Module 25 of 47Lesson 1 of 10 · HL · AI 3.7 — Radian measure

Course map · Topic 3: Geometry and Trigonometry

3.7 Radian measure

HL only11 worked questions

A radian is a second way of measuring angles, based on the radius of a circle. With radians the formulas for arc length and sector area become very short, and many models (Ferris wheels, tides, pendulums) are written in radians.

✏️ Start with the questions. Try each one on paper, then open the solution and compare.

Stuck? Open the notes ↓

✏️ Practice questions

Try each question on paper first. Then open the solution and compare every step.

Question 1●○○ Warm-up✍️ + 🖩 · Paper 1[3 marks]

Convert to radians:

(a) $150^\circ$ , giving your answer as an exact multiple of $\pi$ ;[2]
(b) $40^\circ$ , giving your answer correct to 3 significant figures.[1]

Show the full step-by-step solution

Step 1🧠 ThinkThe one fact you need

Half a turn is

180^\circ

, and it is also

\pi

radians. So to change degrees into radians we find what fraction of

180^\circ

the angle is, and take that fraction of

\pi

. That is the same as multiplying by

\dfrac{\pi}{180}

Part (a)

Step 1✍️ By handMultiply by $\dfrac{\pi}{180}$ and simplify the fraction

150^\circ = 150\times\dfrac{\pi}{180}

=\dfrac{150}{180}\pi

(divide top and bottom by 30)

=\dfrac{5\pi}{6}

Part (b)

Step 1🖩 On the GDCLet the calculator do the decimal

40›×💻 *›π💻 type pi›÷💻 /›180›ctrl+enter💻 Ctrl+Enter

The π key is next to the letters. Pressing ctrl+enter (instead of enter) forces a decimal answer.

40° = 0.698132… rad; also a check of (a):

\dfrac{5\pi}{6}

= 2.61799…

✅ Check it

40^\circ

is a bit less than

57.3^\circ

(one radian), so the answer should be a bit less than 1. ✔

Final answer (a)

\dfrac{5\pi}{6}

(b)

0.698

rad

Question 2●○○ Warm-up✍️ + 🖩 · Paper 1[3 marks]

Convert to degrees: (a)

\dfrac{3\pi}{4}

radians; (b)

2.4

radians, giving your answer correct to 1 decimal place.

Show the full step-by-step solution

Step 1🧠 ThinkGoing the other way

Now we start in radians. Since

\pi

radians is

180^\circ

, we can simply replace $\pi$ by 180. If there is no

\pi

in the angle, multiply by

\dfrac{180}{\pi}

Part (a)

Step 1✍️ By handReplace π by 180°

\dfrac{3\pi}{4} = \dfrac{3\times180^\circ}{4}

=\dfrac{540^\circ}{4} = 135^\circ

Part (b)

Step 1🖩 On the GDCMultiply by 180 and divide by π

2.4›×💻 *›180›÷💻 /›π💻 type pi›ctrl+enter💻 Ctrl+Enter

2.4 rad = 137.509…°

Step 2✍️ By handRound as asked

2.4 \text{ rad} = 137.509\ldots^\circ \approx 137.5^\circ

Final answer (a)

135^\circ

(b)

137.5^\circ

⚠️ Common mistake Multiplying by

\dfrac{\pi}{180}

instead of

\dfrac{180}{\pi}

. Check the size: 2.4 rad is more than 2 × 57° ≈ 115°, so the answer must be over 100°.

Question 3●○○ Warm-up✍️ By hand · Paper 1[4 marks]

A sector of a circle has radius

8\text{ cm}

and angle

1.3

radians at the centre, as shown. Find (a) the length of the arc; (b) the area of the sector.

Show the full step-by-step solution

Step 1🧠 ThinkWhich formulas?

The angle is already in radians (there is no ° sign), so we can use the two radian formulas straight away: arc

l = r\theta

and area

A = \dfrac{1}{2}r^2\theta

. Here

r = 8

and

\theta = 1.3

Part (a)

Step 1✍️ By handArc length

l = r\theta = 8\times1.3

l = 10.4\text{ cm}

Part (b)

Step 1✍️ By handSector area

A = \dfrac{1}{2}\times8^2\times1.3 = \dfrac{1}{2}\times64\times1.3

A = 41.6\text{ cm}^2

✅ Check it The whole circle has area

\pi\times64 \approx 201\text{ cm}^2

. Our angle is

\dfrac{1.3}{2\pi}\approx 21\%

of a full turn, and 21% of 201 is about 42. ✔

Final answer (a)

10.4\text{ cm}

(b)

41.6\text{ cm}^2

Question 4●●○ Standard✍️ By hand · Paper 1[5 marks]

A sector of a circle of radius

6\text{ cm}

has area

30\text{ cm}^2

(a) Find the angle of the sector in radians.[2]
(b) Find the perimeter of the sector.[3]

Show the full step-by-step solution

Step 1🧠 ThinkWork backwards from the area

This time we know the area and want the angle. Put what we know into

A = \dfrac{1}{2}r^2\theta

and solve for

\theta

Part (a)

Step 1✍️ By handSolve for θ

30 = \dfrac{1}{2}\times6^2\times\theta

30 = 18\theta

\theta = \dfrac{30}{18} = \dfrac{5}{3} = 1.67

rad (3 s.f.)A1

Part (b)

Step 1🧠 ThinkPerimeter = all the way round the edge

Walk round the edge of the sector: along one radius (6 cm), round the arc, and back along the other radius (6 cm).

Step 2✍️ By handArc, then perimeter

l = r\theta = 6\times\dfrac{5}{3} = 10\text{ cm}

Perimeter

= 6 + 6 + 10

= 22\text{ cm}

Final answer (a)

\dfrac{5}{3}\approx1.67

rad (b)

22\text{ cm}

⚠️ Common mistake Using the rounded angle 1.67 in part (b) gives

6\times1.67 = 10.02

. Use the exact

\dfrac{5}{3}

and the arc is exactly 10 cm.

Question 5●●○ Standard✍️ By hand · Paper 1[5 marks]

A sector has radius

7\text{ cm}

and perimeter

25\text{ cm}

. Find (a) the angle of the sector in radians; (b) the area of the sector.

Show the full step-by-step solution

Step 1🧠 ThinkTake away the two straight edges

The perimeter is radius + radius + arc. The two radii are

7 + 7 = 14

cm, so whatever is left of the 25 cm must be the arc.

Part (a)

Step 1✍️ By handFind the arc, then the angle

l = 25 - 2\times7 = 11\text{ cm}

r\theta = 11 \Rightarrow 7\theta = 11

\theta = \dfrac{11}{7} = 1.57

rad (3 s.f.)A1

Part (b)

Step 1✍️ By handArea with the exact angle

A = \dfrac{1}{2}\times7^2\times\dfrac{11}{7}

A = \dfrac{1}{2}\times49\times\dfrac{11}{7} = \dfrac{539}{14} = 38.5\text{ cm}^2

💡 Remember

\dfrac{11}{7}\approx 1.571

is very close to

\dfrac{\pi}{2}\approx1.5708

, so this sector is almost a quarter circle. A quarter of

\pi\times49

38.48

— a quick check. ✔

Final answer (a)

\dfrac{11}{7}\approx1.57

rad (b)

38.5\text{ cm}^2

Question 6●●○ Standard🖩 GDC · Paper 1[3 marks]

The height,

h

metres, of a seat on a Ferris wheel at a fair in Gaborone is modelled by

h(t) = 12 - 10\cos(0.5t)

, where

t

is the time in minutes after the ride starts. Find the height of the seat after

3

minutes.

Show the full step-by-step solution

Step 1🧠 ThinkWhy the mode matters

There is no ° sign in

\cos(0.5t)

, so the angle is in radians. The calculator must be in RAD mode, otherwise it works out

\cos(1.5^\circ)

, which is a completely different number.

Step 2🖩 On the GDCSwitch to radians

doc💻 File menu

On the home screen press doc → Settings → Document Settings, and set Angle to Radian. The status bar at the top of every page then shows RAD.

Step 3🖩 On the GDCSubstitute t = 3

12›−›10›×💻 *›trig💻 type sin( cos( tan(›cos›0.5›×💻 *›3›enter💻 Enter

Press trig and choose cos, then type the bracket contents.

RAD mode: h(3) = 11.2926…

Step 4✍️ By handWrite the answer

h(3) = 12 - 10\cos(1.5)

h(3) = 11.3\text{ m}

(3 s.f.)A1

✅ Check it In DEG mode you would get

2.00\text{ m}

— almost at the bottom after 3 minutes. That is not sensible, which warns you the mode is wrong.

Final answer

11.3\text{ m}

⚠️ Common mistake Calculator left in DEG mode, giving

2.00\text{ m}

Question 7●●○ Standard✍️ + 🖩 · Paper 1[5 marks]

The diagram shows a circle with centre O and radius

10\text{ cm}

. The chord AB makes an angle of

1.2

radians at O. Find the area of the shaded segment.

Show the full step-by-step solution

Step 1🧠 ThinkSegment = sector − triangle

The segment is the piece between the chord and the arc. If we take the whole sector OAB and cut off the triangle OAB, the segment is what is left.

Step 2✍️ By handSector area and triangle area

The triangle has two sides of 10 cm with the angle 1.2 rad between them, so use

\dfrac{1}{2}ab\sin C

Sector:

\dfrac{1}{2}\times10^2\times1.2 = 60

Triangle:

\dfrac{1}{2}\times10\times10\times\sin(1.2)

Segment

= 60 - 50\sin(1.2)

Step 3🖩 On the GDCEvaluate in RAD mode

60›−›50›×💻 *›trig💻 type sin( cos( tan(›sin›1.2›enter💻 Enter

Triangle = 46.602…, segment = 13.398…

Step 4✍️ By handAnswer

Segment

= 13.398\ldots \approx 13.4\text{ cm}^2

Final answer

13.4\text{ cm}^2

Question 8●●● Exam-level✍️ + 🖩 · Paper 2[8 marks]

On a farm near Maun, a centre-pivot sprinkler has an arm of length

350\text{ m}

that turns about a fixed point. In one hour the arm turns through an angle of

2.5

radians.

(a) Find the distance travelled by the end of the arm in one hour.[2]
(b) Find the area of land watered in one hour. Give your answer in hectares ( $1\text{ ha} = 10\,000\text{ m}^2$ ).[3]
(c) Write down the angle turned in one hour in degrees.[1]
(d) Find the time, in hours and minutes, the arm takes to turn through one complete circle.[2]

Show the full step-by-step solution

Step 1🧠 ThinkPicture it

The arm is a radius that sweeps round like the hand of a clock. In one hour it sweeps out a sector with

r = 350

and

\theta = 2.5

. The end of the arm moves along the arc; the watered ground is the sector.

Part (a)

Step 1✍️ By handArc length

l = 350\times2.5

l = 875\text{ m}

Part (b)

Step 1✍️ By handSector area, then hectares

A = \dfrac{1}{2}\times350^2\times2.5

A = 153\,125\text{ m}^2

153\,125 \div 10\,000 = 15.3\text{ ha}

(3 s.f.)A1

Part (c)

Step 1🖩 On the GDCConvert 2.5 rad to degrees

2.5›×💻 *›180›÷💻 /›π💻 type pi›ctrl+enter💻 Ctrl+Enter

Area, angle in degrees, and time for a full turn

Step 2✍️ By hand

2.5\text{ rad} = 143.2^\circ

(1 d.p.)A1

Part (d)

Step 1🧠 ThinkHow many lots of 2.5 rad make a full turn?

A full turn is

2\pi

radians. Each hour gives 2.5 rad, so we divide.

Step 2✍️ By handDivide, then change the decimal part into minutes

\dfrac{2\pi}{2.5} = 2.51327\ldots\text{ hours}

0.51327\ldots\times60 = 30.8\text{ minutes}

About 2 hours 31 minutesA1

Final answer (a)

875\text{ m}

(b)

15.3\text{ ha}

(c)

143.2^\circ

(d)

2.51

hours ≈ 2 h 31 min

⚠️ Common mistake In (d) writing 2.51 hours as “2 hours 51 minutes”. The decimal part 0.51 of an hour is

0.51\times60\approx31

minutes.

Question 9●●● Exam-level✍️ + 🖩 · Paper 2[7 marks]

A pendulum of length

80\text{ cm}

swings from point P to point Q. The string turns through an angle of

0.35

radians.

(a) Find the length of the arc PQ travelled by the end of the pendulum.[2]
(b) Find the straight-line distance PQ.[3]
(c) Hence find the percentage by which the arc is longer than the straight line PQ.[2]

Show the full step-by-step solution

Step 1🧠 ThinkArc versus chord

The bob moves along a curved path (an arc). The straight line PQ is a chord. The arc uses

l = r\theta

; the chord is the third side of a triangle with two sides of 80 cm and angle 0.35 rad between them — that needs the cosine rule.

Part (a)

Step 1✍️ By hand

l = 80\times0.35

l = 28\text{ cm}

Part (b)

Step 1✍️ By handCosine rule

PQ^2 = 80^2 + 80^2 - 2\times80\times80\cos(0.35)

M1 A1

Step 2🖩 On the GDCEvaluate in RAD mode

√💻 type sqrt(›80^2+80^2−2×80×80×cos(0.35)›enter💻 Enter

📍 Where to find it on the TI-Nspire

⌨️ sin, cos, tan: click trig on the keypad, or type sin(

PQ = 27.857… cm; then the percentage for (c)

Step 3✍️ By hand

PQ = 27.857\ldots\approx 27.9\text{ cm}

Part (c)

Step 1✍️ By handPercentage difference — divide by the chord

\dfrac{28 - 27.857\ldots}{27.857\ldots}\times100\%

= 0.512\%

(3 s.f.)A1

💡 Remember For small angles the arc and the chord are almost the same length. That is why the answer is tiny — a good sense-check.

Final answer (a)

28\text{ cm}

(b)

27.9\text{ cm}

(c)

0.512\%

Question 10●●● Exam-level✍️ By hand · Paper 1[6 marks]

The minute hand of a clock in Francistown is

12\text{ cm}

long. Consider the time from 10:00 to 10:25.

(a) Show that the minute hand turns through $\dfrac{5\pi}{6}$ radians.[2]
(b) Find the exact distance travelled by the tip of the minute hand.[2]
(c) Find the area swept by the minute hand. Give your answer exactly and to 3 significant figures.[2]

Show the full step-by-step solution

Step 1🧠 ThinkTurn minutes into a fraction of a full turn

The minute hand goes all the way round (

2\pi

radians) in 60 minutes. In 25 minutes it goes

\dfrac{25}{60}

of the way round.

Part (a)

Step 1✍️ By hand

\theta = \dfrac{25}{60}\times2\pi

= \dfrac{50\pi}{60} = \dfrac{5\pi}{6}

A1 AG

Part (b)

Step 1✍️ By hand

l = 12\times\dfrac{5\pi}{6}

l = 10\pi\text{ cm}

(\approx 31.4\text{ cm})

Part (c)

Step 1✍️ By hand

A = \dfrac{1}{2}\times12^2\times\dfrac{5\pi}{6} = 72\times\dfrac{5\pi}{6}

A = 60\pi \approx 188\text{ cm}^2

💡 “Show that” You must show the working that leads to

\dfrac{5\pi}{6}

. Simply writing

\dfrac{5\pi}{6}

earns nothing because it is given.

Final answer (a) shown (b)

10\pi\approx31.4\text{ cm}

(c)

60\pi\approx188\text{ cm}^2

Question 11●●● Exam-level✍️ + 🖩 · Paper 2[7 marks]

Onalenna designs a sector-shaped flower bed of radius

r

metres and angle

\theta

radians. The area of the bed must be

48\text{ m}^2

and the edging around the whole bed is

28\text{ m}

long.

(a) Write down two equations in $r$ and $\theta$ .[2]
(b) Show that $r^2 - 14r + 48 = 0$ .[2]
(c) Hence find the two possible designs, giving $r$ and $\theta$ for each.[3]

Show the full step-by-step solution

Step 1🧠 ThinkTwo facts → two equations

“Area 48” uses

A = \dfrac{1}{2}r^2\theta

. “Edging 28 m” is the perimeter: two radii plus the arc,

2r + r\theta

Part (a)

Step 1✍️ By hand

\dfrac{1}{2}r^2\theta = 48

2r + r\theta = 28

Part (b)

Step 1✍️ By handGet rθ from the perimeter and substitute

Notice

\dfrac{1}{2}r^2\theta = \dfrac{1}{2}r\times(r\theta)

. From the perimeter,

r\theta = 28 - 2r

\dfrac{1}{2}r(28 - 2r) = 48

14r - r^2 = 48

r^2 - 14r + 48 = 0

A1 AG

Part (c)

Step 1🖩 On the GDCSolve the quadratic

menu💻 click menu on keypad›3: Algebra›3: Polynomial Tools›1: Find Roots of Polynomial…

Choose degree 2, Real roots, then enter

a_2 = 1

a_1 = -14

a_0 = 48

. (Or type polyRoots({1,−14,48}).)

r = 6 or r = 8, then θ = (28 − 2r) ÷ r

Step 2✍️ By handFind θ for each radius

r = 6

\theta = \dfrac{28-12}{6} = \dfrac{8}{3}\approx2.67

radA1

r = 8

\theta = \dfrac{28-16}{8} = 1.5

radA1

✅ Check it

r = 8, \theta = 1.5

: area

= \dfrac{1}{2}\times64\times1.5 = 48

✔, edging

= 16 + 12 = 28

✔. Both angles are less than

2\pi

, so both designs are possible.

Final answer (a)

\dfrac{1}{2}r^2\theta = 48,\; 2r + r\theta = 28

(b) shown (c)

r = 6\text{ m}, \theta = \dfrac{8}{3}\approx2.67

rad, or

r = 8\text{ m}, \theta = 1.5

rad

🆘 Stuck on a question? Open the notes, GDC toolkit and common mistakes

💻 Using the TI-Nspire CX Student Software on a laptop

The screens in these notes are drawn exactly as the Student Software shows them. The keypad on the left of the software works like the handheld — click its keys with the mouse.
menu: click menu on the on-screen keypad (or Documents Toolbox ▸ 🔧). The menu numbers and letters are the same as in these notes.
enter = Enter · tab = Tab (next box) · esc = Esc · del = Backspace.
Powers: type ^ (Shift+6), the power, then press → to come back down before typing more. x² = ^2 then →.
Fraction: type / · √: type sqrt( · π: type pi · eˣ: type exp( · store: type :=. For EE (×10ⁿ) and i, click them on the keypad.
New page: Ctrl+I (or the Insert menu) · change page: Ctrl+← / Ctrl+→ · graph entry line: Ctrl+G · var: Ctrl+L.
Radian / Degree: click Settings in the status bar at the bottom of the window. The mode (RAD or DEG) shows in the top bar.
IB exam mode (Press-to-Test): everything in these notes works on the non-CAS TI-Nspire with the IB exam settings. In exam mode there are no dot-product, cross-product, unit-vector or eigenvalue keys and no Diff Eq graphs — the notes show the allowed way (type the components, use polyRoots on the characteristic equation, do Euler steps on a Calculator page).
If you type nDeriv(, the software shows it as centralDiff( — it is the same command.

🎯 What you must know

One radian is the angle at the centre of a circle when the arc length is equal to the radius. It is about $57.3^\circ$ .
A full turn is $360^\circ = 2\pi$ radians, so $180^\circ = \pi$ radians. This one fact does every conversion.
Degrees → radians: multiply by $\dfrac{\pi}{180}$ . Radians → degrees: multiply by $\dfrac{180}{\pi}$ .
Useful exact values: $30^\circ = \dfrac{\pi}{6}$ , $45^\circ = \dfrac{\pi}{4}$ , $60^\circ = \dfrac{\pi}{3}$ , $90^\circ = \dfrac{\pi}{2}$ , $270^\circ = \dfrac{3\pi}{2}$ .
With $\theta$ in radians: arc length $l = r\theta$ and sector area $A = \dfrac{1}{2}r^2\theta$ . These two formulas only work in radians.
Perimeter of a sector $= 2r + r\theta$ (two radii plus the arc). Area of a segment $= \dfrac{1}{2}r^2\theta - \dfrac{1}{2}r^2\sin\theta$ .
If a question has no degree sign (°), the angle is in radians — set the GDC to RAD mode before using sin, cos or tan.

Length of an arc (θ in radians)

l = r\theta

📘 In the formula booklet

Area of a sector (θ in radians)

A = \dfrac{1}{2}r^2\theta

📘 In the formula booklet

Conversion

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

🧠 Learn this one

Area of a triangle (for segments)

A = \dfrac{1}{2}ab\sin C

📘 In the formula booklet

🖩 Your GDC toolkit for this subtopic

Follow the steps on your own TI-Nspire CX II. The screens show what you should see.

Set the whole document to radians: Settings → Document Settings → Angle: Radian

Press doc.
Choose Settings, then Document Settings.
Press tab until Angle is highlighted, press click and choose Radian.
Press tab to OK and press enter. The top of the screen now shows RAD.

Set Angle to Radian

Get a decimal answer instead of an exact one (π stays as a number)

ctrl+enter💻 Ctrl+Enter

Press ctrl then enter to get the answer as a decimal.

40° = 0.698132… rad; also a check of (a):

\dfrac{5\pi}{6}

= 2.61799…

Worked example: Question 1 — Let the calculator do the decimal

The π key is next to the letters. Pressing ctrl+enter (instead of enter) forces a decimal answer.

Degrees → radians on a Calculator page

40›×💻 *›π💻 type pi›÷💻 /›180›ctrl+enter💻 Ctrl+Enter

Type 40.
Press ×.
Press π.
Press ÷.
Type 180.
Press ctrl then enter to get the answer as a decimal.

40° = 0.698132… rad; also a check of (a):

\dfrac{5\pi}{6}

= 2.61799…

Worked example: Question 1 — Let the calculator do the decimal

The π key is next to the letters. Pressing ctrl+enter (instead of enter) forces a decimal answer.

Radians → degrees

2.4›×💻 *›180›÷💻 /›π💻 type pi›ctrl+enter💻 Ctrl+Enter

Type 2.4.
Press ×.
Type 180.
Press ÷.
Press π.
Press ctrl then enter to get the answer as a decimal.

40° = 0.698132… rad; also a check of (a):

\dfrac{5\pi}{6}

= 2.61799…

Worked example: Question 1 — Let the calculator do the decimal

The π key is next to the letters. Pressing ctrl+enter (instead of enter) forces a decimal answer.

⚠️ Where students lose marks

Using $l = r\theta$ or $A = \dfrac{1}{2}r^2\theta$ with $\theta$ in degrees. Convert to radians first.
Leaving the GDC in DEG mode for a radian question: $\cos(1.5)$ becomes $\cos(1.5^\circ)$ and every answer is wrong.
Forgetting the two radii when asked for the perimeter of a sector — the arc alone is not the perimeter.
Rounding $\theta$ early (e.g. to 1.67) and then using the rounded value. Keep the full value in the GDC until the end.

IB DP Mathematics: Applications and Interpretation (AI) — SL and HL HL · AI 3.7 — Radian measure

IB DP Mathematics: Applications and Interpretation (AI) — SL and HL

3.7 Radian measure

✏️ Practice questions

🎯 What you must know

🖩 Your GDC toolkit for this subtopic

⚠️ Where students lose marks