Level 3 - Radian Measure and Arc Length - Circles With and Without Coordinates (Geometry)

Degrees to Radians	Radians to Degrees
$1^{\circ} = \frac{π}{1 8 0} rad$	$1 rad = \frac{1 8 0 ^{\circ}}{π}$

Degrees to Radians

Radians to Degrees

1^{\circ} = \frac{π}{1 8 0} rad

1 rad = \frac{1 8 0 ^{\circ}}{π}

It can be seen in the diagram that

∠ M K N,

which is an inscribed angle, measures

4 0^{\circ} .

By multiplying this value by the conversion factor

\frac{π}{1 8 0 ^{\circ}},

the measure of this angle can be converted from degrees to radians.

m ∠ M K N = 4 0^{\circ} \cdot \frac{π}{1 8 0 ^{\circ}}

MoveLeftFacToNum

$a \cdot \frac{b}{c} = \frac{a \cdot b}{c}$

m ∠ M K N = \frac{4 0 ^{\circ} π}{1 8 0 ^{\circ}}

ReduceFrac

$\frac{a}{b} = \frac{a / 2 0 ^{\circ}}{b / 2 0 ^{\circ}}$

m ∠ M K N = \frac{2 π}{9}

Next, to find the length of

M N,

the measure of the corresponding central angle should be known. Recall that the measure of the inscribed angle is half the measure of the corresponding central angle. In this case,

∠ M K N

corresponds to a central angle

∠ M O N .

Using this information, the measure of

∠ M O N

can be found.

m ∠ M K N = \frac{1}{2} m ∠ M O N

Substitute

$m ∠ M K N = \frac{2 π}{9}$

\frac{2 π}{9} = \frac{1}{2} m ∠ M O N

Solve for

m ∠ M O N

MultEqn

$LHS \cdot 2 = RHS \cdot 2$

\frac{2 π}{9} (2) = m ∠ M O N

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

\frac{2 π ( 2 )}{9} = m ∠ M O N

Multiply

\frac{4 π}{9} = m ∠ M O N

RearrangeEqn

Rearrange equation

m ∠ M O N = \frac{4 π}{9}

Finally, the length of

M N

can be calculated using the corresponding formula.

s = θ r ⇓ M N = m ∠ M O N r

Here,

\frac{4 π}{9}

and

5

can be substituted for

m ∠ M O N

and

r,

respectively.

M N = m ∠ M O N r

SubstituteII

$m ∠ M O N = \frac{4 π}{9}$ , $r = 5$

M N = (\frac{4 π}{9}) (5)

Evaluate right-hand side

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

M N = \frac{4 π ( 5 )}{9}

Multiply

M N = \frac{2 0 π}{9}

UseCalc

Use a calculator

M N = 6.981317 \dots

RoundInt

Round to nearest integer

M N \approx 7

The length of

M N

is approximately

7

inches.

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Challenge

Investigating the Orbit of a Satellite

Discussion

Radian

Pop Quiz

Practice Converting Degrees Into Radians

Pop Quiz

Practice Converting Radians Into Degrees

Discussion

Comparing Radian Measures of Concentric Circles

Discussion

Using Radians to Calculate Arc Lengths

Example

Arc Length Between the Long and Short hand of a Clock

Hint

Solution

Example

Finding the Length of an Arc

Hint

Solution

Example

Finding Angle Measures Given an Arc Length

Hint

Solution

Closure

Finding the Orbit of a Satellite

Hint

Solution

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