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}

Two points,

P

and

Q,

are on the circumference of the unit circle in the first quadrant. They have

x -

coordinates of

0.1

and

0.6,

respectively. How many units is the shortest arc we can draw between

P

and

Q ?

Round the answer to two decimal places.

Let's start by drawing the unit circle and mark the shortest arc that is created by P and Q on the circle's circumference.

To determine the length of this arc, we will use the following formula. s = θ r In this formula, θ is the measure of the central angle in radians. Since the unit circle has a radius of r= 1, the length of the arc is the same as the central angle. s = θ ( 1) ⇔ s = θ We can calculate θ by subtracting θ_Q from θ_P.

Remember that the x-coordinate of a point on the unit circle is the cosine of the central angle. Therefore, P has a cosine value of 0.1 and Q has a cosine value of 0.6. That allows us to write two equations. cos θ_P &= 0.1 ⇔ θ_P = cos^(-1) 0.1 cos θ_Q &= 0.6 ⇔ θ_Q = cos^(-1) 0.6 Let's substitute these identities in the equation describing θ.

To calculate this we need a graphing calculator. Before we can calculate it though, we must make sure the calculator interprets the argument of trigonometric functions as radians. Push MODE and select Radian on the third row.

Now, we can continue calculating the measure of θ.

Five congruent circles are placed next to each other such that their radii form a regular pentagon when connected.

Which is longer, the sum of the marked arcs or the perimeter of the pentagon?

Consider the perimeter of a polygon created by the radii of six congruent circles. Is the total path along the arcs longer or shorter than the perimeter of the polygon created by those six circles?

Since we are not given any lengths, we must find expressions which can be compared. Notice that we are comparing a sum of circle radii to the length of the arcs. These can be compared, even without actual lengths, if we work in radians.

Perimeter of Pentagon

Let's label the radius of each circles a.

As we can see, the perimeter of the pentagon equals the sum of two radii for each of the five circles. With this information, we can determine an expression for the perimeter. 5( 2a) = 10 a

Length of Arcs

To calculate the length of the arcs, we need to find the sum of the angle measures of a pentagon. (n-2)* 180^(∘) ⇓ (5-2)* 180^(∘)=540^(∘) The measures of the interior angles sum to 540^(∘). Since the pentagon is regular, each angle has a measure of one fifth of this. 540^(∘)/5=108^(∘) Let's rewrite this from degrees to radians.

Each angle has a measure of 3π5 radians. Notice that each of these angles also is a central angle in a circle.

When we know the measure of each central angle, we can find the arc length using the following formula. s = θ r In this formula, θ is the measure of the central angle in radians.

The length of one arc is 3aπ5. If we multiply this by 5 we get the combined length of the five arcs. 5s = 5(3aπ/5) ⇓ 5s ≈ 9.424a Since 9.424a<10a, we know that the sum of the arc lengths is less than the perimeter of the pentagon.

Here, we are now dealing with a polygon formed by six rather than five circles. Since the polygon will then have six sides, it is a regular hexagon. Let's draw the figure to aid our solving process.

Again, if we label the radius of each circle as a, we get a total perimeter of 6 times 2a. Perimeter of Hexagon: 6(2a)=12a Now we want to calculate the length of the arcs. A hexagon has a total angle sum of 720^(∘) and each of the six angles measures 120^(∘). Let's rewrite this as radians.

Each angle has a measure of 2π3 radians.

Let's calculate the length of one arc and then multiply this by 6 to account for all of the arcs.

Since 12.57a>12a, we know that the sum of the arc lengths is greater than the perimeter of the hexagon.

Radian Measure and Arc Length

Catch-Up and Review

Investigating the Orbit of a Satellite

Radian

Relationship Between Radians and Degrees

Proof

Practice Converting Degrees Into Radians

Practice Converting Radians Into Degrees

Investigating the Radian Measure for Different Circles

Comparing Radian Measures of Concentric Circles

Using Radians to Calculate Arc Lengths

Arc Length Between the Long and Short hand of a Clock

Hint

Solution

Finding the Length of an Arc

Hint

Solution

Finding Angle Measures Given an Arc Length

Hint

Solution

Finding the Orbit of a Satellite

Hint

Solution

Perimeter of Pentagon

Length of Arcs

Radian Measure and Arc Length

Recommended exercises

	12 Theory slides
	12 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions