Radian measure, arcs, sectors and segments

How can we find the glass area without counting the triangle below the arc?

• A curved window has a straight lower edge.
• This lesson studies sector 扇形: The region bounded by two radii and their circular arc.

The method

• One radian is the central angle with arc length equal to the radius.
• A full turn is 2π radians=360°, so radians=degrees×π/180.
• For r>0 and a nonnegative swept angle θ in radians, arc length s=rθ and sector area A=r²θ/2.
• A sector perimeter is 2r+s, not s alone.
• Specify minor or major region.

A worked case

Where it goes wrong

• Never put 120 directly into rθ: first convert degrees to radians.
• A chord is straight and generally shorter than its minor arc.
• Sector, segment and triangle have different boundaries.
• The minor-segment subtraction shown uses 0≤θ≤π; for a major segment subtract the complementary minor region from the disk.

Applying it

• Draw the radii, chord and required boundary first.
• Recover θ from s/r or 2A/r² when needed.
• Keep π and surds exact; attach length or area units only after choosing the correct formula.
• In the inverse example the sector perimeter is 10+8=18 cm.
• The checked inverse example stays radius 5 cm and arc 8 cm.
• The later problems use other lengths and angles.
• Decide sector or segment before you substitute.
• A major segment is the disk minus the minor segment on the same chord.

In your course

• AQA A-level Mathematics (7357), specification section E.
• Every paper allows a calculator, but method marks still need the expression or equation written before its value.