Measuring Angles

An angle is formed by two rays that start at the same point. The point is the vertex of the angle and the rays are its sides. The measure of an angle tells how far one side has to be turned around the vertex to reach the other side.

Angles are measured in degrees or in radians. This lesson explains both units and how to convert between them, and shows what angles greater than $360^\circ$ and negative angles mean. You can check your answers with the converter at the top of the page.

On this page: Degrees · Minutes and seconds · Angles greater than 360° · Positive and negative angles · Radians · Degrees and radians · Arc length and sector area · Angles outside 0 to 2π · Common mistakes · Practice

Degrees

One full turn is divided into 360 equal parts. Each part is one degree, written $1^\circ.$ So $1^\circ$ is $\frac{1}{360}$ of a full turn, and a full turn is $360^\circ.$

A quarter turn is $90^\circ,$ a half turn is $180^\circ$ and three quarters of a turn is $270^\circ.$ By its measure, an angle is:

• acute if it is between $0^\circ$ and $90^\circ;$
• right if it is exactly $90^\circ;$
• obtuse if it is between $90^\circ$ and $180^\circ;$
• straight if it is exactly $180^\circ;$
• reflex if it is between $180^\circ$ and $360^\circ.$

More about the kinds of angles: Classifying angles.

Measuring an angle with a protractor

1. Put the center of the protractor on the vertex of the angle.
2. Turn the protractor so that its zero line lies along one side of the angle.
3. Read the number where the other side crosses the scale. A protractor has two scales: use the one that starts from 0 on the first side.

In the picture the angle is $50^\circ,$ not $130^\circ\text{:}$ the first side points at the 0 of the outer scale, so we read the outer scale. A quick check helps too: the angle is acute, so its measure must be less than $90^\circ.$

Minutes and Seconds

Parts of a degree are written as decimals, for example $37.5^\circ,$ or in smaller units called minutes and seconds.

One degree is 60 minutes and one minute is 60 seconds: ${1^\circ=60'},$ ${1'=60''},$ so ${1^\circ=3600''}.$

Example 1. Write $37.42^\circ$ in degrees, minutes and seconds.

Solution. The whole part is $37^\circ.$ The rest is ${0.42^\circ}={0.42\cdot60'}={25.2'},$ that is, $25'$ and $0.2'$ more. Then ${0.2'}={0.2\cdot60''}={12''}.$ So ${37.42^\circ=37^\circ25'12''}.$

Example 2. Write $12^\circ45'36''$ as a decimal number of degrees.

Solution. ${12^\circ45'36''}={12^\circ+\frac{45^\circ}{60}+\frac{36^\circ}{3600}}$ $={12^\circ+0.75^\circ+0.01^\circ}={12.76^\circ}.$

Example 3. Find ${48^\circ35'+27^\circ50'}$ and ${90^\circ-34^\circ27'}.$

Solution. We add degrees to degrees and minutes to minutes: ${48^\circ35'+27^\circ50'}={75^\circ85'}.$ Since ${85'=1^\circ25'},$ the sum is $76^\circ25'.$ To subtract, we first write $90^\circ$ as $89^\circ60'$ and get ${89^\circ60'-34^\circ27'}={55^\circ33'}.$

Angles Greater Than 360°

An angle can also describe a rotation. A wheel, the hand of a clock or a figure skater can turn through more than one full turn, and then the angle of rotation is greater than $360^\circ.$ For example, a wheel that makes two full turns rotates through ${2\cdot360^\circ=720^\circ}.$

Example 4. How many turns does an object make when it rotates through a) $380^\circ;$ b) $770^\circ;$ c) $1000^\circ?$

Solution.

a) ${380^\circ=1\cdot360^\circ+20^\circ}\text{:}$ one full turn and $20^\circ$ more. Since $20^\circ$ is ${\frac{20}{360}=\frac{1}{18}}$ of a turn, the object makes $1\frac{1}{18}$ turns.

b) ${770^\circ=2\cdot360^\circ+50^\circ}\text{:}$ two full turns and $50^\circ$ more, as in the picture above, that is ${2\frac{50}{360}=2\frac{5}{36}}$ turns.

c) ${{1000^\circ=2\cdot360^\circ+280^\circ}\text{:}}$ two full turns and $280^\circ$ more, that is ${2\frac{280}{360}=2\frac{7}{9}}$ turns.

Positive and Negative Angles

A rotation has a direction. The side where the rotation starts is called the initial side of the angle, and the side where it stops is the terminal side.

A counterclockwise rotation is measured by a positive angle, and a clockwise rotation by a negative angle.

In a coordinate system an angle is usually drawn in standard position: the vertex is at the origin and the initial side lies along the positive $x$-axis. Then the terminal side of $90^\circ$ points up, the terminal side of $180^\circ$ points left and the terminal side of $270^\circ$ points down. The terminal side of $-90^\circ$ points down as well.

Coterminal angles

The angles $270^\circ$ and $-90^\circ$ are different rotations, but they start from the same initial side and stop on the same terminal side. Such angles are called coterminal. A full turn brings the terminal side back to where it was, so:

Adding or subtracting $360^\circ$ any number of times gives a coterminal angle, and every coterminal angle can be found this way: the angles coterminal with $\alpha$ are ${\alpha+k\cdot360^\circ},$ where $k$ is an integer.

• To find a positive angle coterminal with a negative one, add $360^\circ$ until the result is positive.
• To find a negative angle coterminal with a positive one, subtract $360^\circ$ until the result is negative.
• Every angle is coterminal with exactly one angle from $0^\circ$ up to, but not including, $360^\circ.$ For a positive angle it is the remainder after dividing by $360^\circ.$

Example 5. Find the smallest positive angle coterminal with a) $-35^\circ;$ b) $-60^\circ;$ c) $-180^\circ;$ d) $-670^\circ.$

Solution. We add $360^\circ$ until the angle becomes positive.

a) ${-35^\circ+360^\circ=325^\circ}$

b) ${-60^\circ+360^\circ=300^\circ}$

c) ${-180^\circ+360^\circ=180^\circ}$

d) ${-670^\circ+360^\circ=-310^\circ}$ is still negative, so we add $360^\circ$ once more: ${-310^\circ+360^\circ=50^\circ}.$ In one step, ${-670^\circ+2\cdot360^\circ=50^\circ}.$

Example 6. Find the negative angle closest to zero that is coterminal with a) $80^\circ;$ b) $167^\circ;$ c) $330^\circ;$ d) $1300^\circ.$

Solution. We subtract $360^\circ$ until the angle becomes negative.

a) ${80^\circ-360^\circ=-280^\circ}$

b) ${167^\circ-360^\circ=-193^\circ}$

c) ${330^\circ-360^\circ=-30^\circ}$

d) ${1300^\circ-360^\circ=940^\circ},$ ${940^\circ-360^\circ=580^\circ},$ ${580^\circ-360^\circ=220^\circ},$ ${220^\circ-360^\circ=-140^\circ}.$ In one step, ${1300^\circ-4\cdot360^\circ}={-140^\circ}.$ The third subtraction also shows that ${{1300^\circ=3\cdot360^\circ+220^\circ}\text{:}}$ three full turns and $220^\circ$ more.

Coterminal does not mean equal. The angles $1300^\circ$ and $-140^\circ$ are different rotations: more than three and a half turns one way and less than half a turn the other way. They only end in the same position, so it is wrong to write ${1300^\circ=-140^\circ}.$ But the trigonometric functions have the same values for both angles, for example ${\sin1300^\circ=\sin(-140^\circ)}.$

Radians

The degree is a convenient unit, but the number 360 is a matter of choice. The radian comes from the circle itself.

One radian (1 rad) is the central angle that cuts off an arc as long as the radius of the circle. (A central angle has its vertex at the center of the circle.)

In the picture the arc $AB$ is as long as the radius $r,$ so ${\angle AOB=1\text{ rad}}.$ An arc twice as long belongs to an angle of 2 rad, and so on. In general, a central angle that cuts off an arc of length $s$ in a circle of radius $r$ has the radian measure

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

Here $s$ and $r$ must be in the same unit of length. The units then cancel, so a radian measure is just a number.

The whole circle is an arc of length $2\pi r,$ so a full turn is ${\frac{2\pi r}{r}=2\pi}$ radians.

A full turn is $2\pi$ radians and a half turn is $\pi$ radians: ${360^\circ=2\pi\text{ rad}},$ ${180^\circ=\pi\text{ rad}}.$

From this, ${1\text{ rad}=\frac{180^\circ}{\pi}\approx57.3^\circ}$ and ${1^\circ}={\frac{\pi}{180}\text{ rad}}\approx{0.01745\text{ rad}}.$

Angles in radians are usually written as multiples of $\pi,$ because then the most common angles are exact and short: $\frac{\pi}{2}$ instead of $1.570796\ldots$ The word “rad” is usually left out. An angle written without a unit, such as $\frac{\pi}{3}$ or $2,$ is in radians; an angle in degrees always carries the degree sign.

Converting between degrees and radians

It is enough to remember that $\pi$ radians is $180^\circ.$ When an angle is given as a multiple of $\pi,$ simply replace $\pi$ with $180^\circ\text{:}$ for example, ${\frac{\pi}{6}=\frac{180^\circ}{6}=30^\circ}.$

Degrees	Radians	Degrees	Radians
$30^\circ$	$\frac{\pi}{6}$	$210^\circ$	$\frac{7\pi}{6}$
$45^\circ$	$\frac{\pi}{4}$	$225^\circ$	$\frac{5\pi}{4}$
$60^\circ$	$\frac{\pi}{3}$	$240^\circ$	$\frac{4\pi}{3}$
$90^\circ$	$\frac{\pi}{2}$	$270^\circ$	$\frac{3\pi}{2}$
$120^\circ$	$\frac{2\pi}{3}$	$300^\circ$	$\frac{5\pi}{3}$
$135^\circ$	$\frac{3\pi}{4}$	$315^\circ$	$\frac{7\pi}{4}$
$150^\circ$	$\frac{5\pi}{6}$	$330^\circ$	$\frac{11\pi}{6}$
$180^\circ$	$\pi$	$360^\circ$	$2\pi$

Example 7. Convert $240^\circ,$ $45^\circ,$ $270^\circ,$ $750^\circ$ and $390^\circ$ to radians.

Solution. We multiply each angle by $\frac{\pi}{180^\circ}$ and reduce the fraction.

${240^\circ}={240^\circ\cdot\frac{\pi}{180^\circ}}={\frac{4\pi}{3}}$

${45^\circ=45^\circ\cdot\frac{\pi}{180^\circ}=\frac{\pi}{4}}$

${270^\circ}={270^\circ\cdot\frac{\pi}{180^\circ}}={\frac{3\pi}{2}}$

${750^\circ}={750^\circ\cdot\frac{\pi}{180^\circ}}={\frac{25\pi}{6}}$

${390^\circ}={390^\circ\cdot\frac{\pi}{180^\circ}}={\frac{13\pi}{6}}$

Example 8. Convert to degrees: a) $\frac{5\pi}{4};$ b) $3.12\pi;$ c) $2.4\text{ rad}.$

Solution.

a) ${\frac{5\pi}{4}=\frac{5\cdot180^\circ}{4}=225^\circ}$

b) ${3.12\pi}={3.12\cdot180^\circ}={561.6^\circ}$

c) There is no $\pi$ in this angle, so we multiply by $\frac{180^\circ}{\pi}$ and get ${2.4\cdot\frac{180^\circ}{\pi}}={\frac{432^\circ}{\pi}}\approx{137.5^\circ}.$

Arc length and sector area

Radians make the formulas for a circle short. From ${\theta=\frac{s}{r}}$ the length of an arc is the radius times the angle. A sector with central angle $\theta$ is the fraction $\frac{\theta}{2\pi}$ of the circle, so its area is ${\frac{\theta}{2\pi}\cdot\pi r^2=\frac{1}{2}r^2\theta}.$

For a central angle of $n^\circ$ the same formulas become ${s=\frac{\pi rn}{180}}$ and ${A=\frac{\pi r^2n}{360}}.$

Example 9. A circle has a radius of $6\text{ cm}.$ Find the length of the arc and the area of the sector for a central angle of $120^\circ.$

Solution. In radians, ${120^\circ=\frac{2\pi}{3}}.$ Then ${s}={6\cdot\frac{2\pi}{3}}={4\pi}\approx{12.57\text{ cm}}$ and ${A}={\frac{1}{2}\cdot6^2\cdot\frac{2\pi}{3}}={12\pi}\approx{37.70\text{ cm}^2}.$

Negative angles and angles greater than $2\pi$ radians

Everything said about turns and coterminal angles stays true in radians, with $2\pi$ in place of $360^\circ.$ A full turn is $2\pi,$ the angles $\theta$ and ${\theta+2\pi k}$ are coterminal for every integer $k,$ and a rotation through the angle $\theta$ makes $\frac{\theta}{2\pi}$ turns.

Example 10. Find the smallest positive angle coterminal with a) $-\frac{3\pi}{4};$ b) $-\frac{5\pi}{7}.$

Solution. We add $2\pi$ to each angle.

a) ${-\frac{3\pi}{4}+2\pi}={-\frac{3\pi}{4}+\frac{8\pi}{4}}={\frac{5\pi}{4}}$

b) ${-\frac{5\pi}{7}+2\pi}={-\frac{5\pi}{7}+\frac{14\pi}{7}}={\frac{9\pi}{7}}$

Example 11. How many turns does an object make, and in which direction, when it rotates through a) $-10\pi;$ b) $9\pi;$ c) $\frac{7\pi}{2}?$

Solution. We divide each angle by $2\pi.$

a) ${\frac{-10\pi}{2\pi}=-5}\text{:}$ five full turns clockwise.

b) ${\frac{9\pi}{2\pi}=4.5}\text{:}$ four and a half turns counterclockwise, because ${9\pi=4\cdot2\pi+\pi}.$

c) ${\frac{7\pi}{2}\cdot\frac{1}{2\pi}=\frac{7}{4}=1\frac{3}{4}}\text{:}$ one and three-quarter turns counterclockwise, because ${\frac{7\pi}{2}=2\pi+\frac{3\pi}{2}}.$

Example 12. Find the angle $\theta$ with ${0\le\theta\lt2\pi}$ that is coterminal with a) $\frac{25\pi}{6};$ b) $-\frac{8\pi}{3}.$

Solution.

a) ${\frac{25\pi}{6}}={\frac{24\pi}{6}+\frac{\pi}{6}}={2\cdot2\pi+\frac{\pi}{6}},$ so the angle is $\frac{\pi}{6}.$

b) ${-\frac{8\pi}{3}+2\cdot2\pi}={-\frac{8\pi}{3}+\frac{12\pi}{3}}={\frac{4\pi}{3}}.$

Common Mistakes

Mistake	Wrong	Right
Calculator in the wrong mode	${\sin30^\circ\approx-0.988}\text{:}$ the calculator is in RAD mode and reads 30 as 30 radians	Switch to DEG mode for degrees: ${\sin30^\circ=0.5}$
Reading minutes as decimals	${2^\circ30'=2.30^\circ}$	${2^\circ30'=2.5^\circ},$ because $30'$ is half a degree
Multiplying by the wrong fraction	${60^\circ=60\cdot\frac{180}{\pi}}$	${60^\circ=60^\circ\cdot\frac{\pi}{180^\circ}=\frac{\pi}{3}}\text{:}$ the degrees must cancel
Degrees in the arc length formula	${r=6},$ angle $120^\circ\text{:}$ ${s=6\cdot120=720}$	Convert to radians first: ${s=6\cdot\frac{2\pi}{3}=4\pi}$
Mixing up the directions	A clockwise rotation is positive	Counterclockwise is positive, clockwise is negative
An equals sign between coterminal angles	${-90^\circ=270^\circ}$	$-90^\circ$ and $270^\circ$ are coterminal: they share the terminal side, but they are different angles
Reading the wrong scale of the protractor	An acute angle measured as $130^\circ$	Read the scale that starts from 0 on the first side: $50^\circ$

Practice Problems

1. Convert to radians: $15^\circ,$ $72^\circ,$ $150^\circ,$ $315^\circ,$ $540^\circ.$

Answer

$\frac{\pi}{12},$ $\frac{2\pi}{5},$ $\frac{5\pi}{6},$ $\frac{7\pi}{4},$ $3\pi.$

2. Convert to degrees: $\frac{\pi}{5},$ $\frac{7\pi}{6},$ $\frac{11\pi}{12},$ $3\text{ rad}.$

Answer

$36^\circ,$ $210^\circ,$ $165^\circ,$ ${\frac{540^\circ}{\pi}\approx171.9^\circ}.$

3. a) Write $25.3^\circ$ in degrees and minutes. b) Write $48^\circ21'36''$ as a decimal number of degrees.

Answer

a) ${0.3\cdot60'=18'},$ so ${25.3^\circ=25^\circ18'}.$

b) ${48^\circ+\frac{21^\circ}{60}+\frac{36^\circ}{3600}}={48^\circ+0.35^\circ+0.01^\circ}={48.36^\circ}.$

4. Find ${37^\circ48'+25^\circ35'}.$ Then find the angle that must be added to $37^\circ48'$ to get a right angle.

Answer

${37^\circ48'+25^\circ35'}={62^\circ83'}={63^\circ23'}.$

${90^\circ-37^\circ48'}={89^\circ60'-37^\circ48'}={52^\circ12'}.$

5. A wheel rotates through $1500^\circ.$ How many full turns does it make, and what angle is left?

Answer

${{1500^\circ=4\cdot360^\circ+60^\circ}\text{:}}$ four full turns and $60^\circ$ more, that is $4\frac{1}{6}$ turns.

6. Find the angle $\beta$ with ${0^\circ\le\beta\lt360^\circ}$ that is coterminal with a) $-1000^\circ;$ b) $2024^\circ.$

Answer

a) ${-1000^\circ+3\cdot360^\circ=80^\circ}.$

b) ${2024^\circ-5\cdot360^\circ=224^\circ}.$

7. Find the angle $\theta$ with ${0\le\theta\lt2\pi}$ that is coterminal with a) $\frac{17\pi}{4};$ b) $-\frac{11\pi}{3}.$

Answer

a) ${\frac{17\pi}{4}-2\cdot2\pi}={\frac{17\pi}{4}-\frac{16\pi}{4}}={\frac{\pi}{4}}.$

b) ${-\frac{11\pi}{3}+2\cdot2\pi}={-\frac{11\pi}{3}+\frac{12\pi}{3}}={\frac{\pi}{3}}.$

8. Through what angle does the minute hand of a clock turn in 25 minutes? Give the answer with its sign, in degrees and in radians.

Answer

The minute hand makes a full turn in 60 minutes, so in 25 minutes it turns through ${\frac{25}{60}\cdot360^\circ=150^\circ}.$ It moves clockwise, so the angle is ${-150^\circ=-\frac{5\pi}{6}}.$

9. A wheel makes $2.5$ turns counterclockwise. Find the angle of rotation in degrees and in radians.

Answer

${2.5\cdot360^\circ=900^\circ},$ which is ${2.5\cdot2\pi=5\pi}$ radians.

10. The Earth makes one turn around its axis in about 24 hours. Through what angle does it turn in one hour? Give the answer in degrees and in radians.

Answer

${\frac{360^\circ}{24}=15^\circ},$ which is $\frac{\pi}{12}$ radians.

11. A circle has a radius of $10\text{ cm}.$ Find the arc length and the sector area for a central angle of $54^\circ.$

Answer

In radians, ${54^\circ=\frac{54\pi}{180}=\frac{3\pi}{10}}.$ Then ${s}={10\cdot\frac{3\pi}{10}}={3\pi}\approx{9.42\text{ cm}}$ and ${A}={\frac{1}{2}\cdot10^2\cdot\frac{3\pi}{10}}={15\pi}\approx{47.12\text{ cm}^2}.$

12. An arc of a circle of radius $8\text{ cm}$ is $20\text{ cm}$ long. Find its central angle in radians and in degrees.

Answer

${\theta=\frac{s}{r}=\frac{20}{8}=2.5\text{ rad}},$ which is ${2.5\cdot\frac{180^\circ}{\pi}\approx143.2^\circ}.$

See also: Classifying angles · Angle pairs and parallel lines · Circle · Trigonometry