Circumference and area of a circle

A cir­cle has no straight sides, so you can­not add up sides to find its perime­ter, or count rows of squares to find its area. Yet there are sim­ple for­mu­las for both, and they use one spe­cial num­ber, called pi. In this les­son you will meet pi, find the dis­tance round a cir­cle and the area inside it, and solve prob­lems about wheels, rings and semi­cir­cles.

The num­ber pi#

Take a few round objects: a ban­gle, a plate, a bot­tle cap. Wrap a thread round each one to mea­sure the dis­tance round it, its cir­cum­fer­ence. Then mea­sure its diam­e­ter. Divide the cir­cum­fer­ence by the diam­e­ter.

For every cir­cle, big or small, the answer is the same num­ber, a lit­tle more than 3. This num­ber is called pi and is writ­ten $\pi$.

$$\displaystyle \frac{\text{circumference}}{\text{diameter}} = \pi$$

The dig­its of $\pi$ go on for ever with­out repeat­ing, so we use an approx­i­mate value. In this chap­ter we take $\displaystyle \pi = \frac{22}{7}$, unless a ques­tion says $\pi = 3.14$. Both are close to the true value, $3.14159\ldots$

Cir­cum­fer­ence of a cir­cle#

From $\displaystyle \frac{C}{d} = \pi$ we get $C = \pi d$. The diam­e­ter is twice the radius, $d = 2r$, so

$$C = \pi d = 2\pi r$$

Exam­ple 1#

Find the cir­cum­fer­ence of a cir­cle of radius 7 cm.

$$\displaystyle C = 2 \times \frac{22}{7} \times 7 = 44$$

The cir­cum­fer­ence is 44 cm.

Exam­ple 2#

Find the cir­cum­fer­ence of a cir­cle of diam­e­ter 28 cm.

$$\displaystyle C = \frac{22}{7} \times 28 = 88$$

The cir­cum­fer­ence is 88 cm.

Exam­ple 3#

The cir­cum­fer­ence of a cir­cle is 132 cm. Find its radius.

$$\displaystyle 2 \times \frac{22}{7} \times r = 132 \quad \text{so} \quad \frac{44}{7}r = 132 \quad \text{and} \quad r = \frac{132 \times 7}{44} = 21$$

The radius is 21 cm.

Area of a cir­cle#

Cut a cir­cle into many thin slices, like a cake. Lay them side by side, points up and points down in turn. They make a shape very close to a rec­tan­gle. Its length is half the cir­cum­fer­ence, $\pi r$, and its height is the radius, $r$. The more slices you cut, the closer it gets to a true rec­tan­gle. So

$$\text{area of a circle} = \pi r \times r = \pi r^2$$

Exam­ple 4#

Find the area of a cir­cle of radius 7 cm.

$$\displaystyle A = \frac{22}{7} \times 7 \times 7 = 154$$

The area is 154 cm².

Exam­ple 5#

The area of a cir­cle is 616 cm². Find its radius.

$$\displaystyle \frac{22}{7} \times r^2 = 616 \quad \text{so} \quad r^2 = \frac{616 \times 7}{22} = 196 \quad \text{and} \quad r = 14$$

The radius is 14 cm, because $14 \times 14 = 196$.

Exam­ple 6: Using 3.14#

Find the cir­cum­fer­ence and area of a cir­cle of radius 10 cm. Take $\pi = 3.14$.

$$C = 2 \times 3.14 \times 10 = 62.8 \qquad A = 3.14 \times 10 \times 10 = 314$$

The cir­cum­fer­ence is 62.8 cm and the area is 314 cm².

Semi­cir­cles#

A semi­cir­cle is half a cir­cle. Its curved edge is half the cir­cum­fer­ence, $\pi r$. Its perime­ter also includes the straight diam­e­ter, $2r$.

$$\displaystyle \text{perimeter of a semicircle} = \pi r + 2r \qquad \text{area of a semicircle} = \frac{1}{2}\pi r^2$$

Exam­ple 7#

Find the perime­ter and area of a semi­cir­cle of radius 7 cm.

$$\displaystyle \text{perimeter} = \frac{22}{7} \times 7 + 2 \times 7 = 22 + 14 = 36$$

$$\displaystyle \text{area} = \frac{1}{2} \times \frac{22}{7} \times 7 \times 7 = 77$$

The perime­ter is 36 cm and the area is 77 cm².

Rings#

The region between two cir­cles with the same cen­tre is a ring. Its area is the area of the big cir­cle minus the area of the small one.

Exam­ple 8#

A cir­cu­lar park of radius 14 m has a cir­cu­lar pond of radius 7 m at its cen­tre. Find the area of the park out­side the pond.

$$\displaystyle \frac{22}{7} \times 14 \times 14 - \frac{22}{7} \times 7 \times 7 = 616 - 154 = 462$$

The area is 462 m².

Wheels#

When a wheel turns once, it rolls for­ward a dis­tance equal to its cir­cum­fer­ence.

$$\displaystyle \text{number of turns} = \frac{\text{distance}}{\text{circumference}}$$

Exam­ple 9#

A cycle wheel has diam­e­ter 70 cm. How many times does it turn in going 1.1 km?

Cir­cum­fer­ence $\displaystyle = \frac{22}{7} \times 70 = 220$ cm.

1.1 km = 1100 m = 1,10,000 cm.

$$\displaystyle \text{number of turns} = \frac{110000}{220} = 500$$

The wheel turns 500 times.

Prac­tice#

1. Find the cir­cum­fer­ence of a cir­cle of radius 21 cm.
2. Find the cir­cum­fer­ence of a cir­cle of diam­e­ter 14 m.
3. The cir­cum­fer­ence of a cir­cle is 88 cm. Find its radius.
4. Find the area of a cir­cle of radius 14 cm.
5. Find the area of a cir­cle of diam­e­ter 42 m.
6. The area of a cir­cle is 154 m². Find its diam­e­ter.
7. Find the cir­cum­fer­ence and area of a cir­cle of radius 5 cm, tak­ing $\pi = 3.14$.
8. Find the perime­ter of a semi­cir­cle of radius 14 cm.
9. A cir­cu­lar gar­den of radius 21 m has a path 7 m wide round it, out­side. Find the area of the path.
10. A wheel of radius 35 cm turns 400 times. How far does it go, in metres?

Answers#