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 2πr2\pi r, a full turn is 2π2\pi radians. Radians measure angles in terms of the circle itself, which is why they're the natural unit for calculus and physics—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 π3\dfrac{\pi}{3}.

Example 2

easy
Convert 90°90° to radians.

Example 3

easy
Convert 2π3\frac{2\pi}{3} radians to degrees.

Example 4

medium
Convert π10\dfrac{\pi}{10} radians to degrees.

Example 5

medium
Convert 315∘315^\circ to radians.

Example 6

medium
Convert 11π12\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 20π20\pi cm2^2 and central angle 2π5\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 5π3\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 40°40° subtends an arc in a circle of radius 9. Find the arc length.

Example 16

medium
Convert 7π6\dfrac{7\pi}{6} radians to degrees.

Example 17

medium
Convert 300°300° to radians.

Example 18

easy
Convert 180°180° to radians.

Example 19

easy
How many radians are in one full revolution?

Example 20

easy
Convert π\pi radians to degrees.
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