OMPT drill — Trigonometry

OMPT angles and radians practice: lesson and 10 problems

Q: Are radians on the OMPT-A?

No. Radians belong to the trigonometry chapter, which OMPT-A does not have; its eight chapters run from numbers to differentiation. The same goes for OMPT-C, OMPT-E and OMPT-F. Only OMPT-B, OMPT-D and OMPT-G test angles in radians, and on those three the whole trigonometry chapter is written in radians rather than degrees.

Q: How do I convert degrees to radians for the OMPT?

Use the one fact 180 degrees equals pi radians. Degrees to radians: multiply by pi over 180, so 150 degrees is 5pi/6. Radians to degrees: multiply by 180 over pi, so 3pi/4 is 135 degrees. Simplify the fraction and leave pi in the answer. If the pi does not cancel when you are converting to degrees, the factor was upside down.

Q: Do I need a calculator for radians on the OMPT?

No, and none is provided on the maths variants. Exact answers such as 5pi/6 need no calculator, and the decimal questions are built around landmarks: pi over 2 is about 1.57, pi about 3.14, three pi over 2 about 4.71. Knowing those three numbers is enough to place any angle and to round an arc length sensibly by hand.

Q: How many radian questions are on the OMPT-B?

Directly, my estimate is one or two of the 64 questions: a conversion, a coterminal angle, or an arc length. Indirectly the topic is everywhere in the trigonometry chapter, because every exact value, identity and equation question on OMPT-B states its angles in radians. Treat the chapter as radian fluency training rather than as a small stand-alone topic.

Q: What is the difference between a coterminal angle and a reference angle on the OMPT?

A coterminal angle lands on the same point of the circle and differs from the original by whole turns of 2pi; the OMPT usually wants the one in the interval from 0 to 2pi. A reference angle is the acute angle between the terminal side and the x-axis, and it is what you look up in the exact-value table. 5pi/4 has coterminal partner 13pi/4 and reference angle pi/4.

Q: Does the arc length formula work with degrees on the OMPT?

Not as written. Arc length s equals r times theta, and sector area equals half r squared theta, only when theta is in radians. Given a degree measure, convert it first or use the proportion theta over 360 times the circumference. Feeding 60 degrees straight into r times theta gives an arc many times longer than the whole circle, which is the most common slip on this topic.

Radians are the native language of the OMPT trig chapters. Converting fluently and knowing the unit-circle landmarks by heart is non-negotiable. Keep a unit circle sketch next to you for the first pass, then redo the set without it.

Tested inOMPT-B OMPT-D OMPT-G

OMPT angles and radians: the lesson

What the OMPT tests in angles and radians

A radian measures an angle by arc length: one radian is the angle you get when you walk a distance of one radius along the rim of a circle. The full rim is $2 π$ radii long, so a full turn is $2 π$ radians, a half turn is $π$ , a right angle is $\frac{π}{2}$ . The OMPT trigonometry chapter (OMPT-B, OMPT-D and OMPT-G) speaks radians almost exclusively, and it tests four sub-skills: converting between degrees and radians in both directions, placing an angle like $\frac{7 π}{6}$ in the right quadrant without converting, reducing oversized or negative angles to a coterminal angle in $[0, 2 π)$ , and using the two radian-only formulas, arc length $s = r θ$ and sector area $A = \frac{1}{2} r^{2} θ$ .

The conversion questions are light and quick. The heavy part is the radian fluency every later trig question assumes: a student who has to translate $\frac{5 π}{6}$ into 150° before knowing where it sits pays a speed tax on three whole topics. Questions are phrased plainly: "convert to radians", "give the angle in degrees", "which angle is coterminal with", "find the arc length", and on OMPT-D and OMPT-G the arc-length or angular-speed step is usually buried inside a longer geometry or mechanics problem.

Notation and answer format the OMPT expects

Radian answers are exact multiples of $π$ , typed as a fraction: 5pi/6 or 5/6 pi, never 2.618. The degree sign is written out or given in the question; if the question says "in degrees", type the bare number.
A decimal is expected only when the question says "round to". Then the angle in radians is a plain number such as 2.5, and you must know that 2.5 means 2.5 radians, not degrees.
Coterminal answers are normally wanted in $[0, 2 π)$ or $[0°, 360°)$ unless the question names another interval.
Arc length and sector area come out in the units of the radius (metres, or metres squared). The angle must be in radians before it enters either formula.

Worked example 1: degrees to radians

Convert 150° and 240° to radians.
The single fact is $180° = π$ radians, so degrees to radians means multiply by $\frac{π}{180}$ .
$150° \cdot \frac{π}{180} = \frac{150 π}{180} = \frac{5 π}{6}$ , after dividing top and bottom by 30.
$240° \cdot \frac{π}{180} = \frac{240 π}{180} = \frac{4 π}{3}$ , dividing by 60.
Leave both as exact fractions of $π$ . A decimal like 2.618 for the first is the same angle and useless for the unit-circle work ahead. The trap is the flipped factor: $150 \cdot \frac{180}{π}$ gives about 8594, which no sensible question wants.

Worked example 2: radians to degrees

Convert $\frac{3 π}{4}$ and $\frac{7 π}{12}$ to degrees.
Radians to degrees means multiply by $\frac{180}{π}$ : $\frac{3 π}{4} \cdot \frac{180}{π} = \frac{3 \cdot 180}{4} = 135°$ . The $π$ s cancel; if yours did not, the factor was upside down.
Faster with landmarks: $\frac{π}{4}$ is 45°, so three of them is 135°.
For $\frac{7 π}{12}$ , the landmark is $\frac{π}{12} = 15°$ , so seven of them is 105°. The formula agrees: $\frac{7 \cdot 180}{12} = 105$ .
Check the quadrant: 105° is just past 90°, second quadrant, and $\frac{7 π}{12}$ is just past $\frac{6 π}{12} = \frac{π}{2}$ . Consistent. Landmark thinking beats the formula once the landmarks are memorised.

The full lesson builds the landmark families and the ten-second quadrant test, works through coterminal and reference angles, then spends the radian dividend on arc length, sector area and angular speed, ending with a "how many angles" question and a sector word problem of the kind OMPT-D and OMPT-G hide inside longer items. It closes with the edge cases and the slips I mark most often.

How each OMPT variant tests angles and radians

The chapter weights are the official ones from the omptest.org learning-assessment pages. The question counts are my estimates of how much of that chapter this one topic takes up. The 21-question papers (OMPT-D and OMPT-F) ask long multi-part questions, so a topic there usually shows up as a step inside a bigger problem.

OMPT-A
52 questions
Not tested
OMPT-A has no trigonometry chapter; its eight chapters end at differentiation. Spend the time on functions and derivatives instead.
OMPT-B
64 questions
Testedroughly 1 to 2 of 64 questions directly, plus the radian fluency every other trigonometry question assumes
Official chapter: Trigonometry, 15% of the paper.
Short exact answers: convert an angle, name a coterminal angle, place an angle in a quadrant, or compute an arc length or sector area with the angle in radians.
OMPT-C
30 questions
Not tested
OMPT-C covers numbers, algebra, linear equations and geometry only; its geometry works in degrees, so radians never appear. Study the geometry chapter instead.
OMPT-D
21 questions
Tested1 to 2 of the 21 long questions, usually as a step inside a bigger problem
Official chapter: Trigonometry, 15% of the paper.
Radians appear as the opening step of a geometry or calculus item: a sector perimeter or area, an angle recovered from an arc, or an angle reduced to $[0, 2 π)$ before a trig equation is solved.
OMPT-E
40 questions
Not tested
OMPT-E replaces trigonometry with descriptive statistics, combinatorics and probability. Skip this topic.
OMPT-F
21 questions
Not tested
OMPT-F is six chapters of business-oriented algebra and calculus with no trigonometry. Skip this topic.
OMPT-G
26 questions
Testedroughly 1 to 2 of 26 questions, often inside the geometry or mechanics items
Official chapter: Trigonometric functions, 15% of the paper.
Conversions and arc length, and angular speed in the mechanics chapter: revolutions per minute to radians per second, then rim speed from $v = r ω$ .

OMPT angles and radians practice problems

10 original problems in the formats the OMPT uses: typed numeric answers, multiple choice, and spot-the-error. The last few are harder than the exam average on purpose. Solutions and the common mistake sit behind each problem.

Problem 1Numeric answer

A circular sector has radius $9$ and area $\frac{27 π}{4}$ . Find the central angle $θ$ in radians. Round to two decimals.

Show the worked solution

Answer: 0.52

I start from the sector area formula $A = \frac{1}{2} r^{2} θ$ . Plugging in gives $\frac{27 π}{4} = \frac{1}{2} \cdot 81 \cdot θ$ , so $θ = \frac{27 π}{4} \cdot \frac{2}{81} = \frac{π}{6} \approx 0.52$ rad.

Problem 2Multiple choice

Which angle is coterminal with $- \frac{5 π}{3}$ ?

A $\frac{π}{3}$
B $\frac{2 π}{3}$
C $\frac{5 π}{3}$
D $- \frac{π}{3}$

Show the worked solution

Answer: A — $\frac{π}{3}$

Coterminal angles differ by a full turn of $2 π$ . I add one turn: $- \frac{5 π}{3} + 2 π = \frac{π}{3}$ . That lands in the first quadrant, which matches the terminal side of the original angle.

Problem 3Numeric answer

A bicycle wheel of radius $0.35$ m spins at $240$ revolutions per minute. How fast, in m/s, does a point on the rim move? Round to two decimals.

Show the worked solution

Answer: 8.8

First I convert to angular speed: $240$ rev/min is $240 \cdot 2 π$ radians per minute, so $ω = \frac{480 π}{60} = 8 π$ rad/s. Linear speed is $v = r ω = 0.35 \cdot 8 π = 2.8 π \approx 8.80$ m/s.

Problem 4Multiple choice

What is $\frac{11 π}{12}$ in degrees?

A $15 5^{\circ}$
B $16 5^{\circ}$
C $172. 5^{\circ}$
D $19 5^{\circ}$

Show the worked solution

Answer: B — $16 5^{\circ}$

I multiply by $\frac{18 0 ^{\circ}}{π}$ : $\frac{11 π}{12} \cdot \frac{18 0 ^{\circ}}{π} = 11 \cdot 1 5^{\circ} = 16 5^{\circ}$ . Thinking of $\frac{π}{12}$ as $1 5^{\circ}$ makes twelfths quick to convert.

Problem 5Spot the error

A student converts $10 0^{\circ}$ to radians. Step 1: radians and degrees are proportional, with $18 0^{\circ} = π$ rad. Step 2: so $10 0^{\circ} = 100 \cdot \frac{180}{π}$ . Step 3: this gives approximately $5729.6$ rad. Which step contains the error?

Show the worked solution

Answer: Step 2

Step 1 states the correct proportion. The slip is in Step 2: to go from degrees to radians you multiply by $\frac{π}{180}$ , not $\frac{180}{π}$ . The correct value is $100 \cdot \frac{π}{180} = \frac{5 π}{9} \approx 1.75$ rad. A sanity check helps here: $10 0^{\circ}$ is just over a quarter turn, so the answer must be a bit under $2$ .

Problem 6Numeric answer

A pendulum of length $1.2$ m swings through an angle of $4 0^{\circ}$ . What arc length, in metres, does the tip trace in one swing? Round to two decimals.

Show the worked solution

Answer: 0.84

Arc length needs the angle in radians: $4 0^{\circ} = \frac{40 π}{180} = \frac{2 π}{9}$ . Then $s = r θ = 1.2 \cdot \frac{2 π}{9} = \frac{2.4 π}{9} \approx 0.84$ m.

Problem 7Numeric answer

An arc of length $12$ lies on a circle of radius $5$ . Find the central angle that spans it, in radians.

Show the worked solution

Answer: 2.4

Arc length is $s = r θ$ with $θ$ in radians, so I rearrange: $θ = \frac{s}{r} = \frac{12}{5} = 2.4$ radians. No $π$ appears, and that is fine: a measured arc gives a plain decimal angle. Check: $2.4$ sits between $\frac{π}{2} \approx 1.57$ and $π \approx 3.14$ , so the arc is a bit less than a half circumference, and indeed $12 < 5 π \approx 15.7$ .

Problem 8Multiple choice

Which of these angles lies in the third quadrant?

A $\frac{5 π}{7}$
B $\frac{9 π}{5}$
C $\frac{7 π}{5}$
D $- \frac{π}{5}$

Show the worked solution

Answer: C — $\frac{7 π}{5}$

The third quadrant is $π < θ < \frac{3 π}{2}$ , so I compare each fraction of $π$ with $1$ and $\frac{3}{2}$ . $\frac{5}{7} \approx 0.71$ is between $\frac{1}{2}$ and $1$ : second quadrant. $\frac{9}{5} = 1.8$ is between $\frac{3}{2}$ and $2$ : fourth. $- \frac{π}{5}$ is a small clockwise turn, coterminal with $\frac{9 π}{5}$ : fourth again. $\frac{7}{5} = 1.4$ is between $1$ and $\frac{3}{2}$ : third quadrant. Check in degrees: $\frac{7 π}{5} = 25 2^{\circ}$ , and $180 < 252 < 270$ .

Problem 9Numeric answer

The minute hand of a clock is $12$ cm long. What distance, in cm, does its tip travel between 3:00 and 3:35? Round to one decimal.

Show the worked solution

Answer: 44

Thirty-five minutes is $\frac{35}{60} = \frac{7}{12}$ of a full turn, so the hand sweeps $\frac{7}{12} \cdot 2 π = \frac{7 π}{6}$ radians. The tip traces an arc of length $s = r θ = 12 \cdot \frac{7 π}{6} = 14 π \approx 43.98$ , so $44.0$ cm. Check: a full turn would be $24 π \approx 75.4$ cm and $\frac{7}{12}$ of that is about $44$ cm.

Problem 10Multiple choice

How many integers $k$ with $0 \leq k \leq 24$ make the angle $\frac{k π}{8}$ lie strictly inside the second quadrant?

Show the worked solution

Answer: C — 6

The second quadrant is $\frac{π}{2} < θ < π$ after reducing by full turns. One full turn is $2 π = \frac{16 π}{8}$ , so the position of $\frac{k π}{8}$ depends on $k$ modulo $16$ . Within one turn I need $\frac{π}{2} = \frac{4 π}{8} < \frac{k π}{8} < \frac{8 π}{8}$ , so $4 < k < 8$ : $k = 5, 6, 7$ . Adding a turn gives $k = 21, 22, 23$ , all at most $24$ . Adding another would exceed the range. That is $6$ values. Check the boundaries: $k = 4$ and $k = 8$ sit on the axes and are excluded by the strict inequality; $k = 24$ is $3 π$ , on the negative $x$ -axis, also excluded.

OMPT angles and radians: questions students ask

Are radians on the OMPT-A?

No. Radians belong to the trigonometry chapter, which OMPT-A does not have; its eight chapters run from numbers to differentiation. The same goes for OMPT-C, OMPT-E and OMPT-F. Only OMPT-B, OMPT-D and OMPT-G test angles in radians, and on those three the whole trigonometry chapter is written in radians rather than degrees.

How do I convert degrees to radians for the OMPT?

Use the one fact 180 degrees equals pi radians. Degrees to radians: multiply by pi over 180, so 150 degrees is 5pi/6. Radians to degrees: multiply by 180 over pi, so 3pi/4 is 135 degrees. Simplify the fraction and leave pi in the answer. If the pi does not cancel when you are converting to degrees, the factor was upside down.

Do I need a calculator for radians on the OMPT?

No, and none is provided on the maths variants. Exact answers such as 5pi/6 need no calculator, and the decimal questions are built around landmarks: pi over 2 is about 1.57, pi about 3.14, three pi over 2 about 4.71. Knowing those three numbers is enough to place any angle and to round an arc length sensibly by hand.

How many radian questions are on the OMPT-B?

Directly, my estimate is one or two of the 64 questions: a conversion, a coterminal angle, or an arc length. Indirectly the topic is everywhere in the trigonometry chapter, because every exact value, identity and equation question on OMPT-B states its angles in radians. Treat the chapter as radian fluency training rather than as a small stand-alone topic.

What is the difference between a coterminal angle and a reference angle on the OMPT?

A coterminal angle lands on the same point of the circle and differs from the original by whole turns of 2pi; the OMPT usually wants the one in the interval from 0 to 2pi. A reference angle is the acute angle between the terminal side and the x-axis, and it is what you look up in the exact-value table. 5pi/4 has coterminal partner 13pi/4 and reference angle pi/4.

Does the arc length formula work with degrees on the OMPT?

Not as written. Arc length s equals r times theta, and sector area equals half r squared theta, only when theta is in radians. Given a degree measure, convert it first or use the proportion theta over 360 times the circumference. Feeding 60 degrees straight into r times theta gives an arc many times longer than the whole circle, which is the most common slip on this topic.

Test hubs that cover angles and radians:OMPT-B OMPT-D OMPT-G All 28 topics

OMPT angles and radians practice: lesson and 10 problems

OMPT angles and radians: the lesson

What the OMPT tests in angles and radians

Notation and answer format the OMPT expects

Worked example 1: degrees to radians

Worked example 2: radians to degrees

How each OMPT variant tests angles and radians

OMPT angles and radians practice problems

OMPT angles and radians: questions students ask

Related OMPT topics and test hubs