Example problems and formulas for calculating arc length in circles.

Let’s solve each problem step by step.

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Problem 1:

We are given:
- Radius = 6 meters
- Central angle = 107°

We need to find the arc length (the bolded part of the circle).

Step 1: Recall the formula for arc length

Arc Length = (x / 360) × C
Where:
- x = degree measure of the arc (here, 107°)
- C = circumference of the whole circle = 2πr

Step 2: Plug in the values

First, calculate the circumference:
C = 2 × π × 6 = 12π ≈ 12 × 3.1416 = 37.6992 meters

Now plug into the arc length formula:
Arc Length = (107 / 360) × 37.6992

Calculate 107 ÷ 360 first:
107 ÷ 360 ≈ 0.297222...

Now multiply:
0.297222... × 37.6992 ≈ ?

Let me compute that carefully:

0.297222 × 37.6992
≈ Let’s do it as fractions or use calculator-style steps:

Actually, better to keep it exact until the end:

Arc Length = (107/360) × 2 × π × 6
= (107/360) × 12π
= (107 × 12π) / 360
= (1284π) / 360
Simplify fraction: divide numerator and denominator by 12 → 107π / 30

Now compute numerically:
107 × π ≈ 107 × 3.14159265 ≈ 336.1504
Divide by 30: 336.1504 ÷ 30 ≈ 11.205

Rounded to nearest hundredth: 11.21 meters

Wait — let me double-check with direct multiplication:

(107 / 360) × 2 × π × 6
= (107 / 360) × 12π
= (107 × 12 × π) / 360
= (1284 × π) / 360
1284 ÷ 360 = 3.5666...? No, wait — 1284 ÷ 360 = 3.5666 is wrong.

Wait — 360 × 3 = 1080, 1284 - 1080 = 204 → so 3 + 204/360 = 3 + 17/30 ≈ 3.5667? That can’t be right because we’re multiplying by π.

No — I think I messed up.

Better approach:

Compute step-by-step numerically:

Step 1: 2 × π × 6 = 37.69911184

Step 2: 107 ÷ 360 = 0.2972222222

Step 3: Multiply: 0.2972222222 × 37.69911184 ≈ ?

Let me compute:

0.2972222222 × 37.69911184

Break it down:

0.3 × 37.69911184 = 11.30973355
But we have 0.297222..., which is 0.3 - 0.002777...

So subtract: 0.002777... × 37.69911184 ≈ ?

0.002777... = 1/360? Wait no — 0.002777... = 1/360? Actually 1/360 ≈ 0.002777...

Wait — actually, 107/360 = 1 - 253/360? Not helpful.

Just multiply directly:

Use calculator simulation:

107 * 2 * pi * 6 / 360

= (107 * 12 * pi) / 360

= (1284 * pi) / 360

Now 1284 ÷ 360 = 3.566666...? No:

Wait — 360 × 3 = 1080
1284 - 1080 = 204
So 1284 / 360 = 3 + 204/360 = 3 + 17/30 = 3.566666...

Then multiply by pi: 3.566666... × π ≈ 3.566666 × 3.14159265 ≈ ?

3.566666 × 3 = 10.7
3.566666 × 0.14159265 ≈ ?

Approximate:

3.566666 × 0.14 = 0.499333
3.566666 × 0.00159265 ≈ 0.00568

Total ≈ 0.505

So total ≈ 10.7 + 0.505 = 11.205

Yes — so approximately 11.205 → rounded to nearest hundredth is 11.21

But wait — let's use more precise calculation:

Using exact computation:

Arc Length = (107/360) * 2 * π * 6
= (107/360)*12*π
= (107 * 12 * π) / 360
= (1284 * π) / 360

Reduce fraction: divide numerator and denominator by 12:

1284 ÷ 12 = 107
360 ÷ 12 = 30
→ So (107π)/30

Now compute 107 * π = 107 * 3.1415926535 = let's calculate:

100 * π = 314.15926535
7 * π = 21.9911485745
Sum = 336.1504139245

Divide by 30: 336.1504139245 / 30 = 11.2050137975

Rounded to nearest hundredth: look at thousandths place → 5, so round up → 11.21

✔ Confirmed.

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Problem 2:

Given:
- Radius = 15 inches
- Central angle = 38°

Find arc length.

Step 1: Use same formula

Arc Length = (x / 360) × 2πr

Plug in:
= (38 / 360) × 2 × π × 15

Step 2: Simplify

First, 2 × 15 = 30
So: (38 / 360) × 30π

Or: (38 × 30π) / 360

Simplify: 30/360 = 1/12
So: 38 × π / 12 = (38π)/12

Simplify fraction: divide numerator and denominator by 2 → 19π/6

Now compute numerically:

19 × π ≈ 19 × 3.1415926535 = 59.6902604165

Divide by 6: 59.6902604165 / 6 ≈ 9.948376736

Rounded to nearest hundredth: look at thousandths digit → 8, which is ≥5, so round up the hundredths place (4 becomes 5) → 9.95

Double-check:

(38/360) × 2 × π × 15
= (38/360) × 30π
= (38 × 30π) / 360
= 1140π / 360
Divide numerator and denominator by 60: 19π / 6 → same as above.

19π/6 ≈ 19 × 3.1416 / 6 ≈ 59.6904 / 6 ≈ 9.9484 → rounds to 9.95

✔ Confirmed.

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Final Answer:
1. 11.21 m
2. 9.95 in