Let’s solve each example step by step using the formulas given in the table.
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1. Circumference Example:
Picture shows a circle with radius = 3 inches.
Formula:
Circumference = 2πr or πd
We’ll use 2πr since we’re given radius.
→ 2 × π × 3 = 6π
If we use π ≈ 3.14, then:
→ 6 × 3.14 =
18.84 inches
✔ Final Answer for Circumference:
18.84 in. (or 6π in.)
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2. Arc Length Example:
Picture shows a circle with radius = 7 m, and central angle θ = 100°
Formula:
Arc Length = (2πrθ) / 360
Plug in values:
→ (2 × π × 7 × 100) / 360
= (1400π) / 360
Simplify fraction: divide numerator and denominator by 40 → 35π / 9
Now calculate numerically (using π ≈ 3.14):
→ 35 × 3.14 = 109.9
→ 109.9 ÷ 9 ≈
12.21 meters
✔ Final Answer for Arc Length:
12.21 m (or 35π/9 m)
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3. Area of a Circle Example:
Picture shows radius = 9 cm
Formula:
Area = πr²
→ π × 9² = π × 81 = 81π
Numerically: 81 × 3.14 =
254.34 cm²
✔ Final Answer for Area of Circle:
254.34 cm² (or 81π cm²)
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4. Sector Area Example:
Picture shows radius = 5 ft, central angle θ = 70°
Formula:
Sector Area = (πr²θ) / 360
Plug in:
→ (π × 5² × 70) / 360
= (π × 25 × 70) / 360
= (1750π) / 360
Simplify: divide numerator and denominator by 10 → 175π / 36
Numerically: 175 × 3.14 = 549.5
→ 549.5 ÷ 36 ≈
15.26 ft²
✔ Final Answer for Sector Area:
15.26 ft² (or 175π/36 ft²)
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Think about it... (Similarities between formulas)
All four formulas involve π and the radius (or diameter). The arc length and sector area formulas are just fractions of the full circumference and full area — based on how big the angle is compared to 360°. That’s why they both have “/360” — because a full circle is 360 degrees. So if you take a piece of the circle (like a slice of pie), you’re taking that same fraction of the whole circumference or area.
Example: If your angle is 90°, that’s 90/360 = 1/4 of the circle — so arc length is 1/4 of circumference, and sector area is 1/4 of total area.
This makes sense — bigger angle = longer arc + larger sector area.
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Final Answer:
- Circumference:
18.84 in.
- Arc Length:
12.21 m
- Area of Circle:
254.34 cm²
- Sector Area:
15.26 ft²
*(Answers can also be left in terms of π as shown above)*
Parent Tip: Review the logic above to help your child master the concept of area of sector worksheet.
