Radians, sectors and segments

One radian compares arc length with radius

An angle in radians is θ = s/r, where s is its intercepted arc length and r the circle’s radius. A full turn is 2π radians, corresponding to 360°. Thus 180° is π radians: multiply degrees by π/180 to convert to radians, and radians by 180/π to convert to degrees.

With θ in radians, arc length is s = rθ and sector area is (1/2)r²θ. These simplified formulae do not accept a degree number unchanged. For radius 6 cm and angle π/3, arc length is 2π cm and sector area is 6π cm².

A segment is bounded by a chord and an arc

A sector is bounded by two radii and an arc. For a minor segment with central angle 0 < θ < π, subtract the triangle formed by its two radii and chord:

For r = 6 cm and θ = π/3, this gives 6π-9 square root of 3 ≈ 3.26 cm². Use radian mode for sin(π/3) if entering the angle that way. The triangle and sector must use the same two radii and central angle.

For a requested major segment, subtract the minor segment from the full circle area. Draw the region rather than applying a minor-region formula blindly to the wrong side of the chord.

Check your understanding 1 The formula chooses the angle unit

A sector has radius 8 cm and angle 1.2 radians. Find its arc length, sector area and complete perimeter.

Your answer or drawing on paper

Show answer

Arc length is 8(1.2) = 9.6 cm. Area is (1/2)(64)(1.2) = 38.4 cm². Perimeter is 9.6 + 16 = 25.6 cm. Dividing the 1.2 by 360 would incorrectly treat it as degrees.

Check a numerical angle conversion

Convert 150° to radians and 5π/4 radians to degrees.