From Similarity to Radians

HSG.C.B.5 — Deck 1 of 2

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius

Learning Objectives — Deck 1

By the end of this lesson you will be able to:

1. Explain why arc length is proportional to radius for a fixed central angle, using the similarity of all circles
2. Define radian measure as the constant of proportionality
3. Identify that one radian is the angle whose intercepted arc equals the radius
4. State that a full revolution equals radians
5. Convert fluently between degree and radian measure

Hook: Why ?

A full circle is 360°. But it is also radians.

Why ? And why does it matter?

"There are radians in a full circle because you can fit approximately 6.28 radius-lengths along the circumference."

• The circumference of any circle is
• So the circumference contains radius-lengths
• That ratio — arc length divided by radius — is the key idea of this lesson

Recap: All Circles Are Similar

From HSG.C.A.1: Any two circles are similar.

• A circle with center and radius can be mapped to a circle with center and radius by:
  1. A translation moving to
  2. A dilation with scale factor

Key consequence: Every length on the scaled circle is multiplied by .

• If a chord has length on the first circle, the corresponding chord has length on the second
• The same applies to arc lengths

Same Central Angle, Different Circles

Draw two circles with the same central angle :

The arcs look different in length. Is there a pattern?

• If the outer circle has twice the radius, what happens to its arc?
• The arc on the outer circle is twice as long

Why? The outer circle is a dilation of the inner circle by . All lengths scale by — including arc lengths.

Concentric Arcs: The Proportionality in Action

The Dilation Argument

Setup: Two circles, radii and ; same central angle ; arc lengths and .

Step 1: The outer circle is a dilation of the inner circle by scale factor:

Step 2: Dilations scale all lengths by , so arc lengths scale the same way:

Step 3: Rearrange:

Conclusion: The ratio is the same for both circles when the central angle is the same.

Is Constant for a Given Angle

Theorem (from similarity): For a fixed central angle , the ratio is the same for every circle.

This means:

• The ratio does not depend on the size of the circle
• It depends only on the central angle
• Two arcs subtending the same angle on different circles have the same ratio

This constant ratio is what we are about to name.

Check-In

Two circles have radii cm and cm. Both have a central angle of .

Using the circumference formula and the fact that a arc is of the full circumference:

1. Find the arc length for each circle
2. Compute for each circle
3. What do you notice?

Check-In Answer

Circle with :

Circle with :

Both ratios equal — the same value, confirming that depends only on the angle.

Transition: Naming the Constant

We have proved:

For any fixed central angle, the ratio is the same for every circle.

This ratio is a function of the angle alone. It deserves a name.

We will call it the radian measure of the angle.

This is not a convention chosen for convenience — it is the natural quantity that similarity forces upon us.

Defining Radian Measure

Definition: The radian measure of a central angle is:

where is the intercepted arc length and is the radius of the circle.

Properties:

• is a dimensionless ratio (length ÷ length)
• is independent of which circle we use — any circle gives the same
• Rearranging: and

Example: A angle gives radians

One Radian: When the Arc Equals the Radius

One radian is the angle such that :

Geometric meaning: Lay the radius along the circumference — the angle subtended is exactly 1 radian.

How many radii fit around the circumference?

So radians fill a full revolution — and

Full Circle = Radians

For a full revolution, the arc length equals the circumference:

Applying the definition:

Therefore:

One Radian: Visual Definition

Degree Radian Conversion

The master fact: radians

To convert degrees to radians: multiply by

To convert radians to degrees: multiply by

Don't memorize two formulas — derive both from rad by multiplying or dividing.

M3: Radians Are Not Arbitrary

Misconception: Radians are just an alternative to degrees — a more confusing way to measure the same thing.

Reality: Radians arise directly from the geometry of circles:

• is a ratio of two lengths — it is dimensionless and geometric
• The formulas and are simple because of radians, not despite them
• In degrees, the arc length formula requires an extra factor:
• In calculus: only when is in radians
• Radians are the natural unit — they are what similarity forces on us

M4: Arc Length Depends on Both Angle AND Radius

Misconception: If two arcs subtend the same central angle, they have the same length.

Reality: The angle determines only the ratio , not itself.

The same angle gives arcs that differ by a factor of 100.

Rule: To find , you always need both and .

Worked: Conversion Examples

Degrees → Radians (multiply by ):

Radians → Degrees (multiply by ):

Reference Table: Common Degree-Radian Pairs

Tip: You can reconstruct this table from rad using multiplication and simplification.

Practice: Convert and rad

Convert to radians:

Convert rad to degrees:

Check: Both results match the reference table.

Summary — Deck 1

Five key ideas from this lesson:

1. All circles are similar → arc lengths scale with radius → for fixed
2. The ratio is constant for a fixed angle: we define this as the radian measure
3. One radian = the angle whose intercepted arc equals the radius ()
4. Full circle = radians; the conversion is rad
5. Convert degrees → radians: ; radians → degrees:

Watch out for:

Next: Arc Length and Sector Area

Coming up in Deck 2:

• Arc length formula: — derive it, apply it, check reasonableness
• Sector area formula: — two derivation methods
• Working with degrees: convert first, then apply
• Misconceptions: M1 (degrees in the formula), M2 (arc vs area), M5 (circumference vs area formula)
• Real-world applications: sprinklers, windshield wipers, pizza slices

The formulas in Deck 2 follow directly from . If you understand Deck 1, Deck 2 is just rearranging that one equation.