CHAPTER 11: CIRCULAR MEASURE
Chapter objectives:
• solve problems involving the arc length and sector area of a circle,
including knowledge and use of radian measure
Angular measure
One way of measuring angles is using degrees where one full circle has 360°.
The radian system is a system of measuring the angles in a circle where one
radian equals the angle subtended by an arc equal to the radius of the circle.
One full circle has 2𝜋 radians and so 2𝜋 radians is equal to 360°.
→ 𝜋 radians = 180°
180°
∴ 1 radian = = 57.3°
𝜋
209
,Radians are the form of angular measure used in calculus and for some
equations of circular measure. When working with angles in radians, the
calculator should be in radian mode as trigonometric ratios will have
different values depending on which form of angular measure is used.
Arc length
A minor arc is an arc subtended by an angle less than a straight-line angle i.e.
less than 180° or 𝜋 rad.
A major arc is an arc subtended by an angle more than a straight-line angle
i.e. more than 180° or 𝜋 rad.
An arc of a circle is a portion of the circumference and so it follows that the
length of an arc of a circle is a fraction of the length of the circumference.
Circumference = 2𝜋𝑟
210
, For angular measure in degrees:
𝜽
𝐀𝐜 𝐥𝐞𝐧𝐠𝐭𝐡 = × 𝟐𝝅𝒓
𝟑𝟔𝟎°
360° = 2𝜋 radians, so when 𝜃 is in radians:
𝜃
Arc length = × 2𝜋𝑟
2𝜋
∴ 𝐋𝐞𝐧𝐠𝐭𝐡 𝐨𝐟 𝐚𝐫𝐜 = 𝜽𝒓
Example 11.1
Given the circle below with a radius of 12 cm
a. Express 45° in radians
b. Find the length of minor arc AB
SOLUTION
211
Chapter objectives:
• solve problems involving the arc length and sector area of a circle,
including knowledge and use of radian measure
Angular measure
One way of measuring angles is using degrees where one full circle has 360°.
The radian system is a system of measuring the angles in a circle where one
radian equals the angle subtended by an arc equal to the radius of the circle.
One full circle has 2𝜋 radians and so 2𝜋 radians is equal to 360°.
→ 𝜋 radians = 180°
180°
∴ 1 radian = = 57.3°
𝜋
209
,Radians are the form of angular measure used in calculus and for some
equations of circular measure. When working with angles in radians, the
calculator should be in radian mode as trigonometric ratios will have
different values depending on which form of angular measure is used.
Arc length
A minor arc is an arc subtended by an angle less than a straight-line angle i.e.
less than 180° or 𝜋 rad.
A major arc is an arc subtended by an angle more than a straight-line angle
i.e. more than 180° or 𝜋 rad.
An arc of a circle is a portion of the circumference and so it follows that the
length of an arc of a circle is a fraction of the length of the circumference.
Circumference = 2𝜋𝑟
210
, For angular measure in degrees:
𝜽
𝐀𝐜 𝐥𝐞𝐧𝐠𝐭𝐡 = × 𝟐𝝅𝒓
𝟑𝟔𝟎°
360° = 2𝜋 radians, so when 𝜃 is in radians:
𝜃
Arc length = × 2𝜋𝑟
2𝜋
∴ 𝐋𝐞𝐧𝐠𝐭𝐡 𝐨𝐟 𝐚𝐫𝐜 = 𝜽𝒓
Example 11.1
Given the circle below with a radius of 12 cm
a. Express 45° in radians
b. Find the length of minor arc AB
SOLUTION
211
