Circle and Related Theorems

Lessons

Senior 3 · Mathematics

Circle and Related Theorems

5 lessons

Lesson 1 of 5

Parts of a circle, arcs and sectors

25 min

The circle that waters a farm

A centre-pivot irrigator is a long pipe on wheels, fixed at one end to a pivot. It sprays water as it turns slowly about the pivot, so the ground it waters is a circle, or part of one. Suppose the pipe is 400 m long and the farmer stops it after it has turned through $135^\circ$ . How much land got water, and how far did the wheel at the far end travel? To answer, you need the parts of a circle and two formulas that take a fraction of a whole circle.

Definition

A circle is the set of all points in a plane at the same distance from a fixed point, the centre (usually called $O$ ). That fixed distance is the radius $r$ .

Left: lines of a circle. Right: regions of a circle.

Term	Meaning
Radius	A segment from the centre to the circle; its length is $r$ .
Chord	A segment joining two points of the circle.
Diameter	A chord through the centre; its length is $d = 2 r$ . It is the longest chord.
Circumference	The whole curve, or its length.
Arc	Part of the circumference between two points $A$ and $B$ . The shorter one is the minor arc $A B$ , the longer one the major arc $A B$ .
Semicircle	Half a circle, cut off by a diameter.
Sector	The region between two radii and the arc they cut off (a "slice").
Segment	The region between a chord and the arc it cuts off.
Tangent	A straight line that touches the circle at exactly one point, the point of contact.

Notation: $\angle AOB$ is the angle that the minor arc $A B$ (or the chord $A B$ ) subtends at the centre: the angle between the radii $O A$ and $O B$ .

Important

Circumference and area. For every circle, the circumference divided by the diameter is the same number, $\pi \approx 3.14159$ . So

C = \pi d = 2\pi r \qquad\qquad A = \pi r^2 .

Use the $π \pi$ key on your calculator unless a question says "take $\pi = \frac{22}{7}$ " or "leave your answer in terms of $π \pi$ ".

Worked example 1: a roundabout island

The grass island in the middle of a roundabout is a circle of diameter 24 m. Find (a) the length of the kerb around it and (b) its area, to 1 decimal place.

Solution The radius is $r = \frac{24}{2} = 12$ m.

(a) $C = \pi d = 24\pi = 75.398\ldots \approx 75.4$ m.

(b) $A = \pi r^2 = 144\pi = 452.389\ldots \approx 452.4\ \mathrm{m^2}$ .

Worked example 2: working backwards from the area

A round table top has area $154\ \mathrm{dm^2}$ . Taking $\pi = \frac{22}{7}$ , find its radius and its circumference.

Solution $\pi r^2 = 154$ , so $r^2 = 154 \times \frac{7}{22} = 49$ and $r = 7$ dm (a length is positive).

$C = 2\pi r = 2 \times \frac{22}{7} \times 7 = 44$ dm.

A fraction of the circle: arcs and sectors

A full turn about the centre is $360^\circ$ . If the angle at the centre is $\theta$ , the arc is the fraction $\frac{\theta}{360^\circ}$ of the circumference, and the sector is the same fraction of the area.

Important

Arc length and sector area ( $\theta$ in degrees, radius $r$ ):

l = \frac{\theta}{360^\circ} \times 2\pi r \qquad\qquad A_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2 = \frac{1}{2}\,l\,r .

Why $A = \frac{1}{2}lr$ : divide the first formula into the second: $\frac{A}{l} = \frac{\pi r^2}{2\pi r} = \frac{r}{2}$ .

The perimeter of a sector is the arc plus the two radii: $l + 2 r$ .

Worked example 3: the irrigator

The pipe is 400 m long and turns through $135^\circ$ . Find the distance travelled by the outer wheel and the area watered, in hectares ( $1\ \text{ha} = 10\,000\ \mathrm{m^2}$ ).

Solution The fraction of a full turn is $\frac{135}{360} = \frac{3}{8}$ .

Arc: $l = \frac{3}{8} \times 2\pi \times 400 = 300\pi \approx 942.5$ m.

Sector: $A = \frac{3}{8} \times \pi \times 400^2 = 60\,000\pi \approx 188\,496\ \mathrm{m^2} \approx 18.8$ ha.

Check with $A = \frac{1}{2}lr = \frac{1}{2} \times 300\pi \times 400 = 60\,000\pi$ . ✓

Worked example 4: finding the angle

A sector of radius 15 cm has area $150\ \mathrm{cm^2}$ . Find its arc length and its angle, to 1 decimal place.

Solution From $A = \frac{1}{2}lr$ : $150 = \frac{1}{2} \times l \times 15$ , so $l = 20$ cm.

From $l = \frac{\theta}{360} \times 2\pi r$ : $\theta = \frac{360 \times 20}{2\pi \times 15} = \frac{240}{\pi} \approx 76.4^\circ$ .

Worked example 5: the area of a segment

A chord $A B$ of a circle of radius 6 cm subtends a right angle at the centre. Find the area of the minor segment.

Solution Segment = sector $A O B$ − triangle $A O B$ .

Sector: $\frac{90}{360} \times \pi \times 6^2 = 9\pi \approx 28.27\ \mathrm{cm^2}$ . Triangle: its legs $O A$ and $O B$ are radii at right angles, so its area is $\frac{1}{2} \times 6 \times 6 = 18\ \mathrm{cm^2}$ .

Segment: $9\pi - 18 \approx 10.27\ \mathrm{cm^2}$ .

Warning

Radius or diameter? Read the question twice. Putting the diameter into $\pi r^2$ makes the area four times too big.

A sector's perimeter is not its arc. Add the two radii: $l + 2 r$ .

Tip

Every arc and sector question is one idea: what fraction of the circle is it? Find $\frac{\theta}{360}$ first, then take that fraction of $2\pi r$ or of $\pi r^2$ .

Cognito: Area of a sector and length of an arc of a circle. Watch how each formula takes the fraction of the full circle, the same idea as the tip above, and how minor and major sectors are told apart.

✦

Practice

Use the $π \pi$ key and give answers to 1 decimal place unless told otherwise. The last two are harder.

Find the circumference and the area of a circle of radius 3.5 cm.
A bicycle wheel has diameter 66 cm. How many complete turns does it make in 1 km?
Find the arc length and the area of a sector of radius 8 cm and angle $45^\circ$ . Leave your answers in terms of $π \pi$ .
An arc of length 11 cm lies on a circle of radius 9 cm. Find the angle it subtends at the centre.
Find the perimeter of a sector of radius 6 cm and angle $150^\circ$ .
A chord subtends $90^\circ$ at the centre of a circle of radius 10 cm. Find the area of the minor segment.
A sector of angle $216^\circ$ is cut from a circle of radius 10 cm, and its two straight edges are joined to make a cone. Find the radius of the base of the cone.

Show answersHide answers

$C = 7\pi \approx 22.0$ cm; $A = 12.25\pi \approx 38.5\ \mathrm{cm^2}$ .
One turn is $66\pi \approx 207.35$ cm, and 1 km = 100 000 cm: $\frac{100\,000}{207.35} \approx 482.3$ , so 482 complete turns.
$l = \frac{45}{360} \times 16\pi = 2\pi$ cm; $A = \frac{45}{360} \times 64\pi = 8\pi\ \mathrm{cm^2}$ .
$\theta = \frac{360 \times 11}{2\pi \times 9} \approx 70.0^\circ$ .
$l = \frac{150}{360} \times 12\pi = 5\pi \approx 15.71$ cm, so the perimeter is $5\pi + 12 \approx 27.7$ cm.
$25\pi - 50 \approx 28.5\ \mathrm{cm^2}$ .
The arc, $\frac{216}{360} \times 20\pi = 12\pi$ cm, becomes the base circle: $2\pi R = 12\pi$ , so $R = 6$ cm.

Sources and credits

Written by Ganzaa.
Photo: Center pivot irrigation, Nic McPhee, Wikimedia Commons, CC BY-SA 2.0.
Image: Crops Kansas AST 20010624, NASA, Wikimedia Commons, public domain.
Video: GCSE Maths - Area of a Sector and Length of an Arc of a Circle, Cognito, embedded from YouTube.

The diagrams, worked examples and practice items were written by Ganzaa; the irrigator, roundabout and table measurements are illustrative. This lesson uses CC BY-SA material and is shared under CC BY-SA 4.0.

1The circle that waters a farm

2A fraction of the circle: arcs and sectors

The circle that waters a farm

A fraction of the circle: arcs and sectors