Practice Radian Measure in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

An angle measure defined by the arc length subtended on a unit circle: one radian is the angle that subtends an arc equal in length to the radius.

Imagine wrapping the radius of a circle along its edge like a piece of string. The angle you've swept out is exactly 1 radian. Since the full circumference is 2Ο€r2\pi r, a full turn is 2Ο€2\pi radians. Radians measure angles in terms of the circle itself, which is why they're the natural unit for calculus and physicsβ€”no arbitrary conversion factor like 360360 is needed.

Showing a random 20 of 50 problems.

Example 1

medium
Find the area of a sector with radius 66 and central angle Ο€3\dfrac{\pi}{3}.

Example 2

easy
Convert 90Β°90Β° to radians.

Example 3

easy
Convert 2Ο€3\frac{2\pi}{3} radians to degrees.

Example 4

medium
Convert Ο€10\dfrac{\pi}{10} radians to degrees.

Example 5

medium
Convert 315∘315^\circ to radians.

Example 6

medium
Convert 11Ο€12\dfrac{11\pi}{12} radians to degrees.

Example 7

medium
Find the radius of a circle if a 22-rad central angle subtends an arc of length 99.

Example 8

hard
On a circle of radius 1212, two radii form a central angle of 50∘50^\circ. Find the exact area of the sector.

Example 9

hard
A circular sector has area 20Ο€20\pi cm2^2 and central angle 2Ο€5\dfrac{2\pi}{5} rad. Find the radius.

Example 10

medium
A wheel of radius 0.50.5 m rolls without slipping. Through how many radians does it turn after traveling 44 m?

Example 11

medium
Convert 5Ο€3\frac{5\pi}{3} radians to degrees.

Example 12

medium
A central angle of 1.51.5 radians is subtended on a circle of radius 1010 cm. Find the arc length and the area of the sector.

Example 13

medium
A pendulum swings through an angle of 36∘36^\circ. Express that angle in radians (exact).

Example 14

medium
A wheel turns through 55 radians. Through how many degrees does it turn? (Round to the nearest degree.)

Example 15

medium
A central angle of 40Β°40Β° subtends an arc in a circle of radius 9. Find the arc length.

Example 16

medium
Convert 7Ο€6\dfrac{7\pi}{6} radians to degrees.

Example 17

medium
Convert 300Β°300Β° to radians.

Example 18

easy
Convert 180Β°180Β° to radians.

Example 19

easy
How many radians are in one full revolution?

Example 20

easy
Convert Ο€\pi radians to degrees.
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