What are the different types of angle measures in Geometry?

In Geometry, angles are typically measured in degrees. The main types of angle measures include acute angles (less than 90 degrees), right angles (exactly 90 degrees), obtuse angles (greater than 90 degrees but less than 180 degrees), straight angles (exactly 180 degrees), and reflex angles (greater than 180 degrees but less than 360 degrees). Understanding these types is fundamental in IB MYP Mathematics.

How do you convert angle measures from degrees to radians?

To convert an angle from degrees to radians, multiply the degree measure by π/180. This conversion is crucial in trigonometry, especially in the IB MYP curriculum, where understanding both units is important for solving problems involving circular motion and periodic functions.

Why is understanding angle measures important in the IB MYP Mathematics curriculum?

Understanding angle measures is essential in the IB MYP Mathematics curriculum because it forms the basis for more complex topics in Geometry and Trigonometry. It helps students solve problems involving shapes, understand the properties of polygons, and apply trigonometric ratios in various contexts, which are key skills in both academic and real-world scenarios.

How can I calculate the measure of an angle in a triangle?

To calculate the measure of an angle in a triangle, you can use the fact that the sum of all angles in a triangle is 180 degrees. If you know the measures of two angles, subtract their sum from 180 degrees to find the third angle. This principle is a fundamental concept in Geometry and is widely used in the IB MYP Mathematics curriculum.

What tools can be used to measure angles accurately?

Common tools for measuring angles include a protractor, which is used for direct measurement, and a compass, which can help in constructing angles. In advanced applications, digital tools and software can also be used for precise angle measurement. These tools are often utilized in IB MYP Mathematics to enhance students' understanding of Geometry and Trigonometry.

Why is "Computing $\sin 30°$ on a calculator set to RADIANS." a common mistake in Angle Measures?

Why it happens: Calculator in wrong mode. How to avoid it: Always check the calculator mode. 30° in degree mode = 0.5; in radian mode it's (30\,rad) -0.988.

Why is "Calling the hypotenuse 'opposite' when it's not opposite the angle of interest." a common mistake in Angle Measures?

Why it happens: Confusion about side names. How to avoid it: Hypotenuse = LONGEST side = opposite the RIGHT angle. Opposite = opposite the angle YOU'RE WORKING WITH (). Adjacent = the third side.

IB MYP Mathematics (MYP - M)

Angle Measures

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Short Notes - Angle Measures

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Detailed Study Notes

Detailed notes on Geometry and Trigonometry for IB MYP Mathematics, covering key concepts, explanations, examples, and exam-focused revision points.

Angle Measures — IB MYP Mathematics (Extended): degrees, radians and the basic trig ratios

Angles can be measured in degrees or radians. This MYP Mathematics Extended note covers both systems, conversions, and SOHCAHTOA for right-angled triangles.

At a glance

DEGREES: full turn = $360°$ . Right angle = $90°$ .
RADIANS: full turn = $2\pi$ rad. Right angle = $\pi/2$ .
Conversion: $1\,\text{rad} = \dfrac{180}{\pi}° \approx 57.3°$ .
Multiply degrees by $\dfrac{\pi}{180}$ to convert to radians.
SOHCAHTOA for right triangles: $\sin = \tfrac{O}{H}$ , $\cos = \tfrac{A}{H}$ , $\tan = \tfrac{O}{A}$ .

What you’ll learn

Mapped to the IB MYP Mathematics subject guide (2026 onwards).

MYP Mathematics A — Convert between degrees and radians.
MYP Mathematics A — Apply SOHCAHTOA to right triangles.
MYP Mathematics C — Use angle notation correctly.

Degrees and radians

Two units; same angles.

Degrees. A full turn = $360°$ . Each degree splits into 60 minutes ( $60'$ ) and each minute into 60 seconds ( $60''$ ), but at MYP level we usually just use decimal degrees.

Radians. A full turn = $2\pi$ rad (about 6.28).

A radian is the angle at the centre of a circle subtended by an arc EQUAL IN LENGTH to the radius. So $1\,\text{rad}$ corresponds to an arc of length $r$ .

Degrees	Radians
$0°$	$0$
$30°$	$\pi/6$
$45°$	$\pi/4$
$60°$	$\pi/3$
$90°$	$\pi/2$
$180°$	$\pi$
$270°$	$3\pi/2$
$360°$	$2\pi$

Conversions: $\text{deg} \to \text{rad}: \text{multiply by } \frac{\pi}{180}.$ $\text{rad} \to \text{deg}: \text{multiply by } \frac{180}{\pi}.$

Worked examples:

$75°$ to radians: $75 \times \dfrac{\pi}{180} = \dfrac{75\pi}{180} = \dfrac{5\pi}{12}$ .
$\dfrac{\pi}{3}$ to degrees: $\dfrac{\pi}{3} \times \dfrac{180}{\pi} = 60°$ .

Radians appear at Extended Level because they're cleaner for calculus and circular motion. Degrees stay for everyday work.

Full turn: $360°$ or $2\pi$ rad.
Convert: $\times \dfrac{\pi}{180}$ for deg→rad, $\times \dfrac{180}{\pi}$ for rad→deg.
Common angles: $30°, 45°, 60°, 90°$ in both units.

SOHCAHTOA — basic trig ratios

For RIGHT-angled triangles only.

For a right-angled triangle with one of the non-right angles labelled $\theta$ , the three sides are:

Hypotenuse (H): the longest side, opposite the right angle.
Opposite (O): opposite to $\theta$ .
Adjacent (A): next to $\theta$ (not the hypotenuse).

The three trig ratios: $\sin \theta = \frac{\text{O}}{\text{H}}, \quad \cos \theta = \frac{\text{A}}{\text{H}}, \quad \tan \theta = \frac{\text{O}}{\text{A}}.$

Mnemonic: SOHCAHTOA — Sine = Opp/Hyp, Cosine = Adj/Hyp, Tangent = Opp/Adj.

In any right-angled triangle, label the sides relative to the angle $\theta$ you're working with.

Worked example. In a right triangle, the angle is $30°$ and the hypotenuse is 10 cm. Find the opposite side.

$\sin 30° = \dfrac{O}{10}$ .
$O = 10 \times \sin 30° = 10 \times 0.5 = 5$ cm.

Worked example. A ladder of length 4 m leans against a wall. The base is 1.5 m from the wall. Find the angle between the ladder and the ground.

Adjacent = 1.5 m, Hypotenuse = 4 m.
$\cos \theta = \dfrac{1.5}{4} = 0.375$ .
$\theta = \cos^{-1}(0.375) \approx 68.0°$ .

SOHCAHTOA: $\sin = O/H$ , $\cos = A/H$ , $\tan = O/A$ .
ONLY for right triangles.
Hypotenuse = longest side, opposite right angle.
Use inverse trig ( $\sin^{-1}$ , etc.) to find an angle.

Quick recap

Degrees: full turn $360°$ . Radians: full turn $2\pi$ .
Convert: $\times \pi/180$ or $\times 180/\pi$ .
SOHCAHTOA only for right triangles.
$\sin = O/H$ , $\cos = A/H$ , $\tan = O/A$ .

Memorise this

Verbatim phrases and definitions MYP criterion-A markschemes credit.

$\pi$ rad = $180°$
$\pi/6 = 30°$ , $\pi/4 = 45°$ , $\pi/3 = 60°$ , $\pi/2 = 90°$
SOH-CAH-TOA
Hypotenuse = side opposite right angle

How it’s examined

Criterion A: degree-radian conversions; SOHCAHTOA. Criterion C: clear right-triangle diagrams.

Sources: IB MYP Mathematics subject guide (IBO, official). Last reviewed 2026-05-25.

Take this whole topic with you

Step-by-step worked examples — Angle Measures

Step-by-step solutions to past-paper-style questions on angle measures, written exactly the way a tutor would explain them at the board.

1Convert degrees to radians
Getting started• conversion
Question
Convert $135°$ to radians.
Step-by-step solution
1. Step 1
  Multiply by $\dfrac{\pi}{180}$ : $135 \times \dfrac{\pi}{180}$ .
2. Step 2
  $= \dfrac{135\pi}{180} = \dfrac{3\pi}{4}$ (simplified).
Answer
$\dfrac{3\pi}{4}$ rad.
2Find a side using sine
Building confidence• SOHCAHTOA
Question
In a right triangle, the angle is $40°$ and the hypotenuse is 12 cm. Find the side opposite the $40°$ angle.
Step-by-step solution
1. Step 1
  $\sin 40° = \dfrac{O}{12}$ .
2. Step 2
  $O = 12 \sin 40° \approx 12 \times 0.643 \approx 7.71$ cm.
Answer
$\approx 7.71\,\text{cm}$
3Find the angle using inverse trig
Building confidence• inverse
Question
In a right triangle, opposite = 5, adjacent = 12. Find the angle.
Step-by-step solution
1. Step 1
  $\tan \theta = \dfrac{5}{12} \approx 0.4167$ .
2. Step 2
  $\theta = \tan^{-1}(0.4167) \approx 22.6°$ .
Answer
$\theta \approx 22.6°$
4Ladder problem
Stretch• application
Question
A ladder 6 m long leans against a vertical wall, making a $70°$ angle with the ground. How high up the wall does it reach?
Step-by-step solution
1. Step 1
  Hypotenuse = 6 m, angle = $70°$ . Wall height = opposite side.
2. Step 2
  $\sin 70° = \dfrac{O}{6}$ .
3. Step 3
  $O = 6 \sin 70° \approx 6 \times 0.940 \approx 5.64$ m.
Answer
$\approx 5.64\,\text{m}$ .

Key Definitions and Keywords — Angle Measures

Definitions to memorise and the exact keywords mark schemes credit for angle measures answers — sharpened from recent examiner reports for the 2026 IB MYP Mathematics (Extended) sitting.

Degree
Examiner keyword
$1/360$ th of a full turn. The most common angle unit in school maths.
Radian
Examiner keyword
The angle subtended by an arc equal in length to the radius. $2\pi$ rad = $360°$ .
SOHCAHTOA
Examiner keyword
Mnemonic for the right-triangle trig ratios: Sin = Opp/Hyp, Cos = Adj/Hyp, Tan = Opp/Adj.

Common Mistakes and Misconceptions — Angle Measures

The traps other students keep falling into on angle measures questions — taken from recent IB MYP Mathematics (Extended) examiner reports and mark schemes — and how to avoid them.

✕Computing $\sin 30°$ on a calculator set to RADIANS.
Why it happens
Calculator in wrong mode.
How to avoid it
Always check the calculator mode. $\sin 30°$ in degree mode = 0.5; in radian mode it's $\sin(30\,\text{rad}) \approx -0.988$ .
✕Calling the hypotenuse 'opposite' when it's not opposite the angle of interest.
Why it happens
Confusion about side names.
How to avoid it
Hypotenuse = LONGEST side = opposite the RIGHT angle. Opposite = opposite the angle YOU'RE WORKING WITH ( $\theta$ ). Adjacent = the third side.

Angle Measures — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Degrees

Radians

0°

0

30°

\pi/6

45°

\pi/4

60°

\pi/3

90°

\pi/2

180°

\pi

270°

3\pi/2

360°

2\pi

For a right-angled triangle with one of the non-right angles labelled $\theta$ , the three sides are:

Hypotenuse (H): the longest side, opposite the right angle.
Opposite (O): opposite to $\theta$ .
Adjacent (A): next to $\theta$ (not the hypotenuse).

The three trig ratios: $\sin \theta = \frac{\text{O}}{\text{H}}, \quad \cos \theta = \frac{\text{A}}{\text{H}}, \quad \tan \theta = \frac{\text{O}}{\text{A}}.$

Mnemonic: SOHCAHTOA — Sine = Opp/Hyp, Cosine = Adj/Hyp, Tangent = Opp/Adj.

In any right-angled triangle, label the sides relative to the angle $\theta$ you're working with.

Worked example. In a right triangle, the angle is $30°$ and the hypotenuse is 10 cm. Find the opposite side.

$\sin 30° = \dfrac{O}{10}$ .
$O = 10 \times \sin 30° = 10 \times 0.5 = 5$ cm.

Worked example. A ladder of length 4 m leans against a wall. The base is 1.5 m from the wall. Find the angle between the ladder and the ground.

Adjacent = 1.5 m, Hypotenuse = 4 m.
$\cos \theta = \dfrac{1.5}{4} = 0.375$ .
$\theta = \cos^{-1}(0.375) \approx 68.0°$ .

Angle Measures

Short Notes - Angle Measures

Detailed Study Notes

At a glance

What you’ll learn

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

1Convert degrees to radians

2Find a side using sine

3Find the angle using inverse trig

4Ladder problem

Degree

Radian

SOHCAHTOA

✕Computing sin⁡30°\sin 30°sin30° on a calculator set to RADIANS.

✕Calling the hypotenuse 'opposite' when it's not opposite the angle of interest.

Angle Measures — frequently asked questions

Angle Measures

Short Notes - Angle Measures

Detailed Study Notes

At a glance

What you’ll learn

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

1Convert degrees to radians

2Find a side using sine

3Find the angle using inverse trig

4Ladder problem

Degree

Radian

SOHCAHTOA

✕Computing sin⁡30°\sin 30°sin30° on a calculator set to RADIANS.

✕Calling the hypotenuse 'opposite' when it's not opposite the angle of interest.

Angle Measures — frequently asked questions

✕Computing $\sin 30°$ on a calculator set to RADIANS.

✕Computing $\sin 30°$ on a calculator set to RADIANS.