyle \frac{\theta}{360} \times 2\pi r + 2r$.

Exam­ple 2#

In a cir­cle of radius $21$ cm, an arc sub­tends $60°$ at the cen­tre. Find the length of the arc and the area of the sec­tor.

$$\displaystyle \text{Arc} = \frac{60}{360} \times 2 \times \frac{22}{7} \times 21 = \frac{1}{6} \times 132 = 22 \text{ cm}$$

$$\displaystyle \text{Sector} = \frac{60}{360} \times \frac{22}{7} \times 21 \times 21 = \frac{1}{6} \times 1386 = 231 \text{ cm}^2$$

Exam­ple 3#

The minute hand of a clock is $14$ cm long. Find the area swept by it in $15$ min­utes.

In $60$ min­utes the hand turns $360°$, so in $15$ min­utes it turns $90°$.

$$\displaystyle \text{Area} = \frac{90}{360} \times \frac{22}{7} \times 14 \times 14 = \frac{1}{4} \times 616 = 154 \text{ cm}^2$$

Seg­ments#

A chord cuts a cir­cle into two seg­ments. The smaller is the minor seg­ment, the larger the major seg­ment. The minor seg­ment is what is left of the sec­tor after remov­ing the tri­an­gle formed by the two radii and the chord:

$$\text{Area of minor segment} = \text{Area of sector} - \text{Area of triangle}$$

$$\text{Area of major segment} = \text{Area of circle} - \text{Area of minor segment}$$

Exam­ple 4#

A chord of a cir­cle of radius $10$ cm sub­tends a right angle at the cen­tre. Find the areas of the minor and major seg­ments. (Use $\pi = 3.14$.)

$$\displaystyle \text{Sector} = \frac{90}{360} \times 3.14 \times 100 = 78.5 \text{ cm}^2$$

The tri­an­gle has two radii as its legs, at right angles: area $\displaystyle \frac{1}{2} \times 10 \times 10 = 50$ cm².

$$\text{Minor segment} = 78.5 - 50 = 28.5 \text{ cm}^2$$

$$\text{Major segment} = 3.14 \times 100 - 28.5 = 314 - 28.5 = 285.5 \text{ cm}^2$$

Exam­ple 5#

A chord of a cir­cle of radius $12$ cm sub­tends $60°$ at the cen­tre. Find the area of the minor seg­ment. (Use $\pi = 3.14$ and $\sqrt{3} = 1.73$.)

$$\displaystyle \text{Sector} = \frac{60}{360} \times \pi \times 144 = 24\pi = 24 \times 3.14 = 75.36 \text{ cm}^2$$

The tri­an­gle has two sides of $12$ cm with $60°$ between them, so it is equi­lat­eral, with area $\displaystyle \frac{\sqrt{3}}{4} \times 144 = 36\sqrt{3} = 36 \times 1.73 = 62.28$ cm².

$$\text{Minor segment} = 75.36 - 62.28 = 13.08 \text{ cm}^2$$

Rings#

The region between two cir­cles with the same cen­tre is a ring (annu­lus). Its area is the dif­fer­ence of the two cir­cles:

$$\text{Area of ring} = \pi R^2 - \pi r^2 = \pi(R^2 - r^2) = \pi(R + r)(R - r)$$

Exam­ple 6#

Find the area of a ring between cir­cles of radii $14$ cm and $7$ cm.

$$\displaystyle \text{Area} = \frac{22}{7} \times (14 + 7)(14 - 7) = \frac{22}{7} \times 21 \times 7 = 462 \text{ cm}^2$$

Prac­tice#

Use $\displaystyle \pi = \frac{22}{7}$ unless told oth­er­wise.

1. Find the cir­cum­fer­ence and area of a cir­cle of radius $28$ cm.
2. Find the length of an arc of a cir­cle of radius $14$ cm that sub­tends $45°$ at the cen­tre.
3. Find the area of a sec­tor of a cir­cle of radius $7$ cm with angle $120°$.
4. An arc of length $44$ cm is cut from a cir­cle of radius $42$ cm. Find the angle it sub­tends at the cen­tre.
5. The minute hand of a clock is $21$ cm long. Find the area it sweeps in $20$ min­utes and the dis­tance its tip moves.
6. A chord of a cir­cle of radius $14$ cm sub­tends a right angle at the cen­tre. Find the area of the minor seg­ment.
7. Find the area of a ring between cir­cles of radii $21$ cm and $14$ cm.
8. A wheel has diam­e­ter $70$ cm. How many com­plete turns does it make in cov­er­ing $1.1$ km?
9. Find the perime­ter of a sec­tor of radius $21$ cm with angle $60°$.

Answers#