Arc Length and Radian Measure Worksheet for 10th - 11th Grade ... - Free Printable

Explanation:
We are converting angles between degrees and radians. The key rule is:

\[
\pi \text{ radians} = 180^\circ
\]

So to convert degrees → radians, multiply by \(\frac{\pi}{180}\).
To convert radians → degrees, multiply by \(\frac{180}{\pi}\).

Let’s solve each part one by one.

---

1. Convert \(45^\circ\) to radians

\[
45^\circ \times \frac{\pi}{180^\circ} = \frac{45\pi}{180} = \frac{\pi}{4}
\]

✔ Answer: \(\frac{\pi}{4}\) radians

---

2. Convert \(135^\circ\) to radians

\[
135^\circ \times \frac{\pi}{180^\circ} = \frac{135\pi}{180} = \frac{3\pi}{4}
\]

(Simplify: divide numerator and denominator by 45 → 135÷45=3, 180÷45=4)

✔ Answer: \(\frac{3\pi}{4}\) radians

---

3. Convert \(\frac{\pi}{7}\) to degrees

\[
\frac{\pi}{7} \times \frac{180^\circ}{\pi} = \frac{180}{7}^\circ \approx 25.714^\circ
\]

But the problem says “convert to degrees” — likely expects exact form unless specified otherwise. Since it's a fraction, we can leave as \(\frac{180}{7}^\circ\), but many worksheets expect decimal approximation to nearest degree or tenth.

Looking at other problems (e.g., #5 says “approximately ___”), this one doesn’t say “approximately”, so maybe exact is fine. But let’s check context: #5 explicitly says “approximately”, so #3 probably wants exact: \(\frac{180}{7}^\circ\). However, in many high school worksheets, they expect decimal rounded to nearest tenth or whole number if not exact.

Wait — let’s hold and check all items.

Actually, looking again: Problem 3 says “Convert \(\frac{\pi}{7}\) to degrees” — no “approximate”, so exact is okay. But \(\frac{180}{7}\) is not a nice number. Maybe they want decimal? Let’s see problem 5: “Convert 112° to radians — the number of radians is a complete revolution is approximately ___” — that seems oddly worded. Possibly a typo. Let me re-read carefully.

Wait — the original image text (as transcribed) says:

> 3. Convert π / 7 to degrees, _______ convert 190° to radian measure.

Hmm — maybe it's two separate items on same line? Actually, looking at formatting:

It appears each numbered item has two parts separated by a comma — e.g.:

3. Convert π / 7 to degrees, convert 190° to radian measure.

So #3 is two sub-questions:
- Convert π/7 to degrees
- Convert 190° to radians

Same for others.

Let me reinterpret all lines properly:

1. Convert 45° to radians, convert 135° to radian measure.
2. Convert π/7 to degrees, convert 190° to radian measure.
3. Convert 112° to radians, the number of radians in a complete revolution is approximately ____.
4. Convert π/4 radians to degrees, convert 108° to radian degrees. *(“radian degrees” is likely typo — should be “radians”)*
5. Convert 2π / 4 to degrees, convert 127° to radians.

Yes, that makes sense.

So let’s redo with correct grouping:

---

1.
a) \(45^\circ \to\) radians: \(\frac{\pi}{4}\)
b) \(135^\circ \to\) radians: \(\frac{3\pi}{4}\)

---

2.
a) \(\frac{\pi}{7} \to\) degrees: \(\frac{180}{7}^\circ \approx 25.714^\circ\). Usually round to nearest tenth: 25.7°
b) \(190^\circ \to\) radians: \(190 \times \frac{\pi}{180} = \frac{19\pi}{18}\) radians
(Simplify: 190/180 = 19/18)

✔ So: \(\frac{19\pi}{18}\) rad

---

3.
a) \(112^\circ \to\) radians: \(112 \times \frac{\pi}{180} = \frac{112\pi}{180} = \frac{28\pi}{45}\) (divide numerator/denominator by 4)
b) “the number of radians in a complete revolution is approximately ____”
→ One full circle = \(360^\circ = 2\pi\) radians ≈ \(2 \times 3.1416 = 6.2832\)
So approximately 6.28 (or 6.3 if rounding to nearest tenth)

Most textbooks say ≈ 6.28 or just 2π, but since it says “approximately”, use decimal: 6.28

---

4.
a) \(\frac{\pi}{4}\) rad → degrees: \(\frac{\pi}{4} \times \frac{180}{\pi} = 45^\circ\)
b) \(108^\circ \to\) radians: \(108 \times \frac{\pi}{180} = \frac{108\pi}{180} = \frac{3\pi}{5}\)
(Simplify: divide by 36 → 108÷36=3, 180÷36=5)

✔ So: \(45^\circ\), and \(\frac{3\pi}{5}\) rad

---

5.
a) \(\frac{2\pi}{4} = \frac{\pi}{2}\) rad → degrees: \(\frac{\pi}{2} \times \frac{180}{\pi} = 90^\circ\)
b) \(127^\circ \to\) radians: \(127 \times \frac{\pi}{180} = \frac{127\pi}{180}\)
This doesn’t simplify. As decimal: \(127 \div 180 \approx 0.7056\), so ≈ \(0.706\pi\) rad, or just \(\frac{127\pi}{180}\) rad. If decimal radians: \(127 \times \frac{3.1416}{180} \approx 2.216\) rad (since π≈3.1416, 127×3.1416 = 399.0, ÷180 ≈ 2.2167)

But likely they want exact fraction unless asked for approx.

Now, the worksheet also includes a table at bottom with “Circle of Numbers” and “Percentage Points” — but no instructions, and user only asked to solve the conversion problems. So we focus on the 5 numbered items.

Since the user wants only the final answers (and ExplainAnswer is false), but per instructions we must do full reasoning first, then give Final Answer — and the Final Answer should contain all answers in order.

Let’s list all required answers clearly:

1. a) \(\frac{\pi}{4}\)
b) \(\frac{3\pi}{4}\)

2. a) \(\frac{180}{7}^\circ\) or ≈ 25.7°
b) \(\frac{19\pi}{18}\)

3. a) \(\frac{28\pi}{45}\)
b) approximately 6.28

4. a) \(45^\circ\)
b) \(\frac{3\pi}{5}\)

5. a) \(90^\circ\)
b) \(\frac{127\pi}{180}\) (or ≈ 2.22 rad)

But the worksheet may expect simplified exact forms, and for approximations only where stated.

Since #3b says “approximately”, we give 6.28. Others likely exact.

Also note: In #2, “convert π/7 to degrees” — exact is \(\frac{180}{7}^\circ\), but maybe they want decimal. However, without “approximately”, safest is exact fraction.

But many curricula teach to give decimal to nearest tenth when not a nice angle. Let’s check common practice: π/6 = 30°, π/4 = 45°, π/3 = 60°, π/2 = 90°, etc. π/7 is not standard, so they often ask for approximate.

Given that #5 has 127° (not standard) and no “approx”, but likely they accept exact fraction.

However, the instruction says: “Solve the problem accurately.” So we provide exact values unless approximation is requested.

Let me finalize each answer precisely:

1. \(\frac{\pi}{4},\ \frac{3\pi}{4}\)
2. \(\frac{180}{7}^\circ,\ \frac{19\pi}{18}\)
3. \(\frac{28\pi}{45},\ 6.28\)
4. \(45^\circ,\ \frac{3\pi}{5}\)
5. \(90^\circ,\ \frac{127\pi}{180}\)

But the format of the worksheet seems to have blanks — likely expecting numeric entries.

Since the user only wants the final answer (and no explanation section because ExplainAnswer: false), I will output just the answers in order, as a list matching the blanks.

Assuming each numbered item has two answers, and #3 has a third blank (“approximately ___”), total blanks:

1. two blanks
2. two blanks
3. two blanks (first: 112°→rad, second: approx rev)
4. two blanks
5. two blanks

So 10 answers total.

Let me write them in order:

1. \(\frac{\pi}{4}\), \(\frac{3\pi}{4}\)
2. \(\frac{180}{7}\), \(\frac{19\pi}{18}\)
3. \(\frac{28\pi}{45}\), \(6.28\)
4. \(45\), \(\frac{3\pi}{5}\)
5. \(90\), \(\frac{127\pi}{180}\)

If degrees are expected without the ° symbol in blanks, just numbers.

Also, for #2a, if they want decimal, 25.7 — but since others are exact, and no “approx”, I’ll keep fraction.

However, double-check: In many standardized worksheets, they expect decimal for non-special angles. But the problem didn’t specify. Given the presence of π in answers elsewhere, fractions with π are acceptable.

I think the expected answers are:

1. π/4, 3π/4
2. 180/7, 19π/18
3. 28π/45, 6.28
4. 45, 3π/5
5. 90, 127π/180

Yes.

Final Answer:
π/4, 3π/4, 180/7, 19π/18, 28π/45, 6.28, 45, 3π/5, 90, 127π/180