Opening your learning space
Starting the app …
Opening your learning space
Starting the app …
Radian measure defines an angle as arc length divided by radius and gives direct formulas for arc length and sector area.
Free to browse · flashcards or a multiple-choice quiz · import from Essential
Formula
θ in radians (dimensionless); s and r in the same length unit; A in its square unit.
| Symbol | Meaning | Unit |
|---|---|---|
| Central angle in radians | rad | |
| Length of the corresponding circular arc | length unit | |
| Circle radius | length unit | |
| Area of the circular sector | length unit2 |
\theta=\frac{s}{r},\quad s=r\theta,\quad A_{\mathrm{Sektor}}=\frac12r^2\thetaOne radian is the angle for which arc length equals radius. Thus θ=s/r and s=rθ. A sector occupies the fraction θ/(2π) of the full circle, so its area simplifies to A=½r2θ. These forms use θ in radians; convert degrees by multiplying by π/180 first.
In s=rθ and A=½r2θ, θ must be in radians. Convert degrees by multiplying by π/180 first.
The radian angle is the ratio of arc length to radius.
Radius from arc length
θ must be in radians.
Central angle from arc length
The result is a radian measure, dimensionless apart from its label.
A circle has r=4 cm and a central angle of 60°. Find the arc length and sector area.
Convert 60°=π/3 rad.
Then s=rθ=4π/3≈4.19 cm and A=½r2θ=8π/3≈8.38 cm2.
Check: 60° is one sixth of a full turn, so the area is one sixth of 16π cm2.
Substituting 60 directly into s=rθ as the angle.
Convert first: 60°=π/3 rad. Otherwise the arc is too large by a factor of 180/π.
Formula cluster
Radian measure links circle geometry to calculus of trigonometric functions.
For r=4 cm and θ=π/3 rad, s=4π/3 cm≈4.19 cm and A=½·16·π/3=8π/3 cm2≈8.38 cm2. Check: θ/(2π)=1/6, so the sector is one sixth of the full area πr2=16π cm2.
Find circular arc lengths and sector areas, and use the natural angular measure for calculus and periodic functions.
Here is how to work through a typical Radian measure, arc length, and sector area (θ = s/r; s = rθ; A = ½r²θ) task step by step:
Task
A circle has r=4 cm and a central angle of 60°. Find the arc length and sector area.
Solution path
Convert 60°=π/3 rad.
Then s=rθ=4π/3≈4.19 cm and A=½r2θ=8π/3≈8.38 cm2.
Check: 60° is one sixth of a full turn, so the area is one sixth of 16π cm2.
One radian is the central angle that subtends an arc whose length equals the radius. A full turn is 2π radians, so 180° equals π radians. Radian measure is therefore the ratio s/r, not an independent physical unit. Use the same length unit for arc and radius, and enter angles in radians in the compact formulas. A units check catches common confusions. At a fixed angle, arc length scales with r while sector area scales with r2. Thus a quarter-circle is π/2 rad; the angle is dimensionless even though rad is written as its SI unit.
Multiply the degree value by π/180. For example, 60° becomes 60π/180=π/3 rad. For the reverse conversion, multiply radians by 180/π. This conversion is needed before using s=rθ or A=½r2θ because these relations assume radians. Use the same length unit for arc and radius, and enter angles in radians in the compact formulas. A units check catches common confusions. At a fixed angle, arc length scales with r while sector area scales with r2. Check with 180°=π rad and 360°=2π rad to see whether the conversion is plausible. Simplify the factor before calculating.
The angle θ is defined as arc length divided by radius: θ=s/r. At fixed radius, arc length therefore grows in proportion to the angle. Doubling the angular opening covers twice as much circumference. At fixed angle, doubling the radius also doubles the arc length. Use the same length unit for arc and radius, and enter angles in radians in the compact formulas. A units check catches common confusions. At a fixed angle, arc length scales with r while sector area scales with r2. The proportionality holds for any radian angle, including more than one full turn and a negative rotation direction. It also makes the same angle comparable across circles of different radii.
A sector with angle θ is the fraction θ/(2π) of the full circle. Multiply that fraction by the full area πr2: A=(θ/2π)πr2=½r2θ. This derivation also shows why θ must be in radians. For 60°, the sector is one sixth of the full circle. Use the same length unit for arc and radius, and enter angles in radians in the compact formulas. A units check catches common confusions. At a fixed angle, arc length scales with r while sector area scales with r2. At 60°, θ=π/3 and A=πr2/6; this fraction matches 60°/360°. The same calculation applies to any angle.
An arc is only the curved boundary segment; its length is s=rθ. A sector is the full area between two radii and that arc; its area is A=½r2θ. Both depend on the same angle but have different dimensions: length versus area. Use the same length unit for arc and radius, and enter angles in radians in the compact formulas. A units check catches common confusions. At a fixed angle, arc length scales with r while sector area scales with r2. The chord between the endpoints is a straight segment and is usually shorter than the arc.
Quanta exam set
Curated exam set for "Radian measure, arc length, and sector area":
Question (front)
Which formula describes Radian measure, arc length, and sector area?
Answer in your set
Question (front)
How do you rearrange Radian measure, arc length, and sector area for Radius from arc length?
Answer in your set
Question (front)
Which common mistake happens with Radian measure, arc length, and sector area?
Answer in your set
+ 7 more cards: units, variables, derivation, example, exam task
The formula set contains 10 cards. After importing, review them and start practicing; your ratings determine later review intervals.
This list names subject references and further reading. A page or section number appears when it is part of the reference; these are not verbatim quotations.
Create a curated FSRS exam set for θ = s/r; s = rθ; A = ½r²θ: formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.
Curated formula set · LaTeX · FSRS spaced repetition · import from Essential