Radians and circular measure
Use radians to connect an angle directly to lengths and areas.
What you will learn
- Convert degrees to radians.
- Calculate arc length and sector area.
A radian is a length ratio
An angle of one radian subtends an arc equal in length to the circle's radius. Since a full circumference is 2πr, a full turn is 2π radians. Thus 180° corresponds to π radians.
The formula s=rθ works when θ is in radians. Degrees must be converted first. Keep exact multiples of π until the end if exact form is requested.
Sector area is a fraction of a circle
The sector occupies the fraction θ/(2π) of a full circle. Multiplying that by πr² gives 1/2 r²θ. If asked for a sector perimeter, include both radii as well as the arc.
Worked example
A sector has radius 9 cm and angle π/3 radians. Find its arc length and perimeter.
Show the worked solution
- Use s=rθ=9×π/3=3π cm.
- The boundary also includes two radii totalling 18 cm.
- Add the straight and curved parts.
Answer Arc length 3π cm; perimeter 18+3π cm
Common mistakes
- Using degrees directly in s=rθ gives an incorrect result.
- An arc length is not the entire sector perimeter.
What can you explain now?
Find the area of a sector of radius 4 cm and angle π/2.
Compare with the explanation
4π cm²
Use 1/2×16×π/2=4π.
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