Day 133 – Radian measure and proportionality to radius

Introduction We are used to measuring angles using degrees a system which was developed by ancient Babylonians. However, there are a number of ways in which we can measure angles. We will introduce radian measure of an angle which we can use instead of degrees. In this lesson, we will define the radian measure of the angle as the constant of proportionality.

Vocabulary Sector of a circle This is a part of a circle that is enclosed by an arc and the radii at the endpoints of the arc. Arc length This is the distance along the curved line

We have learned that all circles are similar since a translation followed by a dilation will always map two circles onto each other. However, two sectors of a circle may not be similar depending on the size of the angle between the radii of the sectors. Two sectors of circles are similar if the angles between their radii are equal.

Consider two sectors of a circle below The two sectors above are similar because the angles between their radii are equal. A translation to the right followed a dilation about point O with a scale factor of 𝑟 𝑅 will map the sectors together. L and 𝑙 denote the arc lengths. R r O 𝑂′ L 𝑙

A sector is a fraction of a circle A sector is a fraction of a circle. Sectors (a) and (b) are 90 360 = 1 4 of the circle. Since the circumference of a circle is given by 𝐶 =2𝜋𝑟, where C is the circumference of the circle and 𝑟 the radius, the arc length is given by, 𝑙= 𝜃 360 ×2𝜋𝑟, where 𝑙 is the arc length and 𝜃 is the angle between the radii.

Consider two similar sectors below Consider two similar sectors below. Since the sectors are similar, 𝑅 𝑟 = 𝐿 𝑙 , cross multiplying the characters we get, 𝑅𝑙=𝐿𝑟, dividing both sides by R𝑟 we get, 𝑙 𝑟 = 𝐿 𝑅 𝑅 𝑂 𝐿 𝜃 𝑟 𝑂 𝑙 𝜃

Thus for a given angle, the ratio of the arc length subtending the angle to the center of the circle is always constant. This constant, A, is called the radian measure. 𝑟𝑎𝑑𝑖𝑎𝑛 𝑚𝑒𝑎𝑠𝑢𝑟𝑒= 𝐴𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ, 𝑙 𝑅𝑎𝑑𝑖𝑢𝑠, 𝑟 𝑙 𝑟 =𝐴, 𝐿 𝑅 =𝐴 The two equations above can be written as, 𝑙=𝐴𝑟, 𝐿=𝐴𝑅 since 𝐴 is a constant, we can write, 𝑙∝𝑟, 𝐿∝𝑅 Therefore, for a given angle, the arc length is directly proportional to the radius of the circle and the radian measure is the constant of proportionality.

Unit of radian measure The radian measure is a ratio of two quantities with same units: arc length and radius. The units will cancel out and therefore the radian measure has no units.

Example 2 A sector of a circle is enclosed by an arc of length 14 𝑖𝑛 and radii each of length 7 𝑖𝑛. If the angle between the radii is 𝜃. Calculate the radian measure of 𝜃. If a similar sector has an arc length of 20 𝑖𝑛, find its radius.

Solution a) radian measure= arc length radius = 14 𝑖𝑛 7 𝑖𝑛 =2 b) the radian measure 2 is constant for all sectors with an arc subtending angle 𝜃 at the center. 2= 20 𝑖𝑛 radius radius= 20 𝑖𝑛 2 =10 𝑖𝑛 𝜃 7 𝑖𝑛 14 𝑖𝑛

Example 2 A sector of a circle has a radius of 10 𝑖𝑛 and an angle of 60 ° between its radii. Find the length of its arc hence find the radian measure of 60 ° . Leave your answer in terms of 𝜋. Solution Arc length, 𝑙= 𝜃 360 ×2𝜋𝑟 = 60 360 ×2×𝜋×10= 10𝜋 3 𝑖𝑛 radian measure= Arc length, 𝑙 Radius, 𝑟 = 10𝜋 3 10 = 10𝜋 3 × 1 10 = 𝜋 3

homework A sector of a circle has an arc length of 27 𝑖𝑛 and a radius of 9 𝑖𝑛. Find the radian measure of the angle subtended by its arc at the center. Find the arc length of another sector with the same central angle and a radius of 2 𝑖𝑛.

Answers to homework Radian measure = 3 Arc length =6 𝑖𝑛

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