Let’s go through each part of the table step by step. We’ll fill in the “Example” column for each row using the formulas and pictures given.
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1. Circumference
Formula:
C = 2πr or πd
Picture shows a circle with radius labeled “3 in.”
So, use C = 2πr
→ r = 3 inches
→ C = 2 × π × 3 = 6π inches
If you want a decimal answer (using π ≈ 3.14):
→ 6 × 3.14 = 18.84 inches
✔ Example Answer:
6π in or about 18.84 in
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2. Arc Length
Formula:
Arc Length = (2πrθ) / 360
Picture shows a circle with radius “6” and central angle “60°”
Plug into formula:
→ r = 6, θ = 60
→ Arc Length = (2 × π × 6 × 60) / 360
First multiply numerator: 2 × 6 × 60 = 720 → so 720π / 360
Simplify: 720 ÷ 360 = 2 → so 2π
Decimal? 2 × 3.14 = 6.28
✔ Example Answer:
2π units or about 6.28 units
*(Note: The picture doesn’t specify units, so we just say “units”)*
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3. Area of a Circle
Formula:
A = πr²
Picture shows radius = “4 cm”
So:
→ A = π × (4)² = π × 16 = 16π cm²
Decimal: 16 × 3.14 = 50.24 cm²
✔ Example Answer:
16π cm² or about 50.24 cm²
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4. Sector Area
Formula:
Sector Area = (πr²θ) / 360
Picture shows radius = “5 ft”, angle = “90°”
Plug in:
→ r = 5, θ = 90
→ Sector Area = (π × 5² × 90) / 360
= (π × 25 × 90) / 360
Multiply 25 × 90 = 2250 → so 2250π / 360
Simplify fraction:
Divide numerator and denominator by 90:
2250 ÷ 90 = 25, 360 ÷ 90 = 4 → so 25π/4
Or as mixed number: 6.25π
Decimal: 6.25 × 3.14 = 19.625 ft²
✔ Example Answer:
25π/4 ft² or 6.25π ft² or about 19.625 ft²
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Think about it... What similarities do you see between the formulas? Why do these similarities exist?
Look at arc length and sector area:
- Arc Length = (2πrθ)/360 → this is like taking the full circumference (2πr) and multiplying by the fraction of the circle (θ/360)
- Sector Area = (πr²θ)/360 → this is like taking the full area (πr²) and multiplying by the same fraction (θ/360)
👉 So both are just “part of the whole” — they take the full measurement (circumference or area) and scale it down by how big the angle is compared to a full circle (360°).
That’s why they look similar — because they’re both based on proportions!
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Final Answer:
Circumference Example: 6π in or about 18.84 in
Arc Length Example: 2π units or about 6.28 units
Area of Circle Example: 16π cm² or about 50.24 cm²
Sector Area Example: 25π/4 ft² or 6.25π ft² or about 19.625 ft²
Similarities in Formulas: Both arc length and sector area use the idea of taking the full circle measurement (circumference or area) and multiplying by the fraction θ/360 to get the part that matches the central angle. This makes sense because a sector or arc is just a piece of the whole circle.
Parent Tip: Review the logic above to help your child master the concept of arc length and sector area worksheet.
