The circle revisited AI HL

IB Mathematics: Applications and Interpretation HL · Unit 3: Geometry and Trigonometry

Lesson 3.7 · The Circle Revisited

Understanding radians and using them to measure arcs and sectors.

Faith connection

Psalm 150 ends with an invitation to worship: “Let every thing that hath breath praise the LORD. Praise ye the LORD” (Psalm 150:6, KJV). When we revisit a mathematical idea, we often see more than we noticed the first time. As you deepen your understanding of circles, pause to praise God for the gift of learning and for the breath with which you can thank him.

Learning objectives

By the end of this lesson, you should be able to:

define one radian using a circle’s radius and arc length;
convert angles between degrees and radians, in exact and decimal form;
derive and use $s=r\theta$ for arc length;
derive and use $K=\frac12r^2\theta$ for sector area;
solve problems involving minor and major arcs and sectors;
interpret radians in contexts involving rotation and circular motion.

1. What exactly is a radian?

Imagine a circle of radius $r$ . Mark an arc along its circumference that also has length $r$ . The angle at the centre subtending that arc is one radian.

More generally, the angle in radians is the ratio of arc length to radius:

$\theta=\frac{s}{r}$

This ratio has no physical units because both $s$ and $r$ are lengths in the same units. We nevertheless write “rad” to identify the angle measure.

An arc equal in length to the radius subtends an angle of one radian.

Worked example · Use the definition.

On a circle of radius $8$ cm, an arc of length $8$ cm subtends $8/8=1$ radian. An arc of length $20$ cm on the same circle subtends $20/8=2.5$ radians.

2. From a full circle to degree conversion

A complete circumference has length $2\pi r$ . Dividing by the radius shows that a complete turn is $2\pi$ radians. The same turn is $360^\circ$ , so:

$2\pi\text{ rad}=360^\circ$

$\pi\text{ rad}=180^\circ$

To convert:

$\text{radians}=\text{degrees}\times\frac{\pi}{180}$

$\text{degrees}=\text{radians}\times\frac{180}{\pi}$

Worked example · Exact and decimal answers.

Convert $72^\circ$ to radians:

$72^\circ=72\frac{\pi}{180}=\frac{2\pi}{5}\text{ rad}\approx1.257\text{ rad}$

Convert $2.4$ radians to degrees:

$2.4\frac{180}{\pi}\approx137.5^\circ$

Keep a multiple of $\pi$ when the question asks for an exact answer.

Degrees	Radians
$30^\circ$	$\pi/6$
$45^\circ$	$\pi/4$
$60^\circ$	$\pi/3$
$90^\circ$	$\pi/2$
$180^\circ$	$\pi$
$360^\circ$	$2\pi$

3. Arc length in radians

From the radian definition $\theta=s/r$ , multiply by $r$ :

$s=r\theta$

The radius and arc length must use the same length unit, and $\theta$ must be in radians.

Worked example · Exact arc length.

A circle has radius $9$ cm. A central angle is $5\pi/6$ radians. The corresponding arc length is:

$s=9\left(\frac{5\pi}{6}\right)=\frac{15\pi}{2}\text{ cm}\approx23.6\text{ cm}$

If an angle is given in degrees, convert it first or use the equivalent fraction-of-a-circle formula:

$s=\frac{\theta^\circ}{360^\circ}\,2\pi r$

4. Sector area in radians

A sector covering $\theta$ radians occupies the fraction $\theta/(2\pi)$ of a full circle. Multiply this fraction by the circle’s area:

$K=\frac{\theta}{2\pi}\times\pi r^2$

$K=\frac12r^2\theta$

Again, $\theta$ must be in radians. The answer is in square units.

Worked example · Exact sector area.

For radius $9$ cm and central angle $5\pi/6$ radians:

$K=\frac12(9)^2\left(\frac{5\pi}{6}\right) =\frac{135\pi}{4}\text{ cm}^2\approx106\text{ cm}^2$

Arc length and sector area are different quantities: the first measures a curved boundary; the second measures a region.

5. Find an unknown angle or radius

Rearrange the formulas according to the information given:

$\theta=\frac{s}{r},\qquad r=\frac{s}{\theta},\qquad \theta=\frac{2K}{r^2}$

If both arc length and sector area are known, combine the formulas by replacing $\theta$ with $s/r$ :

$K=\frac12rs\qquad\Longrightarrow\qquad r=\frac{2K}{s}$

Worked example · Determine both unknowns.

A sector has arc length $27$ cm and area $121.5$ cm $^2$ . Then:

$r=\frac{2(121.5)}{27}=9\text{ cm}$

$\theta=\frac{27}{9}=3\text{ rad}$

Check: $\frac12(9)^2(3)=121.5$ cm $^2$ .

6. Minor and major arcs and sectors

Two points on a circle usually divide it into a minor arc and a major arc. If the minor central angle is $\theta$ , the corresponding major angle is $2\pi-\theta$ . Their arc lengths add to $2\pi r$ and their sector areas add to $\pi r^2$ .

Worked example.

A circle has radius $6$ cm and minor central angle $\pi/3$ radians. The major angle is:

$2\pi-\frac{\pi}{3}=\frac{5\pi}{3}$

Its major arc length and major sector area are:

$s_{\mathrm{major}}=6\left(\frac{5\pi}{3}\right)=10\pi\text{ cm}$

$K_{\mathrm{major}}=\frac12(6)^2\left(\frac{5\pi}{3}\right) =30\pi\text{ cm}^2$

7. Circular motion

Radians are useful when something rotates. If an object turns through $\theta$ radians along a circle of radius $r$ , it travels an arc distance $s=r\theta$ . If its angular speed is $\omega$ radians per second, then after $t$ seconds $\theta=\omega t$ , provided the speed stays constant.

$s=r\omega t$

Worked example · A rotating point.

A point lies $0.35$ m from the centre of a rotating wheel. The wheel turns at $2.4$ rad/s. In $5$ seconds, the point turns through $\theta=2.4(5)=12$ rad and travels:

$s=r\theta=0.35(12)=4.2\text{ m}$

The arc distance can exceed one circumference because the wheel can turn more than once.

Calculator and exam tips

In AI HL, use radians unless an angle is explicitly given in degrees or the question says otherwise.
When evaluating $\sin\theta$ or $\cos\theta$ for an angle in radians, select radian mode.
Multiplying by $\pi/180$ converts degrees to radians; multiplying by $180/\pi$ converts radians to degrees.
Check whether the question asks for a minor or major arc or sector.
Use centimetres or metres consistently, then square the unit for area.
Keep exact multiples of $\pi$ when an exact answer is requested.

IB-style Paper 1 · Skill practice

Question 1 · Degree conversion

Express $210^\circ$ as an exact multiple of $\pi$ radians.

Show solution

$210^\circ=210\frac{\pi}{180}=\frac{7\pi}{6}\text{ rad}$

Question 2 · Radian conversion

Convert $7\pi/12$ radians to degrees. Then convert $1.8$ radians to degrees to one decimal place.

Show solution

$(7\pi/12)(180/\pi)=105^\circ$ . Also, $1.8(180/\pi)\approx103.1^\circ$ .

Question 3 · Arc and sector

A circle has radius $12$ cm and central angle $5\pi/8$ radians. Find the exact arc length and sector area.

Show solution

$s=12(5\pi/8)=15\pi/2$ cm. The sector area is $K=\frac12(12)^2(5\pi/8)=45\pi$ cm $^2$ .

Question 4 · Unknown angle

An arc of length $18$ cm lies on a circle of radius $8$ cm. Find its central angle in radians and in degrees to one decimal place.

Show solution

$\theta=s/r=18/8=2.25$ rad. In degrees this is $2.25(180/\pi)\approx128.9^\circ$ .

Question 5 · Sector perimeter

A sector has radius $7$ cm and central angle $1.6$ radians. Find its perimeter and area.

Show solution

The arc length is $7(1.6)=11.2$ cm. The perimeter is $11.2+2(7)=25.2$ cm. The area is $\frac12(7)^2(1.6)=39.2$ cm $^2$ .

Question 6 · Arc length and area

A sector has arc length $16$ m and area $64$ m $^2$ . Find the radius and central angle.

Show solution

$r=2K/s=2(64)/16=8$ m. Then $\theta=s/r=16/8=2$ rad.

Applications and interpretation

Application 1 · A windscreen wiper

The tip of a windscreen wiper is $35$ cm from its pivot. During one sweep, it turns through $110^\circ$ .

Convert the angle to exact radians.
Find the distance travelled by the tip.
Find the area of the sector swept out by a line from the pivot to the tip.

Show solution

$110^\circ=11\pi/18$ rad. Therefore:

$s=35\left(\frac{11\pi}{18}\right)\approx67.2\text{ cm}$

$K=\frac12(35)^2\left(\frac{11\pi}{18}\right)\approx1176\text{ cm}^2$

This is a geometric sector area; the actual glass cleaned also depends on the wiper blade’s shape and width.

Application 2 · A circular running path

A runner follows a circular path of radius $40$ m and travels an arc subtending $1.8$ radians at the centre.

Find the distance travelled.
Find the area of the corresponding sector.
Find what fraction of one complete lap the runner has travelled.

Show solution

The distance is $s=40(1.8)=72$ m. The sector area is $\frac12(40)^2(1.8)=1440$ m $^2$ . The lap fraction is $1.8/(2\pi)=0.9/\pi\approx0.286$ , or about $28.6\%$ of one lap.

Application 3 · A Ferris wheel

A Ferris wheel has radius $15$ m and makes one complete revolution every $12$ minutes at a constant rate.

Find its angular speed in radians per minute.
Find the angle turned through in $5$ minutes.
Find the arc distance travelled by a passenger in $5$ minutes.

Show solution

The angular speed is $\omega=2\pi/12=\pi/6$ rad/min. In $5$ minutes, the wheel turns through $5\pi/6$ rad. The passenger travels an arc distance $s=15(5\pi/6)=25\pi/2\approx39.3$ m.

Application 4 · A circular garden

A circular garden has radius $9$ m. A path follows the boundary of a sector whose central angle is $2.2$ radians.

Find the curved part of the path.
If the two radii also form part of the path, find its total length.
Find the enclosed sector area.

Show solution

The arc is $9(2.2)=19.8$ m. Including both radii gives $19.8+18=37.8$ m of path. The sector area is $\frac12(9)^2(2.2)=89.1$ m $^2$ .

Application 5 · Two sectors from one circle

A circle of radius $10$ cm is divided by two radii into a minor sector with angle $72^\circ$ and a major sector.

Express both angles in radians.
Find both sector areas.
Check that they add to the area of the whole circle.

Show solution

The minor angle is $72\pi/180=2\pi/5$ rad. The major angle is $2\pi-2\pi/5=8\pi/5$ rad. The minor area is $\frac12(10)^2(2\pi/5)=20\pi$ cm $^2$ ; the major area is $\frac12(10)^2(8\pi/5)=80\pi$ cm $^2$ . Their sum is $100\pi$ cm $^2$ , matching $\pi(10)^2$ .

Paper 1 checklist

Write angles in radians before using $s=r\theta$ or $K=\frac12r^2\theta$ . Show whether an answer is exact or rounded, distinguish arc length from sector perimeter, and check minor and major answers against the complete circle.

Lesson 3.7 · The Circle Revisited

Faith connection

Learning objectives

1. What exactly is a radian?

2. From a full circle to degree conversion

3. Arc length in radians

4. Sector area in radians

5. Find an unknown angle or radius

6. Minor and major arcs and sectors

7. Circular motion

Calculator and exam tips

IB-style Paper 1 · Skill practice

Question 1 · Degree conversion

Question 2 · Radian conversion

Question 3 · Arc and sector

Question 4 · Unknown angle

Question 5 · Sector perimeter

Question 6 · Arc length and area

Applications and interpretation

Application 1 · A windscreen wiper

Application 2 · A circular running path

Application 3 · A Ferris wheel

Application 4 · A circular garden

Application 5 · Two sectors from one circle

Paper 1 checklist

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IB Mathematics: Applications and Interpretation HL 6 Sections80 LessonsLifetime

Curriculum

The circle revisited AI HL

Faith connection

Learning objectives

1. What exactly is a radian?

2. From a full circle to degree conversion

3. Arc length in radians

4. Sector area in radians

5. Find an unknown angle or radius

6. Minor and major arcs and sectors

7. Circular motion

Calculator and exam tips

IB-style Paper 1 · Skill practice

Question 1 · Degree conversion

Question 2 · Radian conversion

Question 3 · Arc and sector

Question 4 · Unknown angle

Question 5 · Sector perimeter

Question 6 · Arc length and area

Applications and interpretation

Application 1 · A windscreen wiper

Application 2 · A circular running path

Application 3 · A Ferris wheel

Application 4 · A circular garden

Application 5 · Two sectors from one circle

Paper 1 checklist

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